References

This list of publications closely related to parallel-in-time integration is probably not complete. Please feel free to add any missing publications through a pull request on GitHub .

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2024

BossuytEtAl2024

I. Bossuyt, S. Vandewalle, and G. Samaey, “Micro-macro Parareal, from ODEs to SDEs and back again,” arXiv:2401.01798v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2401.01798v1
BibTeX entry BossuytEtAl2024

@unpublished{BossuytEtAl2024, author = {Bossuyt, Ignace and Vandewalle, Stefan and Samaey, Giovanni}, howpublished = {arXiv:2401.01798v1 [math.NA]}, title = {Micro-macro Parareal, from ODEs to SDEs and back again}, url = {http://arxiv.org/abs/2401.01798v1}, year = {2024} }
Abstract for BibTeX entry BossuytEtAl2024

In this paper, we are concerned with the micro-macro Parareal algorithm for the simulation of initial-value problems. In this algorithm, a coarse (fast) solver is applied sequentially over the time domain, and a fine (time-consuming) solver is applied as a corrector in parallel over smaller chunks of the time interval. Moreover, the coarse solver acts on a reduced state variable, which is coupled to the fine state variable through appropriate coupling operators. We first provide a contribution to the convergence analysis of the micro-macro Parareal method for multiscale linear ordinary differential equations (ODEs). Then, we extend a variant of the micro-macro Parareal algorithm for scalar stochastic differential equations (SDEs) to higher-dimensional SDEs.
CaoEtAl2024

R. Cao, S. Hou, and L. Ma, “A Pipeline-Based ODE Solving Framework,” IEEE Access, vol. 12, pp. 37995–38004, 2024, doi: 10.1109/ACCESS.2024.3375305.
BibTeX entry CaoEtAl2024

@article{CaoEtAl2024, author = {Cao, Ruixia and Hou, Shangjun and Ma, Lin}, doi = {10.1109/ACCESS.2024.3375305}, journal = {IEEE Access}, number = {}, pages = {37995-38004}, title = {A Pipeline-Based ODE Solving Framework}, volume = {12}, year = {2024} }
Abstract for BibTeX entry CaoEtAl2024

The traditional parallel solving methods of ordinary differential equations (ODE) are mainly classified into task-parallelism, data-parallelism, and instruction-level parallelism. Based on the RIDC (revisionist integral deferred correction) algorithm, a hybrid solver dispatched on both CPU and GPU is proposed, which realizes computing in a pipeline form and a remarkable parallelism is obtained both inside a single equation and among many different equations. The proposed framework can make full use of the multi-core advantage of GPU, which is conducive to load balancing within computing nodes. The efficiency and accuracy of the framework are verified in several experiments.
FreeseEtAl2024

P. Freese, S. Götschel, T. Lunet, D. Ruprecht, and M. Schreiber, “Parallel performance of shared memory parallel spectral deferred corrections,” arXiv:2403.20135v1 [cs.CE], 2024 [Online]. Available at: http://arxiv.org/abs/2403.20135v1
BibTeX entry FreeseEtAl2024

@unpublished{FreeseEtAl2024, author = {Freese, Philip and Götschel, Sebastian and Lunet, Thibaut and Ruprecht, Daniel and Schreiber, Martin}, howpublished = {arXiv:2403.20135v1 [cs.CE]}, title = {Parallel performance of shared memory parallel spectral deferred corrections}, url = {http://arxiv.org/abs/2403.20135v1}, year = {2024} }
Abstract for BibTeX entry FreeseEtAl2024

We investigate parallel performance of parallel spectral deferred corrections, a numerical approach that provides small-scale parallelism for the numerical solution of initial value problems. The scheme is applied to the shallow water equation and uses an IMEX splitting that integrates fast modes implicitly and slow modes explicitly in order to be efficient. We describe parallel \textttOpenMP-based implementations of parallel SDC in two well established simulation codes: the finite volume based operational ocean model \textttICON-O and the spherical harmonics based research code \textttSWEET. The implementations are benchmarked on a single node of the JUSUF (\textttSWEET) and JUWELS (\textttICON-O) system at Jülich Supercomputing Centre. We demonstrate a reduction of time-to-solution across a range of accuracies. For \textttICON-O, we show speedup over the currently used Adams–Bashforth-2 integrator with \textttOpenMP loop parallelization. For \textttSWEET, we show speedup over serial spectral deferred corrections and a second order implicit-explicit integrator.
FungEtAl2024

P. Y. Fung and S. Hon, “Block ω-circulant preconditioners for parabolic optimal control problems,” arXiv:2406.00952v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2406.00952v1
BibTeX entry FungEtAl2024

@unpublished{FungEtAl2024, author = {Fung, Po Yin and Hon, Sean}, howpublished = {arXiv:2406.00952v1 [math.NA]}, title = {Block $ω$-circulant preconditioners for parabolic optimal control problems}, url = {http://arxiv.org/abs/2406.00952v1}, year = {2024} }
Abstract for BibTeX entry FungEtAl2024

In this work, we propose a class of novel preconditioned Krylov subspace methods for solving an optimal control problem of parabolic equations. Namely, we develop a family of block ω-circulant based preconditioners for the all-at-once linear system arising from the concerned optimal control problem, where both first order and second order time discretization methods are considered. The proposed preconditioners can be efficiently diagonalized by fast Fourier transforms in a parallel-in-time fashion, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix are clustered around \pm 1, which leads to rapid convergence when the minimal residual method is used. When the generalized minimal residual method is deployed, the efficacy of the proposed preconditioners are justified in the way that the singular values of the preconditioned matrices are proven clustered around unity. Numerical results are provided to demonstrate the effectiveness of our proposed solvers.
GattiglioEtAl2024

G. Gattiglio, L. Grigoryeva, and M. Tamborrino, “Nearest Neighbors GParareal: Improving Scalability of Gaussian Processes for Parallel-in-Time Solvers,” arXiv:2405.12182v1 [stat.CO], 2024 [Online]. Available at: http://arxiv.org/abs/2405.12182v1
BibTeX entry GattiglioEtAl2024

@unpublished{GattiglioEtAl2024, author = {Gattiglio, Guglielmo and Grigoryeva, Lyudmila and Tamborrino, Massimiliano}, howpublished = {arXiv:2405.12182v1 [stat.CO]}, title = {Nearest Neighbors GParareal: Improving Scalability of Gaussian Processes for Parallel-in-Time Solvers}, url = {http://arxiv.org/abs/2405.12182v1}, year = {2024} }
Abstract for BibTeX entry GattiglioEtAl2024

With the advent of supercomputers, multi-processor environments and parallel-in-time (PinT) algorithms offer ways to solve initial value problems for ordinary and partial differential equations (ODEs and PDEs) over long time intervals, a task often unfeasible with sequential solvers within realistic time frames. A recent approach, GParareal, combines Gaussian Processes with traditional PinT methodology (Parareal) to achieve faster parallel speed-ups. The method is known to outperform Parareal for low-dimensional ODEs and a limited number of computer cores. Here, we present Nearest Neighbors GParareal (nnGParareal), a novel data-enriched PinT integration algorithm. nnGParareal builds upon GParareal by improving its scalability properties for higher-dimensional systems and increased processor count. Through data reduction, the model complexity is reduced from cubic to log-linear in the sample size, yielding a fast and automated procedure to integrate initial value problems over long time intervals. First, we provide both an upper bound for the error and theoretical details on the speed-up benefits. Then, we empirically illustrate the superior performance of nnGParareal, compared to GParareal and Parareal, on nine different systems with unique features (e.g., stiff, chaotic, high-dimensional, or challenging-to-learn systems).
GuEtAl2024

X.-M. Gu, J. Liu, and C. W. Oosterlee, “Parallel-in-Time Iterative Methods for Pricing American Options,” arXiv:2405.08280v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2405.08280v1
BibTeX entry GuEtAl2024

@unpublished{GuEtAl2024, author = {Gu, Xian-Ming and Liu, Jun and Oosterlee, Cornelis W.}, howpublished = {arXiv:2405.08280v1 [math.NA]}, title = {Parallel-in-Time Iterative Methods for Pricing American Options}, url = {http://arxiv.org/abs/2405.08280v1}, year = {2024} }
Abstract for BibTeX entry GuEtAl2024

For pricing American options, %after suitable discretization in space and time, a sequence of discrete linear complementarity problems (LCPs) or equivalently Hamilton-Jacobi-Bellman (HJB) equations need to be solved in a sequential time-stepping manner. In each time step, the policy iteration or its penalty variant is often applied due to their fast convergence rates. In this paper, we aim to solve for all time steps simultaneously, by applying the policy iteration to an “all-at-once form" of the HJB equations, where two different parallel-in-time preconditioners are proposed to accelerate the solution of the linear systems within the policy iteration. Our proposed methods are generally applicable for such all-at-once forms of the HJB equation, arising from option pricing problems with optimal stopping and nontrivial underlying asset models. Numerical examples are presented to show the feasibility and robust convergence behavior of the proposed methodology.
HeinzelreiterEtAl2024

B. Heinzelreiter and J. W. Pearson, “Diagonalization-Based Parallel-in-Time Preconditioners for Instationary Fluid Flow Control Problems,” arXiv:2405.18964v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2405.18964v1
BibTeX entry HeinzelreiterEtAl2024

@unpublished{HeinzelreiterEtAl2024, author = {Heinzelreiter, Bernhard and Pearson, John W.}, howpublished = {arXiv:2405.18964v1 [math.NA]}, title = {Diagonalization-Based Parallel-in-Time Preconditioners for Instationary Fluid Flow Control Problems}, url = {http://arxiv.org/abs/2405.18964v1}, year = {2024} }
Abstract for BibTeX entry HeinzelreiterEtAl2024

We derive a new parallel-in-time approach for solving large-scale optimization problems constrained by time-dependent partial differential equations arising from fluid dynamics. The solver involves the use of a block circulant approximation of the original matrices, enabling parallelization-in-time via the use of fast Fourier transforms, and we devise bespoke matrix approximations which may be applied within this framework. These make use of permutations, saddle-point approximations, commutator arguments, as well as inner solvers such as the Uzawa method, Chebyshev semi-iteration, and multigrid. Theoretical results underpin our strategy of applying a block circulant strategy, and numerical experiments demonstrate the effectiveness and robustness of our approach on Stokes and Oseen problems. Noteably, satisfying results for the strong and weak scaling of our methods are provided within a fully parallel architecture.

J. Huang, L. Ju, and Y. Xu, “A parareal exponential integrator finite element method for semilinear parabolic equations,” Numerical Methods for Partial Differential Equations, May 2024, doi: 10.1002/num.23116. [Online]. Available at: http://dx.doi.org/10.1002/num.23116

IbrahimEtAl2024

A. Q. Ibrahim, S. Götschel, and D. Ruprecht, “Space-time parallel scaling of Parareal with a Fourier Neural Operator as coarse propagator,” arXiv:2404.02521v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2404.02521v1
BibTeX entry IbrahimEtAl2024

@unpublished{IbrahimEtAl2024, author = {Ibrahim, Abdul Qadir and Götschel, Sebastian and Ruprecht, Daniel}, howpublished = {arXiv:2404.02521v1 [math.NA]}, title = {Space-time parallel scaling of Parareal with a Fourier Neural Operator as coarse propagator}, url = {http://arxiv.org/abs/2404.02521v1}, year = {2024} }
Abstract for BibTeX entry IbrahimEtAl2024

Iterative parallel-in-time algorithms like Parareal can extend scaling beyond the saturation of purely spatial parallelization when solving initial value problems. However, they require the user to build coarse models to handle the inevitably serial transport of information in time.This is a time consuming and difficult process since there is still only limited theoretical insight into what constitutes a good and efficient coarse model. Novel approaches from machine learning to solve differential equations could provide a more generic way to find coarse level models for parallel-in-time algorithms. This paper demonstrates that a physics-informed Fourier Neural Operator (PINO) is an effective coarse model for the parallelization in time of the two-asset Black-Scholes equation using Parareal. We demonstrate that PINO-Parareal converges as fast as a bespoke numerical coarse model and that, in combination with spatial parallelization by domain decomposition, it provides better overall speedup than both purely spatial parallelization and space-time parallelizaton with a numerical coarse propagator.
JackamanEtAl2024

J. Jackaman and S. MacLachlan, “Space-time waveform relaxation multigrid for Navier-Stokes,” arXiv:2407.13997v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2407.13997v1
BibTeX entry JackamanEtAl2024

@unpublished{JackamanEtAl2024, author = {Jackaman, James and MacLachlan, Scott}, howpublished = {arXiv:2407.13997v1 [math.NA]}, title = {Space-time waveform relaxation multigrid for Navier-Stokes}, url = {http://arxiv.org/abs/2407.13997v1}, year = {2024} }
Abstract for BibTeX entry JackamanEtAl2024

Space-time finite-element discretizations are well-developed in many areas of science and engineering, but much work remains within the development of specialized solvers for the resulting linear and nonlinear systems. In this work, we consider the all-at-once solution of the discretized Navier-Stokes equations over a space-time domain using waveform relaxation multigrid methods. In particular, we show how to extend the efficient spatial multigrid relaxation methods from [37] to a waveform relaxation method, and demonstrate the efficiency of the resulting monolithic Newton-Krylov-multigrid solver. Numerical results demonstrate the scalability of the solver for varying discretization order and physical parameters.

N. Janssens and J. Meyers, “Parallel-in-time multiple shooting for optimal control problems governed by the Navier–Stokes equations,” Computer Physics Communications, vol. 296, p. 109019, Mar. 2024, doi: 10.1016/j.cpc.2023.109019. [Online]. Available at: http://dx.doi.org/10.1016/j.cpc.2023.109019

F. C. Joseph and G. Gurrala, “Adaptive Homotopy Based Coarse Solver for Parareal-in-Time Transient Stability Simulations,” IEEE Transactions on Power Systems, pp. 1–12, 2024, doi: 10.1109/tpwrs.2024.3424555. [Online]. Available at: http://dx.doi.org/10.1109/TPWRS.2024.3424555

KaiserEtAl2024

L. Kaiser, R. Tsai, and C. Klingenberg, “Efficient Numerical Wave Propagation Enhanced By An End-to-End Deep Learning Model,” arXiv:2402.02304v4 [math.AP], 2024 [Online]. Available at: http://arxiv.org/abs/2402.02304v4
BibTeX entry KaiserEtAl2024

@unpublished{KaiserEtAl2024, author = {Kaiser, Luis and Tsai, Richard and Klingenberg, Christian}, howpublished = {arXiv:2402.02304v4 [math.AP]}, title = {Efficient Numerical Wave Propagation Enhanced By An End-to-End Deep Learning Model}, url = {http://arxiv.org/abs/2402.02304v4}, year = {2024} }
Abstract for BibTeX entry KaiserEtAl2024

Recent advances in wave modeling use sufficiently accurate fine solver outputs to train a neural network that enhances the accuracy of a fast but inaccurate coarse solver. In this paper we build upon the work of Nguyen and Tsai (2023) and present a novel unified system that integrates a numerical solver with a deep learning component into an end-to-end framework. In the proposed setting, we investigate refinements to the network architecture and data generation algorithm. A stable and fast solver further allows the use of Parareal, a parallel-in-time algorithm to correct high-frequency wave components. Our results show that the cohesive structure improves performance without sacrificing speed, and demonstrate the importance of temporal dynamics, as well as Parareal, for accurate wave propagation.

A. Kumar, “Investigation of Second Order Taylor Series in the Coarse Operator of Parareal Algorithm for Power System Simulation,” IEEE Transactions on Circuits and Systems II: Express Briefs, pp. 1–1, 2024, doi: 10.1109/tcsii.2024.3381372. [Online]. Available at: http://dx.doi.org/10.1109/TCSII.2024.3381372

KwokEtAl2024

F. Kwok and D. N. Tognon, “A parallel in time algorithm based ParaExp for optimal control problems,” arXiv:2406.11478v1 [cs.DC], 2024 [Online]. Available at: http://arxiv.org/abs/2406.11478v1
BibTeX entry KwokEtAl2024

@unpublished{KwokEtAl2024, author = {Kwok, Felix and Tognon, Djahou N}, howpublished = {arXiv:2406.11478v1 [cs.DC]}, title = {A parallel in time algorithm based ParaExp for optimal control problems}, url = {http://arxiv.org/abs/2406.11478v1}, year = {2024} }
Abstract for BibTeX entry KwokEtAl2024

We propose a new parallel-in-time algorithm for solving optimal control problems constrained bypartial differential equations. Our approach, which is based on a deeper understanding of ParaExp,considers an overlapping time-domain decomposition in which we combine the solution of homogeneous problems using exponential propagation with the local solutions of inhomogeneous problems.The algorithm yields a linear system whose matrix-vector product can be fully performed in parallel.We then propose a preconditioner to speed up the convergence of GMRES in the special cases ofthe heat and wave equations. Numerical experiments are provided to illustrate the efficiency of ourpreconditioners.

F. Li and Y. Xu, “A Diagonalization-Based Parallel-in-Time Algorithm for Crank-Nicolson’s Discretization of the Viscoelastic Equation,” East Asian Journal on Applied Mathematics, vol. 14, no. 1, pp. 47–78, Jun. 2024, doi: 10.4208/eajam.2022-304.070323. [Online]. Available at: http://dx.doi.org/10.4208/eajam.2022-304.070323

Z. Miao, B. W. null, and Y. Jiang, “Energy-Preserving Parareal-RKN Algorithms for Hamiltonian Systems,” Numerical Mathematics: Theory, Methods and Applications, vol. 17, no. 1, pp. 121–144, Jun. 2024, doi: 10.4208/nmtma.oa-2023-0081. [Online]. Available at: http://dx.doi.org/10.4208/nmtma.oa-2023-0081

Z. Miao, R.-H. Zhang, W.-W. Han, and Y.-L. Jiang, “Analysis of a fractional-step parareal algorithm for the incompressible Navier-Stokes equations,” Computers & Mathematics with Applications, vol. 161, pp. 78–89, May 2024, doi: 10.1016/j.camwa.2024.02.035. [Online]. Available at: http://dx.doi.org/10.1016/j.camwa.2024.02.035

MuralikrishnanEtAl2024

S. Muralikrishnan and R. Speck, “ParaPIF: A Parareal Approach for Parallel-in-Time Integration of Particle-in-Fourier schemes,” arXiv:2407.00485v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2407.00485v1
BibTeX entry MuralikrishnanEtAl2024

@unpublished{MuralikrishnanEtAl2024, author = {Muralikrishnan, Sriramkrishnan and Speck, Robert}, howpublished = {arXiv:2407.00485v1 [math.NA]}, title = {ParaPIF: A Parareal Approach for Parallel-in-Time Integration of Particle-in-Fourier schemes}, url = {http://arxiv.org/abs/2407.00485v1}, year = {2024} }
Abstract for BibTeX entry MuralikrishnanEtAl2024

We propose ParaPIF, a parareal based time parallelization scheme for the particle-in-Fourier (PIF) discretization of the Vlasov-Poisson system used in kinetic plasma simulations. Our coarse propagators are based on the coarsening of the numerical discretization scheme combined with, if possible, temporal coarsening rather than coarsening of particles and/or Fourier modes, which are not possible or effective for PIF schemes. Specifically, we use PIF with a coarse tolerance for nonuniform FFTs or even the standard particle-in-cell schemes as coarse propagators for the ParaPIF algorithm. We state and prove the convergence of the algorithm and verify the results numerically with Landau damping, two-stream instability, and Penning trap test cases in 3D-3V. We also implement the space-time parallelization of the PIF schemes in the open-source, performance-portable library IPPL and conduct scaling studies up to 1536 A100 GPUs on the JUWELS booster supercomputer. The space-time parallelization utilizing the ParaPIF algorithm for the time parallelization provides up to 4-6 times speedup compared to spatial parallelization alone and achieves a push rate of around 1 billion particles per second for the benchmark plasma mini-apps considered.
PamelaEtAl2024

S. J. P. Pamela et al., “Neural-Parareal: Dynamically Training Neural Operators as Coarse Solvers for Time-Parallelisation of Fusion MHD Simulations,” arXiv:2405.01355v1 [physics.plasm-ph], 2024 [Online]. Available at: http://arxiv.org/abs/2405.01355v1
BibTeX entry PamelaEtAl2024

@unpublished{PamelaEtAl2024, author = {Pamela, S. J. P. and Carey, N. and Brandstetter, J. and Akers, R. and Zanisi, L. and Buchanan, J. and Gopakumar, V. and Hoelzl, M. and Huijsmans, G. and Pentland, K. and James, T. and Antonucci, G. and the JOREK Team}, howpublished = {arXiv:2405.01355v1 [physics.plasm-ph]}, title = {Neural-Parareal: Dynamically Training Neural Operators as Coarse Solvers for Time-Parallelisation of Fusion MHD Simulations}, url = {http://arxiv.org/abs/2405.01355v1}, year = {2024} }
Abstract for BibTeX entry PamelaEtAl2024

The fusion research facility ITER is currently being assembled to demonstrate that fusion can be used for industrial energy production, while several other programmes across the world are also moving forward, such as EU-DEMO, CFETR, SPARC and STEP. The high engineering complexity of a tokamak makes it an extremely challenging device to optimise, and test-based optimisation would be too slow and too costly. Instead, digital design and optimisation must be favored, which requires strongly-coupled suites of High-Performance Computing calculations. In this context, having surrogate models to provide quick estimates with uncertainty quantification is essential to explore and optimise new design options. Furthermore, these surrogates can in turn be used to accelerate simulations in the first place. This is the case of Parareal, a time-parallelisation method that can speed-up large HPC simulations, where the coarse-solver can be replaced by a surrogate. A novel framework, Neural-Parareal, is developed to integrate the training of neural operators dynamically as more data becomes available. For a given input-parameter domain, as more simulations are being run with Parareal, the large amount of data generated by the algorithm is used to train new surrogate models to be used as coarse-solvers for future Parareal simulations, leading to progressively more accurate coarse-solvers, and thus higher speed-up. It is found that such neural network surrogates can be much more effective than traditional coarse-solver in providing a speed-up with Parareal. This study is a demonstration of the convergence of HPC and AI which simply has to become common practice in the world of digital engineering design.

B. Park, “Stochastic Power System Dynamic Simulation Using Parallel-in-Time Algorithm,” IEEE Access, vol. 12, pp. 28500–28510, 2024, doi: 10.1109/access.2024.3367358. [Online]. Available at: http://dx.doi.org/10.1109/ACCESS.2024.3367358

PeterssonEtAl2024

N. A. Petersson, S. Günther, and S. W. Chung, “A time-parallel multiple-shooting method for large-scale quantum optimal control,” arXiv:2407.13950v1 [quant-ph], 2024 [Online]. Available at: http://arxiv.org/abs/2407.13950v1
BibTeX entry PeterssonEtAl2024

@unpublished{PeterssonEtAl2024, author = {Petersson, N. Anders and Günther, Stefanie and Chung, Seung Whan}, howpublished = {arXiv:2407.13950v1 [quant-ph]}, title = {A time-parallel multiple-shooting method for large-scale quantum optimal control}, url = {http://arxiv.org/abs/2407.13950v1}, year = {2024} }
Abstract for BibTeX entry PeterssonEtAl2024

Quantum optimal control plays a crucial role in quantum computing by employing control pulses to steer quantum systems and realize logical gate transformations, essential for quantum algorithms. However, this optimization task is computationally demanding due to challenges such as the exponential growth in computational complexity with the system’s dimensionality and the deterioration of optimization convergence for multi-qubit systems. Various methods have been developed, including gradient-based and gradient-free reduced-space methods and full-space collocation methods. This paper introduces an intermediate approach based on multiple-shooting, aiming to balance solution accuracy and computational efficiency of previous approaches. Unlike conventional reduced-space methods, multiple-shooting divides the time domain into windows and introduces optimization variables for the initial state in each window. This enables parallel computation of state evolution across windows, significantly accelerating objective function and gradient evaluations. The initial state matrix in each window is only guaranteed to be unitary upon convergence of the optimization algorithm. For this reason the conventional gate trace infidelity is replaced by a generalized infidelity that is convex for non-unitary state matrices. Continuity of the state across window boundaries is enforced by equality constraints. A quadratic penalty optimization method is used to solve the constrained optimal control problem, and an efficient adjoint technique is employed to calculate the gradients in each iteration. We demonstrate the effectiveness of the proposed method through numerical experiments on quantum Fourier transform gates in systems with 2, 3, and 4 qubits. Parallel scalability and optimization performance are evaluated, highlighting the method’s potential for optimizing control pulses in multi-qubit quantum systems.

Y. Poirier, J. Salomon, A. Babarit, P. Ferrant, and G. Ducrozet, “Acceleration of a wave-structure interaction solver by the Parareal method,” Engineering Analysis with Boundary Elements, vol. 167, p. 105870, Oct. 2024, doi: 10.1016/j.enganabound.2024.105870. [Online]. Available at: http://dx.doi.org/10.1016/j.enganabound.2024.105870

SarpeEtAl2024

J. Sarpe, A. Klaedtke, and H. D. Gersem, “Periodic Adjoint Sensitivity Analysis,” arXiv:2405.19048v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2405.19048v1
BibTeX entry SarpeEtAl2024

@unpublished{SarpeEtAl2024, author = {Sarpe, Julian and Klaedtke, Andreas and Gersem, Herbert De}, howpublished = {arXiv:2405.19048v1 [math.NA]}, title = {Periodic Adjoint Sensitivity Analysis}, url = {http://arxiv.org/abs/2405.19048v1}, year = {2024} }
Abstract for BibTeX entry SarpeEtAl2024

This paper proposes the utilization of a periodic Parareal with a periodic coarse problem to efficiently perform adjoint sensitivity analysis for the steady state of time-periodic nonlinear circuits. In order to implement this method, a modified formulation for adjoint sensitivity analysis based on the transient approach is derived.
Scheiber2024

E. Scheiber, “A Convergence Theorem for the Parareal Algorithm Revisited,” arXiv:2405.06954v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2405.06954v1
BibTeX entry Scheiber2024

@unpublished{Scheiber2024, author = {Scheiber, Ernest}, howpublished = {arXiv:2405.06954v1 [math.NA]}, title = {A Convergence Theorem for the Parareal Algorithm Revisited}, url = {http://arxiv.org/abs/2405.06954v1}, year = {2024} }
Abstract for BibTeX entry Scheiber2024

The subject of the paper is to verify the convergence conditions for the parareal algorithm using Gander and Hairer’s theorem . The analysis is conducted in the case where the coarse integrator is the Euler method and the high-accuracy integrator is an explicit Runge-Kutta type method.
SchnaubeltEtAl2024

E. Schnaubelt, M. Wozniak, J. Dular, I. C. Garcia, A. Verweij, and S. Schöps, “Parallel-in-Time Integration of Transient Phenomena in No-Insulation Superconducting Coils Using Parareal,” arXiv:2404.13333v1 [cs.CE], 2024 [Online]. Available at: http://arxiv.org/abs/2404.13333v1
BibTeX entry SchnaubeltEtAl2024

@unpublished{SchnaubeltEtAl2024, author = {Schnaubelt, Erik and Wozniak, Mariusz and Dular, Julien and Garcia, Idoia Cortes and Verweij, Arjan and Schöps, Sebastian}, howpublished = {arXiv:2404.13333v1 [cs.CE]}, title = {Parallel-in-Time Integration of Transient Phenomena in No-Insulation Superconducting Coils Using Parareal}, url = {http://arxiv.org/abs/2404.13333v1}, year = {2024} }
Abstract for BibTeX entry SchnaubeltEtAl2024

High-temperature superconductors (HTS) have the potential to enable magnetic fields beyond the current limits of low-temperature superconductors in applications like accelerator magnets. However, the design of HTS-based magnets requires computationally demanding transient multi-physics simulations with highly non-linear material properties. To reduce the solution time, we propose using Parareal (PR) for parallel-in-time magneto-thermal simulation of magnets based on HTS, particularly, no-insulation coils without turn-to-turn insulation. We propose extending the classical PR method to automatically find a time partitioning using a first coarse adaptive propagator. The proposed PR method is shown to reduce the computing time when fine engineering tolerances are required despite the highly nonlinear character of the problem. The full software stack used is open-source.
SouzaEtAl2024

G. R. de Souza, S. Pezzuto, and R. Krause, “High-order parallel-in-time method for the monodomain equation in cardiac electrophysiology,” arXiv:2405.19994v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2405.19994v1
BibTeX entry SouzaEtAl2024

@unpublished{SouzaEtAl2024, author = {de Souza, Giacomo Rosilho and Pezzuto, Simone and Krause, Rolf}, howpublished = {arXiv:2405.19994v1 [math.NA]}, title = {High-order parallel-in-time method for the monodomain equation in cardiac electrophysiology}, url = {http://arxiv.org/abs/2405.19994v1}, year = {2024} }
Abstract for BibTeX entry SouzaEtAl2024

Simulation of the monodomain equation, crucial for modeling the heart’s electrical activity, faces scalability limits when traditional numerical methods only parallelize in space. To optimize the use of large multi-processor computers by distributing the computational load more effectively, time parallelization is essential. We introduce a high-order parallel-in-time method addressing the substantial computational challenges posed by the stiff, multiscale, and nonlinear nature of cardiac dynamics. Our method combines the semi-implicit and exponential spectral deferred correction methods, yielding a hybrid method that is extended to parallel-in-time employing the PFASST framework. We thoroughly evaluate the stability, accuracy, and robustness of the proposed parallel-in-time method through extensive numerical experiments, using practical ionic models such as the ten-Tusscher-Panfilov. The results underscore the method’s potential to significantly enhance real-time and high-fidelity simulations in biomedical research and clinical applications.
SterckEtAl2024

H. D. Sterck, R. D. Falgout, O. A. Krzysik, and J. B. Schroder, “Parallel-in-time solution of scalar nonlinear conservation laws,” arXiv:2401.04936v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2401.04936v1
BibTeX entry SterckEtAl2024

@unpublished{SterckEtAl2024, author = {Sterck, H. De and Falgout, R. D. and Krzysik, O. A. and Schroder, J. B.}, howpublished = {arXiv:2401.04936v1 [math.NA]}, title = {Parallel-in-time solution of scalar nonlinear conservation laws}, url = {http://arxiv.org/abs/2401.04936v1}, year = {2024} }
Abstract for BibTeX entry SterckEtAl2024

We consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially non-oscillatory (WENO) reconstructions, and in time with high-order explicit Runge-Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of non-differentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton’s method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for non-conservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley–Leverett equations. For many test problems, the solver converges in just a handful of iterations with convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.
SterckEtAl2024b

H. D. Sterck, R. D. Falgout, O. A. Krzysik, and J. B. Schroder, “Parallel-in-time solution of hyperbolic PDE systems via characteristic-variable block preconditioning,” arXiv:2407.03873v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2407.03873v1
BibTeX entry SterckEtAl2024b

@unpublished{SterckEtAl2024b, author = {Sterck, H. De and Falgout, R. D. and Krzysik, O. A. and Schroder, J. B.}, howpublished = {arXiv:2407.03873v1 [math.NA]}, title = {Parallel-in-time solution of hyperbolic PDE systems via characteristic-variable block preconditioning}, url = {http://arxiv.org/abs/2407.03873v1}, year = {2024} }
Abstract for BibTeX entry SterckEtAl2024b

We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that inter-variable coupling for characteristic variables is weak relative to intra-variable coupling, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small. For an \ell-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of \ell scalar linear(ized)-advection-like problems, each being associated with a different characteristic wave-speed in the underlying linear(ized) PDE. We approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media, and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions.
YamaleevEtAl2024

N. K. Yamaleev and S. Paudel, “A New Parallel-in-time Direct Inverse Method for Nonlinear Differential Equations,” arXiv:2406.00878v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2406.00878v1
BibTeX entry YamaleevEtAl2024

@unpublished{YamaleevEtAl2024, author = {Yamaleev, Nail K. and Paudel, Subhash}, howpublished = {arXiv:2406.00878v1 [math.NA]}, title = {A New Parallel-in-time Direct Inverse Method for Nonlinear Differential Equations}, url = {http://arxiv.org/abs/2406.00878v1}, year = {2024} }
Abstract for BibTeX entry YamaleevEtAl2024

We present a new approach to parallelization of the first-order backward difference discretization (BDF1) of the time derivative in partial differential equations, such as the nonlinear heat and viscous Burgers equations. The time derivative term is discretized by using the method of lines based on the implicit BDF1 scheme, while the inviscid and viscous terms are approximated by conventional 2nd-order 3-point central discretizations of the 1st- and 2nd-order derivatives in each spatial direction. The global system of nonlinear discrete equations in the space-time domain is solved by the Newton method for all time levels simultaneously. For the BDF1 discretization, this all-at-once system at each Newton iteration is block bidiagonal, which can be inverted directly in a blockwise manner, thus leading to a set of fully decoupled equations associated with each time level. This allows for an efficient parallel-in-time implementation of the implicit BDF1 discretization for nonlinear differential equations. The proposed parallel-in-time method preserves a quadratic rate of convergence of the Newton method of the sequential BDF1 scheme, so that the computational cost of solving each block matrix in parallel is nearly identical to that of the sequential counterpart at each time step. Numerical results show that the new parallel-in-time BDF1 scheme provides the speedup of up to 28 on 32 computing cores for the 2-D nonlinear partial differential equations with both smooth and discontinuous solutions.

R. Yoda, M. Bolten, K. Nakajima, and A. Fujii, “Coarse-grid operator optimization in multigrid reduction in time for time-dependent Stokes and Oseen problems,” Japan Journal of Industrial and Applied Mathematics, Apr. 2024, doi: 10.1007/s13160-024-00652-8. [Online]. Available at: http://dx.doi.org/10.1007/s13160-024-00652-8

ZhangEtAl2024

L. Zhang and Q. Zhang, “Convergence analysis of the parareal algorithms for stochastic Maxwell equations driven by additive noise,” arXiv:2407.10907v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2407.10907v1
BibTeX entry ZhangEtAl2024

@unpublished{ZhangEtAl2024, author = {Zhang, Liying and Zhang, Qi}, howpublished = {arXiv:2407.10907v1 [math.NA]}, title = {Convergence analysis of the parareal algorithms for stochastic Maxwell equations driven by additive noise}, url = {http://arxiv.org/abs/2407.10907v1}, year = {2024} }
Abstract for BibTeX entry ZhangEtAl2024

In this paper, we investigate the strong convergence analysis of parareal algorithms for stochastic Maxwell equations with the damping term driven by additive noise. The proposed parareal algorithms proceed as two-level temporal parallelizable integrators with the stochastic exponential integrator as the coarse propagator and both the exact solution integrator and the stochastic exponential integrator as the fine propagator. It is proved that the convergence order of the proposed algorithms linearly depends on the iteration number. Numerical experiments are performed to illustrate the convergence order of the algorithms for different choices of the iteration number, the damping coefficient and the scale of noise.
ZhaoEtAl2024

Y.-L. Zhao, X.-M. Gu, and C. W. Oosterlee, “A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization,” arXiv:2401.16113v1 [math.NA], 2024 [Online]. Available at: http://arxiv.org/abs/2401.16113v1
BibTeX entry ZhaoEtAl2024

@unpublished{ZhaoEtAl2024, author = {Zhao, Yong-Liang and Gu, Xian-Ming and Oosterlee, Cornelis W.}, howpublished = {arXiv:2401.16113v1 [math.NA]}, title = {A parallel preconditioner for the all-at-once linear system from evolutionary PDEs with Crank-Nicolson discretization}, url = {http://arxiv.org/abs/2401.16113v1}, year = {2024} }
Abstract for BibTeX entry ZhaoEtAl2024

The Crank-Nicolson (CN) method is a well-known time integrator for evolutionary partial differential equations (PDEs) arising in many real-world applications. Since the solution at any time depends on the solution at previous time steps, the CN method will be inherently difficult to parallelize. In this paper, we consider a parallel method for the solution of evolutionary PDEs with the CN scheme. Using an all-at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can prove that most eigenvalues of preconditioned matrices are equal to 1 and the others lie in the set: \left{z∈\mathbbC: 1/(1 + α) < |z| < 1/(1 - α) \rm and \Ree(z) > 0\right}, where 0 < α< 1 is a free parameter. Meanwhile, the efficient implementation of this proposed preconditioner is described and a mesh-independent convergence rate of the preconditioned GMRES method is derived under certain conditions. Finally, we will verify our theoretical findings via numerical experiments on financial option pricing partial differential equations.

M. Zhen, X. Liu, X. Ding, and J. Cai, “High-order space–time parallel computing of the Navier–Stokes equations,” Computer Methods in Applied Mechanics and Engineering, vol. 423, p. 116880, Apr. 2024, doi: 10.1016/j.cma.2024.116880. [Online]. Available at: http://dx.doi.org/10.1016/j.cma.2024.116880

M. Zhen, X. Ding, K. Qu, J. Cai, and S. Pan, “Enhancing the Convergence of the Multigrid-Reduction-in-Time Method for the Euler and Navier–Stokes Equations,” Journal of Scientific Computing, vol. 100, no. 2, Jun. 2024, doi: 10.1007/s10915-024-02596-0. [Online]. Available at: http://dx.doi.org/10.1007/s10915-024-02596-0

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2023

A. Barman and A. Sharma, “A Space-Time framework for compressible flow simulations using Finite Volume Method,” in AIAA AVIATION 2023 Forum, 2023, doi: 10.2514/6.2023-3431 [Online]. Available at: https://doi.org/10.2514/6.2023-3431

M. Bolten, S. Friedhoff, and J. Hahne, “Task graph-based performance analysis of parallel-in-time methods,” Parallel Computing, vol. 118, p. 103050, Nov. 2023, doi: 10.1016/j.parco.2023.103050. [Online]. Available at: https://doi.org/10.1016/j.parco.2023.103050

BoschEtAl2023

N. Bosch, A. Corenflos, F. Yaghoobi, F. Tronarp, P. Hennig, and S. Särkkä, “Parallel-in-Time Probabilistic Numerical ODE Solvers,” arXiv:2310.01145v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2310.01145v1
BibTeX entry BoschEtAl2023

@unpublished{BoschEtAl2023, author = {Bosch, Nathanael and Corenflos, Adrien and Yaghoobi, Fatemeh and Tronarp, Filip and Hennig, Philipp and Särkkä, Simo}, howpublished = {arXiv:2310.01145v1 [math.NA]}, title = {Parallel-in-Time Probabilistic Numerical ODE Solvers}, url = {http://arxiv.org/abs/2310.01145v1}, year = {2023} }
Abstract for BibTeX entry BoschEtAl2023

Probabilistic numerical solvers for ordinary differential equations (ODEs) treat the numerical simulation of dynamical systems as problems of Bayesian state estimation. Aside from producing posterior distributions over ODE solutions and thereby quantifying the numerical approximation error of the method itself, one less-often noted advantage of this formalism is the algorithmic flexibility gained by formulating numerical simulation in the framework of Bayesian filtering and smoothing. In this paper, we leverage this flexibility and build on the time-parallel formulation of iterated extended Kalman smoothers to formulate a parallel-in-time probabilistic numerical ODE solver. Instead of simulating the dynamical system sequentially in time, as done by current probabilistic solvers, the proposed method processes all time steps in parallel and thereby reduces the span cost from linear to logarithmic in the number of time steps. We demonstrate the effectiveness of our approach on a variety of ODEs and compare it to a range of both classic and probabilistic numerical ODE solvers.
BossuytEtAl2023

I. Bossuyt, S. Vandewalle, and G. Samaey, “Monte-Carlo/Moments micro-macro Parareal method for unimodal and bimodal scalar McKean-Vlasov SDEs,” arXiv:2310.11365v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2310.11365v1
BibTeX entry BossuytEtAl2023

@unpublished{BossuytEtAl2023, author = {Bossuyt, Ignace and Vandewalle, Stefan and Samaey, Giovanni}, howpublished = {arXiv:2310.11365v1 [math.NA]}, title = {Monte-Carlo/Moments micro-macro Parareal method for unimodal and bimodal scalar McKean-Vlasov SDEs}, url = {http://arxiv.org/abs/2310.11365v1}, year = {2023} }
Abstract for BibTeX entry BossuytEtAl2023

We propose a micro-macro parallel-in-time Parareal method for scalar McKean-Vlasov stochastic differential equations (SDEs). In the algorithm, the fine Parareal propagator is a Monte Carlo simulation of an ensemble of particles, while an approximate ordinary differential equation (ODE) description of the mean and the variance of the particle distribution is used as a coarse Parareal propagator to achieve speedup. We analyse the convergence behaviour of our method for a linear problem and provide numerical experiments indicating the parallel weak scaling of the algorithm on a set of examples. We show that convergence typically takes place in a low number of iterations, depending on the quality of the ODE predictor. For bimodal SDEs, we avoid quality deterioration of the coarse predictor (compared to unimodal SDEs) through the usage of multiple ODEs, each describing the mean and variance of the particle distribution in locally unimodal regions of the phase space. The benefit of the proposed algorithm can be viewed through two lenses: (i) through the parallel-in-time lens, speedup is obtained through the use of a very cheap coarse integrator (an ODE moment model), and (ii) through the moment models lens, accuracy is iteratively gained through the use of parallel machinery as a corrector. In contrast to the isolated use of a moment model, the proposed method (iteratively) converges to the true distribution generated by the SDE.
BouillonEtAl2023

A. Bouillon, G. Samaey, and K. Meerbergen, “On generalized preconditioners for time-parallel parabolic optimal control,” arXiv:2302.06406v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2302.06406v1
BibTeX entry BouillonEtAl2023

@unpublished{BouillonEtAl2023, author = {Bouillon, Arne and Samaey, Giovanni and Meerbergen, Karl}, howpublished = {arXiv:2302.06406v1 [math.NA]}, title = {On generalized preconditioners for time-parallel parabolic optimal control}, url = {http://arxiv.org/abs/2302.06406v1}, year = {2023} }
Abstract for BibTeX entry BouillonEtAl2023

The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.
BouillonEtAl2023b

A. Bouillon, G. Samaey, and K. Meerbergen, “Diagonalization-based preconditioners and generalized convergence bounds for ParaOpt,” arXiv:2304.09235v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2304.09235v1
BibTeX entry BouillonEtAl2023b

@unpublished{BouillonEtAl2023b, author = {Bouillon, Arne and Samaey, Giovanni and Meerbergen, Karl}, howpublished = {arXiv:2304.09235v1 [math.NA]}, title = {Diagonalization-based preconditioners and generalized convergence bounds for ParaOpt}, url = {http://arxiv.org/abs/2304.09235v1}, year = {2023} }
Abstract for BibTeX entry BouillonEtAl2023b

The ParaOpt algorithm was recently introduced as a time-parallel solver for optimal-control problems with a terminal-cost objective, and convergence results have been presented for the linear diffusive case with implicit-Euler time integrators. We reformulate ParaOpt for tracking problems and provide generalized convergence analyses for both objectives. We focus on linear diffusive equations and prove convergence bounds that are generic in the time integrators used. For large problem dimensions, ParaOpt’s performance depends crucially on having a good preconditioner to solve the arising linear systems. For the case where ParaOpt’s cheap, coarse-grained propagator is linear, we introduce diagonalization-based preconditioners, inspired by recent advances in the ParaDiag family of methods. These preconditioners not only lead to a weakly-scalable ParaOpt version, but are themselves invertible in parallel, making maximal use of available concurrency. They have proven convergence properties in the linear diffusive case that are generic in the time discretization used, similarly to our ParaOpt results. Numerical results confirm that the iteration count of the iterative solvers used for ParaOpt’s linear systems becomes constant in the limit of an increasing processor count. The paper is accompanied by a sequential MATLAB implementation.

L. D’Amore and R. Cacciapuoti, “Space-Time Decomposition of Kalman Filter,” Numerical Mathematics: Theory, Methods and Applications, vol. 0, no. 0, pp. 0–0, Sep. 2023, doi: 10.4208/nmtma.oa-2022-0203. [Online]. Available at: https://doi.org/10.4208/nmtma.oa-2022-0203

R. Cacciapuoti and L. D’Amore, “Scalability analysis of a two level domain decomposition approach in space and time solving data assimilation models,” Concurrency and Computation: Practice and Experience, Nov. 2023, doi: 10.1002/cpe.7937. [Online]. Available at: https://doi.org/10.1002/cpe.7937

CaldasEtAl2023

J. G. Caldas Steinstraesser, P. da Silva Peixoto, and M. Schreiber, “Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT,” arXiv:2306.09497v1 [math.NA], 2023 [Online]. Available at: https://arxiv.org/abs/2306.09497v1
BibTeX entry CaldasEtAl2023

@unpublished{CaldasEtAl2023, author = {Caldas Steinstraesser, Jo\~{a}o Guilherme and da Silva Peixoto, Pedro and Schreiber, Martin}, howpublished = {arXiv:2306.09497v1 [math.NA]}, title = {Parallel-in-time integration of the shallow water equations on the rotating sphere using Parareal and MGRIT}, url = {https://arxiv.org/abs/2306.09497v1}, year = {2023} }
Abstract for BibTeX entry CaldasEtAl2023

Despite the growing interest in parallel-in-time methods as an approach to accelerate numerical simulations in atmospheric modelling, improving their stability and convergence remains a substantial challenge for their application to operational models. In this work, we study the temporal parallelization of the shallow water equations on the rotating sphere combined with time-stepping schemes commonly used in atmospheric modelling due to their stability properties, namely an Eulerian implicit-explicit (IMEX) method and a semi-Lagrangian semi-implicit method (SL-SI-SETTLS). The main goal is to investigate the performance of parallel-in-time methods, namely Parareal and Multigrid Reduction in Time (MGRIT), when these well-established schemes are used on the coarse discretization levels and provide insights on how they can be improved for better performance. We begin by performing an analytical stability study of Parareal and MGRIT applied to a linearized ordinary differential equation depending on the choice of coarse scheme. Next, we perform numerical simulations of two standard tests to evaluate the stability, convergence and speedup provided by the parallel-in-time methods compared to a fine reference solution computed serially. We also conduct a detailed investigation on the influence of artificial viscosity and hyperviscosity approaches, applied on the coarse discretization levels, on the performance of the temporal parallelization. Both the analytical stability study and the numerical simulations indicate a poorer stability behaviour when SL-SI-SETTLS is used on the coarse levels, compared to the IMEX scheme. With the IMEX scheme, a better trade-off between convergence, stability and speedup compared to serial simulations can be obtained under proper parameters and artificial viscosity choices, opening the perspective of the potential competitiveness for realistic models.

B. Carrel, M. J. Gander, and B. Vandereycken, “Low-rank Parareal: a low-rank parallel-in-time integrator,” BIT Numerical Mathematics, vol. 63, no. 1, Feb. 2023, doi: 10.1007/s10543-023-00953-3. [Online]. Available at: https://doi.org/10.1007%2Fs10543-023-00953-3

ChenEtAl2023

Z. Chen and Y. Liu, “Efficient and Parallel Solution of High-order Continuous Time Galerkin for Dissipative and Wave Propagation Problems,” arXiv:2303.05008v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2303.05008v1
BibTeX entry ChenEtAl2023

@unpublished{ChenEtAl2023, author = {Chen, Zhiming and Liu, Yong}, howpublished = {arXiv:2303.05008v1 [math.NA]}, title = {Efficient and Parallel Solution of High-order Continuous Time Galerkin for Dissipative and Wave Propagation Problems}, url = {http://arxiv.org/abs/2303.05008v1}, year = {2023} }
Abstract for BibTeX entry ChenEtAl2023

We propose efficient and parallel algorithms for the implementation of the high-order continuous time Galerkin method for dissipative and wave propagation problems. By using Legendre polynomials as shape functions, we obtain a special structure of the stiffness matrix which allows us to extend the diagonal Padé approximation to solve ordinary differential equations with source terms. The unconditional stability, hp error estimates, and hp superconvergence at the nodes of the continuous time Galerkin method are proved. Numerical examples confirm our theoretical results.

T. Cheng, H. Yang, J. Huang, and C. Yang, “Nonlinear parallel-in-time simulations of multiphase flow in porous media,” Journal of Computational Physics, p. 112515, Sep. 2023, doi: 10.1016/j.jcp.2023.112515. [Online]. Available at: https://doi.org/10.1016/j.jcp.2023.112515

Cyr2023

E. C. Cyr, “A 2-Level Domain Decomposition Preconditioner for KKT Systems with Heat-Equation Constraints,” arXiv:2305.04421v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2305.04421v1
BibTeX entry Cyr2023

@unpublished{Cyr2023, author = {Cyr, Eric C.}, howpublished = {arXiv:2305.04421v1 [math.NA]}, title = {A 2-Level Domain Decomposition Preconditioner for KKT Systems with Heat-Equation Constraints}, url = {http://arxiv.org/abs/2305.04421v1}, year = {2023} }
Abstract for BibTeX entry Cyr2023

Solving optimization problems with transient PDE-constraints is computationally costly due to the number of nonlinear iterations and the cost of solving large-scale KKT matrices. These matrices scale with the size of the spatial discretization times the number of time steps. We propose a new two level domain decomposition preconditioner to solve these linear systems when constrained by the heat equation. Our approach leverages the observation that the Schur-complement is elliptic in time, and thus amenable to classical domain decomposition methods. Further, the application of the preconditioner uses existing time integration routines to facilitate implementation and maximize software reuse. The performance of the preconditioner is examined in an empirical study demonstrating the approach is scalable with respect to the number of time steps and subdomains.

C. Dajana, C. Eduardo, and V. Carmine, “Non-stationary wave relaxation methods for general linear systems of Volterra equations: convergence and parallel GPU implementation,” Numerical Algorithms, Jun. 2023, doi: 10.1007/s11075-023-01567-0. [Online]. Available at: https://doi.org/10.1007/s11075-023-01567-0

DanieliEtAl2023

F. Danieli, B. S. Southworth, and J. B. Schroder, “Space-Time Block Preconditioning for Incompressible Resistive Magnetohydrodynamics,” arXiv:2309.00768v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2309.00768v1
BibTeX entry DanieliEtAl2023

@unpublished{DanieliEtAl2023, author = {Danieli, Federico and Southworth, Ben S. and Schroder, Jacob B.}, howpublished = {arXiv:2309.00768v1 [math.NA]}, title = {Space-Time Block Preconditioning for Incompressible Resistive Magnetohydrodynamics}, url = {http://arxiv.org/abs/2309.00768v1}, year = {2023} }
Abstract for BibTeX entry DanieliEtAl2023

This work develops a novel all-at-once space-time preconditioning approach for resistive magnetohydrodynamics (MHD), with a focus on model problems targeting fusion reactor design. We consider parallel-in-time due to the long time domains required to capture the physics of interest, as well as the complexity of the underlying system and thereby computational cost of long-time integration. To ameliorate this cost by using many processors, we thus develop a novel approach to solving the whole space-time system that is parallelizable in both space and time. We develop a space-time block preconditioning for resistive MHD, following the space-time block preconditioning concept first introduced by Danieli et al. in 2022 for incompressible flow, where an effective preconditioner for classic sequential time-stepping is extended to the space-time setting. The starting point for our derivation is the continuous Schur complement preconditioner by Cyr et al. in 2021, which we proceed to generalise in order to produce, to our knowledge, the first space-time block preconditioning approach for the challenging equations governing incompressible resistive MHD. The numerical results are promising for the model problems of island coalescence and tearing mode, with the overhead computational cost associated with space-time preconditioning versus sequential time-stepping being modest and primarily in the range of 2x-5x, which is low for parallel-in-time schemes in general. Additionally, the scaling results for inner (linear) and outer (nonlinear) iterations are flat in the case of fixed time-step size and only grow very slowly in the case of time-step refinement.
Erlangga2023

Y. A. Erlangga, “Parallel-in-time Multilevel Krylov Methods: A Prototype,” arXiv:2401.00228v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2401.00228v1
BibTeX entry Erlangga2023

@unpublished{Erlangga2023, author = {Erlangga, Yogi A.}, howpublished = {arXiv:2401.00228v1 [math.NA]}, title = {Parallel-in-time Multilevel Krylov Methods: A Prototype}, url = {http://arxiv.org/abs/2401.00228v1}, year = {2023} }
Abstract for BibTeX entry Erlangga2023

This paper presents a parallel-in-time multilevel iterative method for solving differential algebraic equation, arising from a discretization of linear time-dependent partial differential equation. The core of the method is the multilevel Krylov method, introduced by Erlangga and Nabben \it [SIAM J. Sci. Comput., 30(2008), pp. 1572–1595]. In the method, special time restriction and interpolation operators are proposed to coarsen the time grid and to map functions between fine and coarse time grids. The resulting Galerkin coarse-grid system can be interpreted as time integration of an equivalent differential algebraic equation associated with a larger time step and a modified θ-scheme. A perturbed coarse time-grid matrix is used on the coarsest level to decouple the coarsest-level system, allowing full parallelization of the method. Within this framework, spatial coarsening can be included in a natural way, reducing further the size of the coarsest grid problem to solve. Numerical results are presented for the 1- and 2-dimensional heat equation using \it simulated parallel implementation, suggesting the potential computational speed-up of up to 9 relative to the single-processor implementation and the speed-up of about 3 compared to the sequential θ-scheme.

L. Fang, S. Vandewalle, and J. Meyers, “An SQP-based multiple shooting algorithm for large-scale PDE-constrained optimal control problems,” Journal of Computational Physics, vol. 477, p. 111927, Mar. 2023, doi: 10.1016/j.jcp.2023.111927. [Online]. Available at: https://doi.org/10.1016/j.jcp.2023.111927

FangEtAl2023b

R. Fang and R. Tsai, “Stabilization of parareal algorithms for long time computation of a class of highly oscillatory Hamiltonian flows using data,” arXiv:2309.01225v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2309.01225v1
BibTeX entry FangEtAl2023b

@unpublished{FangEtAl2023b, author = {Fang, Rui and Tsai, Richard}, howpublished = {arXiv:2309.01225v1 [math.NA]}, title = {Stabilization of parareal algorithms for long time computation of a class of highly oscillatory Hamiltonian flows using data}, url = {http://arxiv.org/abs/2309.01225v1}, year = {2023} }
Abstract for BibTeX entry FangEtAl2023b

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lion, Maday, and Turinici in 2001, for multiscale Hamiltonian systems. The first method involves constructing a correction operator to improve a given inaccurate coarse solver through solving a Procrustes problem using data collected online along parareal trajectories. The second method involves constructing an efficient, high-fidelity solver by a neural network trained with offline generated data. For the second method, we address the issues of effective data generation and proper loss function design based on the Hamiltonian function. We show proof-of-concept by applying the proposed methods to a Fermi-Pasta-Ulum (FPU) problem. The numerical results demonstrate that the Procrustes parareal method is able to produce solutions that are more stable in energy compared to the standard parareal. The neural network solver can achieve comparable or better runtime performance compared to numerical solvers of similar accuracy. When combined with the standard parareal algorithm, the improved neural network solutions are slightly more stable in energy than the improved numerical coarse solutions.

S. Frei and A. Heinlein, “Towards parallel time-stepping for the numerical simulation of atherosclerotic plaque growth,” Journal of Computational Physics, vol. 491, p. 112347, Oct. 2023, doi: 10.1016/j.jcp.2023.112347. [Online]. Available at: https://doi.org/10.1016%2Fj.jcp.2023.112347

GanderEtAl2023

M. J. Gander and D. Palitta, “A new ParaDiag time-parallel time integration method,” arXiv:2304.12597v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2304.12597v1
BibTeX entry GanderEtAl2023

@unpublished{GanderEtAl2023, author = {Gander, Martin J. and Palitta, Davide}, howpublished = {arXiv:2304.12597v1 [math.NA]}, title = {A new ParaDiag time-parallel time integration method}, url = {http://arxiv.org/abs/2304.12597v1}, year = {2023} }
Abstract for BibTeX entry GanderEtAl2023

Time-parallel time integration has received a lot of attention in the high performance computing community over the past two decades. Indeed, it has been shown that parallel-in-time techniques have the potential to remedy one of the main computational drawbacks of parallel-in-space solvers. In particular, it is well-known that for large-scale evolution problems space parallelization saturates long before all processing cores are effectively used on today’s large scale parallel computers. Among the many approaches for time-parallel time integration, ParaDiag schemes have proved themselves to be a very effective approach. In this framework, the time stepping matrix or an approximation thereof is diagonalized by Fourier techniques, so that computations taking place at different time steps can be indeed carried out in parallel. We propose here a new ParaDiag algorithm combining the Sherman-Morrison-Woodbury formula and Krylov techniques. A panel of diverse numerical examples illustrates the potential of our new solver. In particular, we show that it performs very well compared to different ParaDiag algorithms recently proposed in the literature.
GanderEtAl2023b

M. J. Gander, T. Lunet, D. Ruprecht, and R. Speck, “A Unified Analysis Framework for Iterative Parallel-in-Time Algorithms,” SIAM Journal on Scientific Computing, vol. 45, no. 5, pp. A2275–A2303, 2023, doi: 10.1137/22M1487163. [Online]. Available at: https://doi.org/10.1137/22M1487163
BibTeX entry GanderEtAl2023b

@article{GanderEtAl2023b, author = {Gander, Martin J. and Lunet, Thibaut and Ruprecht, Daniel and Speck, Robert}, doi = {10.1137/22M1487163}, journal = {SIAM Journal on Scientific Computing}, number = {5}, pages = {A2275-A2303}, title = {A Unified Analysis Framework for Iterative Parallel-in-Time Algorithms}, url = {https://doi.org/10.1137/22M1487163}, volume = {45}, year = {2023} }
Abstract for BibTeX entry GanderEtAl2023b

Abstract. Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notation, and the convergence estimates that are available are difficult to compare. We describe Parareal, PFASST, MGRIT, and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge superlinearly, and we also compare them numerically. The generating function framework provides further opportunities to explore and analyze existing and new methods. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available”, as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/parallel-in-Time/gfm.
GanglEtAl2023

P. Gangl, M. Gobrial, and O. Steinbach, “A space-time finite element method for the eddy current approximation of rotating electric machines,” arXiv:2307.00278v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2307.00278v1
BibTeX entry GanglEtAl2023

@unpublished{GanglEtAl2023, author = {Gangl, Peter and Gobrial, Mario and Steinbach, Olaf}, howpublished = {arXiv:2307.00278v1 [math.NA]}, title = {A space-time finite element method for the eddy current approximation of rotating electric machines}, url = {http://arxiv.org/abs/2307.00278v1}, year = {2023} }
Abstract for BibTeX entry GanglEtAl2023

In this paper we formulate and analyze a space-time finite element method for the numerical simulation of rotating electric machines where the finite element mesh is fixed in space-time domain. Based on the Babuška–Nečas theory we prove unique solvability both for the continuous variational formulation and for a standard Galerkin finite element discretization in the space-time domain. This approach allows for an adaptive resolution of the solution both in space and time, but it requires the solution of the overall system of algebraic equations. While the use of parallel solution algorithms seems to be mandatory, this also allows for a parallelization simultaneously in space and time. This approach is used for the eddy current approximation of the Maxwell equations which results in an elliptic-parabolic interface problem. Numerical results for linear and nonlinear constitutive material relations confirm the applicability, efficiency and accuracy of the proposed approach.
GaraiEtAl2023b

G. Garai and B. C. Mandal, “Linear and Nonlinear Parareal Methods for the Cahn-Hilliard Equation,” arXiv:2304.14074v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2304.14074v1
BibTeX entry GaraiEtAl2023b

@unpublished{GaraiEtAl2023b, author = {Garai, Gobinda and Mandal, Bankim C.}, howpublished = {arXiv:2304.14074v1 [math.NA]}, title = {Linear and Nonlinear Parareal Methods for the Cahn-Hilliard Equation}, url = {http://arxiv.org/abs/2304.14074v1}, year = {2023} }
Abstract for BibTeX entry GaraiEtAl2023b

In this paper, we propose, analyze and implement efficient time parallel methods for the Cahn-Hilliard (CH) equation. It is of great importance to develop efficient numerical methods for the CH equation, given the range of applicability of the CH equation has. The CH equation generally needs to be simulated for a very long time to get the solution of phase coarsening stage. Therefore it is desirable to accelerate the computation using parallel method in time. We present linear and nonlinear Parareal methods for the CH equation depending on the choice of fine approximation. We illustrate our results by numerical experiments.

G. Garai and B. C. Mandal, “Diagonalization based Parallel-in-Time method for a class of fourth order time dependent PDEs,” Mathematics and Computers in Simulation, Aug. 2023, doi: 10.1016/j.matcom.2023.07.028. [Online]. Available at: https://doi.org/10.1016%2Fj.matcom.2023.07.028

J. Hahne, B. Polenz, I. Kulchytska-Ruchka, S. Friedhoff, S. Ulbrich, and S. Schöps, “Parallel-in-time optimization of induction motors,” Journal of Mathematics in Industry, vol. 13, no. 1, Jun. 2023, doi: 10.1186/s13362-023-00134-5. [Online]. Available at: https://doi.org/10.1186/s13362-023-00134-5

S. Hon and S. Serra-Capizzano, “A block Toeplitz preconditioner for all-at-once systems from linear wave equations,” ETNA - Electronic Transactions on Numerical Analysis, vol. 58, pp. 177–195, 2023, doi: 10.1553/etna_vol58s177. [Online]. Available at: https://doi.org/10.1553/etna_vol58s177

HonEtAl2023b

S. Hon, J. Dong, and S. Serra-Capizzano, “A preconditioned MINRES method for optimal control of wave equations and its asymptotic spectral distribution theory,” arXiv:2307.12850v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2307.12850v1
BibTeX entry HonEtAl2023b

@unpublished{HonEtAl2023b, author = {Hon, Sean and Dong, Jiamei and Serra-Capizzano, Stefano}, howpublished = {arXiv:2307.12850v1 [math.NA]}, title = {A preconditioned MINRES method for optimal control of wave equations and its asymptotic spectral distribution theory}, url = {http://arxiv.org/abs/2307.12850v1}, year = {2023} }
Abstract for BibTeX entry HonEtAl2023b

In this work, we propose a novel preconditioned Krylov subspace method for solving an optimal control problem of wave equations, after explicitly identifying the asymptotic spectral distribution of the involved sequence of linear coefficient matrices from the optimal control problem. Namely, we first show that the all-at-once system stemming from the wave control problem is associated to a structured coefficient matrix-sequence possessing an eigenvalue distribution. Then, based on such a spectral distribution of which the symbol is explicitly identified, we develop an ideal preconditioner and two parallel-in-time preconditioners for the saddle point system composed of two block Toeplitz matrices. For the ideal preconditioner, we show that the eigenvalues of the preconditioned matrix-sequence all belong to the set \left(-\frac32,-\frac12\right)⋃\left(\frac12,\frac32\right) well separated from zero, leading to mesh-independent convergence when the minimal residual method is employed. The proposed parallel-in-time preconditioners can be implemented efficiently using fast Fourier transforms or discrete sine transforms, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix-sequences are clustered around \pm 1, which leads to rapid convergence. When these parallel-in-time preconditioners are not fast diagonalizable, we further propose modified versions which can be efficiently inverted. Several numerical examples are reported to verify our derived localization and spectral distribution result and to support the effectiveness of our proposed preconditioners and the related advantages with respect to the relevant literature.

A. Q. Ibrahim, S. Götschel, and D. Ruprecht, “Parareal with a Physics-Informed Neural Network as Coarse Propagator,” in Euro-Par 2023: Parallel Processing, Springer Nature Switzerland, 2023, pp. 649–663 [Online]. Available at: https://doi.org/10.1007/978-3-031-39698-4_44

Y. Jiang and J. Liu, “Fast parallel-in-time quasi-boundary value methods for backward heat conduction problems,” Applied Numerical Mathematics, vol. 184, pp. 325–339, Feb. 2023, doi: 10.1016/j.apnum.2022.10.006. [Online]. Available at: https://doi.org/10.1016%2Fj.apnum.2022.10.006

Y. Jiang, J. Liu, and X.-S. Wang, “A direct parallel-in-time quasi-boundary value method for inverse space-dependent source problems,” Journal of Computational and Applied Mathematics, vol. 423, p. 114958, May 2023, doi: 10.1016/j.cam.2022.114958. [Online]. Available at: https://doi.org/10.1016%2Fj.cam.2022.114958

JinEtAl2023

B. Jin, Q. Lin, and Z. Zhou, “Learning Coarse Propagators in Parareal Algorithm,” arXiv:2311.15320v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2311.15320v1
BibTeX entry JinEtAl2023

@unpublished{JinEtAl2023, author = {Jin, Bangti and Lin, Qingle and Zhou, Zhi}, howpublished = {arXiv:2311.15320v1 [math.NA]}, title = {Learning Coarse Propagators in Parareal Algorithm}, url = {http://arxiv.org/abs/2311.15320v1}, year = {2023} }
Abstract for BibTeX entry JinEtAl2023

The parareal algorithm represents an important class of parallel-in-time algorithms for solving evolution equations and has been widely applied in practice. To achieve effective speedup, the choice of the coarse propagator in the algorithm is vital. In this work, we investigate the use of learned coarse propagators. Building upon the error estimation framework, we present a systematic procedure for constructing coarse propagators that enjoy desirable stability and consistent order. Additionally, we provide preliminary mathematical guarantees for the resulting parareal algorithm. Numerical experiments on a variety of settings, e.g., linear diffusion model, Allen-Cahn model, and viscous Burgers model, show that learning can significantly improve parallel efficiency when compared with the more ad hoc choice of some conventional and widely used coarse propagators.
KraftEtAl2023

R. Kraft, S. Koraltan, M. Gattringer, F. Bruckner, D. Suess, and C. Abert, “Parallel-in-Time Integration of the Landau-Lifshitz-Gilbert Equation with the Parallel Full Approximation Scheme in Space and Time,” arXiv:2310.11819v1 [physics.comp-ph], 2023 [Online]. Available at: http://arxiv.org/abs/2310.11819v1
BibTeX entry KraftEtAl2023

@unpublished{KraftEtAl2023, author = {Kraft, Robert and Koraltan, Sabri and Gattringer, Markus and Bruckner, Florian and Suess, Dieter and Abert, Claas}, howpublished = {arXiv:2310.11819v1 [physics.comp-ph]}, title = {Parallel-in-Time Integration of the Landau-Lifshitz-Gilbert Equation with the Parallel Full Approximation Scheme in Space and Time}, url = {http://arxiv.org/abs/2310.11819v1}, year = {2023} }
Abstract for BibTeX entry KraftEtAl2023

Speeding up computationally expensive problems, such as numerical simulations of large micromagnetic systems, requires efficient use of parallel computing infrastructures. While parallelism across space is commonly exploited in micromagnetics, this strategy performs poorly once a minimum number of degrees of freedom per core is reached. We use magnum.pi, a finite-element micromagnetic simulation software, to investigate the Parallel Full Approximation Scheme in Space and Time (PFASST) as a space- and time-parallel solver for the Landau-Lifshitz-Gilbert equation (LLG). Numerical experiments show that PFASST enables efficient parallel-in-time integration of the LLG, significantly improving the speedup gained from using a given number of cores as well as allowing the code to scale beyond spatial limits.
LevequeEtAl2023

S. Leveque, L. Bergamaschi, Á. Martínez, and J. W. Pearson, “Fast Iterative Solver for the All-at-Once Runge–Kutta Discretization,” arXiv:2303.02090v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2303.02090v1
BibTeX entry LevequeEtAl2023

@unpublished{LevequeEtAl2023, author = {Leveque, Santolo and Bergamaschi, Luca and Martínez, Ángeles and Pearson, John W.}, howpublished = {arXiv:2303.02090v1 [math.NA]}, title = {Fast Iterative Solver for the All-at-Once Runge--Kutta Discretization}, url = {http://arxiv.org/abs/2303.02090v1}, year = {2023} }
Abstract for BibTeX entry LevequeEtAl2023

In this article, we derive fast and robust preconditioned iterative methods for the all-at-once linear systems arising upon discretization of time-dependent PDEs. The discretization we employ is based on a Runge–Kutta method in time, for which the development of robust solvers is an emerging research area in the literature of numerical methods for time-dependent PDEs. By making use of classical theory of block matrices, one is able to derive a preconditioner for the systems considered. An approximate inverse of the preconditioner so derived consists in a fixed number of linear solves for the system of the stages of the method. We thus propose a preconditioner for the latter system based on a singular value decomposition (SVD) of the (real) Runge–Kutta matrix A_\mathrmRK = U ΣV^⊤. Supposing A_\mathrmRK is invertible, we prove that the spectrum of the system for the stages preconditioned by our SVD-based preconditioner is contained within the right-half of the unit circle, under suitable assumptions on the matrix U^⊤V (which is well defined due to the polar decomposition of A_\mathrmRK). We show the numerical efficiency of our SVD-based preconditioner by solving the system of the stages arising from the discretization of the heat equation and the Stokes equations, with sequential time-stepping. Finally, we provide numerical results of the all-at-once approach for both problems.
Li2023

G. Li, “Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain,” arXiv:2307.06529v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2307.06529v1
BibTeX entry Li2023

@unpublished{Li2023, author = {Li, Guanglian}, howpublished = {arXiv:2307.06529v1 [math.NA]}, title = {Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain}, url = {http://arxiv.org/abs/2307.06529v1}, year = {2023} }
Abstract for BibTeX entry Li2023

We present the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm, recently proposed in [Li and Hu, \it J. Comput. Phys., 2021], for efficiently solving subdiffusion equations with heterogeneous coefficients in long time. This algorithm combines the benefits of multiscale methods, which can handle heterogeneity in the spatial domain, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. Our algorithm overcomes the challenge posed by the nonlocality of the fractional derivative in previous parabolic problem work by constructing an auxiliary problem on each coarse temporal subdomain to completely uncouple the temporal variable. We prove the approximation properties of the correction operator and derive a new summation of exponential to generate a single-step time stepping scheme, with the number of terms of \mathcalO(|\log\tau_f|^2) independent of the final time, where \tau_f is the fine-scale time step size. We establish the convergence rate of our algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step size, and the fine-scale time step size. Finally, we present several numerical tests that demonstrate the effectiveness of our algorithm and validate our theoretical results.

J. Li and Y. Jiang, “Analysis of a New Accelerated Waveform Relaxation Method Based on the Time-Parallel Algorithm,” Journal of Scientific Computing, vol. 96, no. 3, Jul. 2023, doi: 10.1007/s10915-023-02285-4. [Online]. Available at: https://doi.org/10.1007/s10915-023-02285-4

LinEtAl2023

X.-lei Lin and S. Hon, “A block α-circulant based preconditioned MINRES method for wave equations,” arXiv:2306.03574v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2306.03574v1
BibTeX entry LinEtAl2023

@unpublished{LinEtAl2023, author = {Lin, Xue-lei and Hon, Sean}, howpublished = {arXiv:2306.03574v1 [math.NA]}, title = {A block $α$-circulant based preconditioned MINRES method for wave equations}, url = {http://arxiv.org/abs/2306.03574v1}, year = {2023} }
Abstract for BibTeX entry LinEtAl2023

In this work, we propose an absolute value block α-circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Since the original block α-circulant preconditioner shown successful by many recently is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Motivated by the absolute value block circulant preconditioner proposed in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput., 40(2):A1012-A1033, 2018], we propose an absolute value version of the block α-circulant preconditioner. Our proposed preconditioner is the first Hermitian positive definite variant of the block α-circulant preconditioner, which fills the gap between block α-circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for properly chosen αthe MINRES solver with the proposed preconditioner has a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved.

Z. Miao and Y.-L. Jiang, “A Fast Simulation Approach to Switched Systems,” IEEE Transactions on Circuits and Systems II: Express Briefs, pp. 1–1, 2023, doi: 10.1109/tcsii.2023.3332694. [Online]. Available at: http://dx.doi.org/10.1109/TCSII.2023.3332694

P. Munch, I. Dravins, M. Kronbichler, and M. Neytcheva, “Stage-Parallel Fully Implicit Runge–Kutta Implementations with Optimal Multilevel Preconditioners at the Scaling Limit,” SIAM Journal on Scientific Computing, pp. S71–S96, Jul. 2023, doi: 10.1137/22m1503270. [Online]. Available at: https://doi.org/10.1137%2F22m1503270

V.-T. Nguyen and L. Grigori, “Interpretation of parareal as a two-level additive Schwarz in time preconditioner and its acceleration with GMRES,” Numerical Algorithms, Mar. 2023, doi: 10.1007/s11075-022-01492-8. [Online]. Available at: https://doi.org/10.1007/s11075-022-01492-8

H. Nguyen and R. Tsai, “Numerical wave propagation aided by deep learning,” Journal of Computational Physics, vol. 475, p. 111828, Feb. 2023, doi: 10.1016/j.jcp.2022.111828. [Online]. Available at: https://doi.org/10.1016%2Fj.jcp.2022.111828

PhilippiEtAl2023

B. Philippi and T. Slawig, “A Micro-Macro Parareal Implementation for the Ocean-Circulation Model FESOM2,” arXiv:2306.17269v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2306.17269v1
BibTeX entry PhilippiEtAl2023

@unpublished{PhilippiEtAl2023, author = {Philippi, B. and Slawig, T.}, howpublished = {arXiv:2306.17269v1 [math.NA]}, title = {A Micro-Macro Parareal Implementation for the Ocean-Circulation Model FESOM2}, url = {http://arxiv.org/abs/2306.17269v1}, year = {2023} }
Abstract for BibTeX entry PhilippiEtAl2023

A micro-macro variant of the parallel-in-time algorithm Parareal has been applied to the ocean-circulation and sea-ice model model FESOM2. The state-of-the-art software in climate research has been developed by the Alfred-Wegener-Institut (AWI) in Bremen, Germany. The algorithm requires two meshes of low and high spatial resolution to define the coarse and fine propagator. As a first assessment we refined the PI mesh, increasing its resolution by factor 4. The main objective of this study was to demonstrate that micro-macro Parareal can provide convergence in diagnostic variables in complex climate research problems. After the introduction to FESOM2 we show how to generate the refined mesh and which interpolation methods were chosen. With the convergence results presented we discuss the success of this attempt and which steps have to be taken to extend the approach to current research problems.
PhilippiEtAl2023b

B. Philippi, M. S. Miraz, and T. Slawig, “A Micor-Macro parallel-in-time Implementation for the 2D Navier-Stokes Equations,” arXiv:2309.03037v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2309.03037v1
BibTeX entry PhilippiEtAl2023b

@unpublished{PhilippiEtAl2023b, author = {Philippi, Benedict and Miraz, Mahfuz Sarker and Slawig, Thomas}, howpublished = {arXiv:2309.03037v1 [math.NA]}, title = {A Micor-Macro parallel-in-time Implementation for the 2D Navier-Stokes Equations}, url = {http://arxiv.org/abs/2309.03037v1}, year = {2023} }
Abstract for BibTeX entry PhilippiEtAl2023b

In this paper the Micro-Macro Parareal algorithm was adapted to PDEs. The parallel-in-time approach requires two meshes of different spatial resolution in order to compute approximations in an iterative way to a predefined reference solution. When fast convergence in few iterations can be accomplished the algorithm is able to generate wall-time reduction in comparison to the serial computation. We chose the laminar flow around a cylinder benchmark on 2-dimensional domain which was simulated with the open-source software OpenFoam. The numerical experiments presented in this work aim to approximate states local in time and space and the diagnostic lift coefficient. The Reynolds number is gradually increased from 100 to 1,000, before the transition to turbulent flows sets in. After the results are presented the convergence behavior is discussed with respect to the Reynolds number and the applied interpolation schemes.
SarpeEtAl2023

J. Sarpe, A. Klaedtke, and H. D. Gersem, “A Parallel-In-Time Adjoint Sensitivity Analysis for a B6 Bridge-Motor Supply Circuit,” arXiv:2307.00802v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2307.00802v1
BibTeX entry SarpeEtAl2023

@unpublished{SarpeEtAl2023, author = {Sarpe, Julian and Klaedtke, Andreas and Gersem, Herbert De}, howpublished = {arXiv:2307.00802v1 [math.NA]}, title = {A Parallel-In-Time Adjoint Sensitivity Analysis for a B6 Bridge-Motor Supply Circuit}, url = {http://arxiv.org/abs/2307.00802v1}, year = {2023} }
Abstract for BibTeX entry SarpeEtAl2023

This paper presents a parallel-in-time adjoint sensitivity analysis which combines a transient adjoint sensitivity analysis with the parareal approach in order to significantly speed up the simulation. The adjoint method is the method of choice to calculate the sensitivities in a many-parameter setting. In order to obtain sensitivity information that is time-dependent, multiple adjoint problems must be solved. This slows down the simulation wall-clock time and leaves a large optimization potential for the analysis. The parareal is applied to the adjoint solution, significantly speeding up the adjoint solution for every timestep respectively.

J. Schleuß, K. Smetana, and L. ter Maat, “Randomized Quasi-Optimal Local Approximation Spaces in Time,” SIAM Journal on Scientific Computing, vol. 45, no. 3, pp. A1066–A1096, May 2023, doi: 10.1137/22m1481002. [Online]. Available at: https://doi.org/10.1137%2F22m1481002

X. Shan and M. B. van Gijzen, “Parareal Method for Anisotropic Diffusion Denoising,” in Parallel Processing and Applied Mathematics, Springer International Publishing, 2023, pp. 313–322 [Online]. Available at: https://doi.org/10.1007/978-3-031-30445-3_26

B. Song, J.-Y. Wang, and Y.-L. Jiang, “Analysis of a New Krylov subspace enhanced parareal algorithm for time-periodic problems,” Numerical Algorithms, Nov. 2023, doi: 10.1007/s11075-023-01704-9. [Online]. Available at: http://dx.doi.org/10.1007/s11075-023-01704-9

Y. Takahashi, K. Fujiwara, and T. Iwashita, “Parallel-in-Space-and-Time Finite-Element Method for Time-Periodic Magnetic Field Problems with Hysteresis,” IEEE Transactions on Magnetics, pp. 1–1, 2023, doi: 10.1109/tmag.2023.3307498. [Online]. Available at: https://doi.org/10.1109/tmag.2023.3307498

Trotti2023

K. Trotti, “A domain splitting strategy for solving PDEs,” arXiv:2303.01163v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2303.01163v1
BibTeX entry Trotti2023

@unpublished{Trotti2023, author = {Trotti, Ken}, howpublished = {arXiv:2303.01163v1 [math.NA]}, title = {A domain splitting strategy for solving PDEs}, url = {http://arxiv.org/abs/2303.01163v1}, year = {2023} }
Abstract for BibTeX entry Trotti2023

In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the d-dimensional advection-diffusion equation, our proposal is a two-level algorithm that merges the solutions obtained from the discretization of the equation over highly anisotropic submeshes to compute an initial approximation of the fine solution. The algorithm then iteratively refines the initial guess by leveraging the structure of the residual. Performing costly calculations on anisotropic submeshes enable us to reduce the dimensionality of the problem by one, and the merging process, which involves the computation of solutions over disjoint domains, allows for parallel implementation.

D. A. Vargas, R. D. Falgout, S. Günther, and J. B. Schroder, “Multigrid Reduction in Time for Chaotic Dynamical Systems,” SIAM Journal on Scientific Computing, vol. 45, no. 4, pp. A2019–A2042, Aug. 2023, doi: 10.1137/22m1518335. [Online]. Available at: https://doi.org/10.1137%2F22m1518335

Y. Wang, “Parallel Numerical Picard Iteration Methods,” Journal of Scientific Computing, vol. 95, no. 1, Mar. 2023, doi: 10.1007/s10915-023-02156-y. [Online]. Available at: https://doi.org/10.1007/s10915-023-02156-y

M. Wang and S. Zhang, “A Preconditioner for Galerkin–Legendre Spectral All-at-Once System from Time-Space Fractional Diffusion Equation,” Symmetry, vol. 15, no. 12, p. 2144, Dec. 2023, doi: 10.3390/sym15122144. [Online]. Available at: http://dx.doi.org/10.3390/sym15122144

S.-L. Wu, Z. Wang, and T. Zhou, “PinT Preconditioner for Forward-Backward Evolutionary Equations,” SIAM Journal on Matrix Analysis and Applications, vol. 44, no. 4, pp. 1771–1798, Nov. 2023, doi: 10.1137/22m1516476. [Online]. Available at: http://dx.doi.org/10.1137/22M1516476

H. Yamazaki, C. J. Cotter, and B. A. Wingate, “Time-parallel integration and phase averaging for the nonlinear shallow-water equations on the sphere,” Quarterly Journal of the Royal Meteorological Society, Jul. 2023, doi: 10.1002/qj.4517. [Online]. Available at: https://doi.org/10.1002%2Fqj.4517

X. Yue, Z. Wang, and S.-L. Wu, “Convergence Analysis of a Mixed Precision Parareal Algorithm,” SIAM Journal on Scientific Computing, vol. 45, no. 5, pp. A2483–A2510, Sep. 2023, doi: 10.1137/22m1510169. [Online]. Available at: https://doi.org/10.1137/22m1510169

J. Zeifang, A. T. Manikantan, and J. Schütz, “Time parallelism and Newton-adaptivity of the two-derivative deferred correction discontinuous Galerkin method,” Applied Mathematics and Computation, vol. 457, p. 128198, Nov. 2023, doi: 10.1016/j.amc.2023.128198. [Online]. Available at: https://doi.org/10.1016/j.amc.2023.128198

ZhouEtAl2023

Q. Zhou, Y. Liu, and S.-L. Wu, “Parareal algorithm via Chebyshev-Gauss spectral collocation method,” arXiv:2304.10152v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2304.10152v1
BibTeX entry ZhouEtAl2023

@unpublished{ZhouEtAl2023, author = {Zhou, Quan and Liu, Yicheng and Wu, Shu-Lin}, howpublished = {arXiv:2304.10152v1 [math.NA]}, title = {Parareal algorithm via Chebyshev-Gauss spectral collocation method}, url = {http://arxiv.org/abs/2304.10152v1}, year = {2023} }
Abstract for BibTeX entry ZhouEtAl2023

We present the Parareal-CG algorithm for time-dependent differential equations in this work. The algorithm is a parallel in time iteration algorithm utilizes Chebyshev-Gauss spectral collocation method for fine propagator F and backward Euler method for coarse propagator G. As far as we know, this is the first time that the spectral method used as the F propagator of the parareal algorithm. By constructing the stable function of the Chebyshev-Gauss spectral collocation method for the symmetric positive definite (SPD) problem, we find out that the Parareal-CG algorithm and the Parareal-TR algorithm, whose F propagator is chosen to be a trapezoidal ruler, converge similarly, i.e., the Parareal-CG algorithm converge as fast as Parareal-Euler algorithm with sufficient Chebyhsev-Gauss points in every coarse grid. Numerical examples including ordinary differential equations and time-dependent partial differential equations are given to illustrate the high efficiency and accuracy of the proposed algorithm.
ZhouEtAl2023b

Z. Zhou, H. Gu, G. Ju, and W. Xing, “A Parallel-in-time Method Based on Preconditioner for Biot’s Model,” arXiv:2310.10430v1 [math.NA], 2023 [Online]. Available at: http://arxiv.org/abs/2310.10430v1
BibTeX entry ZhouEtAl2023b

@unpublished{ZhouEtAl2023b, author = {Zhou, Zeyuan and Gu, Huipeng and Ju, Guoliang and Xing, Wei}, howpublished = {arXiv:2310.10430v1 [math.NA]}, title = {A Parallel-in-time Method Based on Preconditioner for Biot's Model}, url = {http://arxiv.org/abs/2310.10430v1}, year = {2023} }
Abstract for BibTeX entry ZhouEtAl2023b

We proposed a parallel-in-time method based on preconditioner for Biot’s consolidation model in poroelasticity. In order to achieve a fast and stable convergence for the matrix system of the Biot’s model, we design two preconditioners with approximations of the Schur complement. The parallel-in-time method employs an inverted time-stepping scheme that iterates to solve the preconditioned linear system in the outer loop and advances the time step in the inner loop. This allows us to parallelize the iterations with a theoretical parallel efficiency that approaches 1 as the number of time steps and spatial steps grows. We demonstrate the stability, accuracy, and linear speedup of our method on HPC platform through numerical experiments.

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2022

AgbohEtAl2022

W. C. Agboh, D. Ruprecht, and M. R. Dogar, “Combining Coarse and Fine Physics for Manipulation using Parallel-in-Time Integration,” in Robotics Research, 2022, pp. 725–740, doi: 10.1007/978-3-030-95459-8_44 [Online]. Available at: https://doi.org/10.1007/978-3-030-95459-8_44
BibTeX entry AgbohEtAl2022

@inproceedings{AgbohEtAl2022, author = {Agboh, Wisdom C. and Ruprecht, Daniel and Dogar, Mehmet R.}, booktitle = {Robotics Research}, doi = {10.1007/978-3-030-95459-8_44}, editor = {Asfour, Tamim and Yoshida, Eiichi and Park, Jaeheung and Christensen, Henrik and Khatib, Oussama}, pages = {725 -- 740}, publisher = {Springer International Publishing}, title = {Combining Coarse and Fine Physics for Manipulation using Parallel-in-Time Integration}, url = {https://doi.org/10.1007/978-3-030-95459-8_44}, year = {2022} }
Abstract for BibTeX entry AgbohEtAl2022

We present a method for fast and accurate physics-based predictions during non-prehensile manipulation planning and control. Given an initial state and a sequence of controls, the problem of predicting the resulting sequence of states is a key component of a variety of model-based planning and control algorithms. We propose combining a coarse (i.e. computationally cheap but not very accurate) predictive physics model, with a fine (i.e. computationally expensive but accurate) predictive physics model, to generate a hybrid model that is at the required speed and accuracy for a given manipulation task. Our approach is based on the Parareal algorithm, a parallel-in-time integration method used for computing numerical solutions for general systems of ordinary differential equations. We use Parareal to combine a coarse pushing model with an off-the-shelf physics engine to deliver physics-based predictions that are as accurate as the physics engine but runs in substantially less wall-clock time, thanks to Parareal being amenable to parallelization. We use these physics-based predictions in a model-predictive-control framework based on trajectory optimization, to plan pushing actions that avoid an obstacle and reach a goal location. We show that by combining the two physics models, we can achieve the same success rates as the planner that uses the off-the-shelf physics engine directly, but significantly faster. We present experiments in simulation and on a real robotic setup.

A. Arrarás, F. J. Gaspar, L. Portero, and C. Rodrigo, “Space-Time Parallel Methods for Evolutionary Reaction-Diffusion Problems,” in Domain Decomposition Methods in Science and Engineering XXVI, Springer International Publishing, 2022, pp. 643–651 [Online]. Available at: https://doi.org/10.1007/978-3-030-95025-5_70

D. Q. Bui, C. Japhet, Y. Maday, and P. Omnes, “Coupling Parareal with Optimized Schwarz Waveform Relaxation for Parabolic Problems,” SIAM Journal on Numerical Analysis, vol. 60, no. 3, pp. 913–939, May 2022, doi: 10.1137/21m1419428. [Online]. Available at: https://doi.org/10.1137/21m1419428

T. Cheng, N. Lin, and V. Dinavahi, “Hybrid Parallel-in-Time-and-Space Transient Stability Simulation of Large-Scale AC/DC Grids,” IEEE Transactions on Power Systems, pp. 1–1, 2022, doi: 10.1109/tpwrs.2022.3153450. [Online]. Available at: https://doi.org/10.1109/tpwrs.2022.3153450

S. Costanzo, T. Sayadi, M. F. de Pando, P. J. Schmid, and P. Frey, “Parallel-in-time adjoint-based optimization – application to unsteady incompressible flows,” Journal of Computational Physics, p. 111664, Oct. 2022, doi: 10.1016/j.jcp.2022.111664. [Online]. Available at: https://doi.org/10.1016/j.jcp.2022.111664

L. D’Amore, E. Constantinescu, and L. Carracciuolo, “A Scalable Space-Time Domain Decomposition Approach for Solving Large Scale Nonlinear Regularized Inverse Ill Posed Problems in 4D Variational Data Assimilation,” Journal of Scientific Computing, vol. 91, no. 2, Apr. 2022, doi: 10.1007/s10915-022-01826-7. [Online]. Available at: https://doi.org/10.1007/s10915-022-01826-7

F. Danieli, B. S. Southworth, and A. J. Wathen, “Space-Time Block Preconditioning for Incompressible Flow,” SIAM Journal on Scientific Computing, vol. 44, no. 1, pp. A337–A363, Feb. 2022, doi: 10.1137/21m1390773. [Online]. Available at: https://doi.org/10.1137%2F21m1390773

F. Danieli and S. MacLachlan, “Multigrid reduction in time for non-linear hyperbolic equations,” ETNA - Electronic Transactions on Numerical Analysis, vol. 58, pp. 43–65, 2022, doi: 10.1553/etna_vol58s43. [Online]. Available at: https://doi.org/10.1553%2Fetna_vol58s43

FreiEtAl2022b

S. Frei and A. Heinlein, “Efficient coarse correction for parallel time-stepping in plaque growth simulations,” arXiv:2207.02081v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2207.02081v1
BibTeX entry FreiEtAl2022b

@unpublished{FreiEtAl2022b, author = {Frei, Stefan and Heinlein, Alexander}, howpublished = {arXiv:2207.02081v1 [math.NA]}, title = {Efficient coarse correction for parallel time-stepping in plaque growth simulations}, url = {http://arxiv.org/abs/2207.02081v1}, year = {2022} }
Abstract for BibTeX entry FreiEtAl2022b

In order to make the numerical simulation of atherosclerotic plaque growth feasible, a temporal homogenization approach is employed. The resulting macro-scale problem for the plaque growth can be further accelerated by using parallel time integration schemes, such as the parareal algorithm. However, the parallel scalability is dominated by the computational cost of the coarse propagator. Therefore, in this paper, an interpolation-based coarse propagator, which uses growth values from previously computed micro-scale problems, is introduced. For a simple model problem, it is shown that this approach reduces both the computational work for a single parareal iteration as well as the required number of parareal iterations.

I. C. Garcia, I. Kulchytska-Ruchka, and S. Schöps, “Parareal for index two differential algebraic equations,” Numerical Algorithms, Mar. 2022, doi: 10.1007/s11075-022-01267-1. [Online]. Available at: https://doi.org/10.1007%2Fs11075-022-01267-1

GoryninaEtAl2022

O. Gorynina, F. Legoll, T. Lelievre, and D. Perez, “Combining machine-learned and empirical force fields with the parareal algorithm: application to the diffusion of atomistic defects,” arXiv:2212.10508v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2212.10508v1
BibTeX entry GoryninaEtAl2022

@unpublished{GoryninaEtAl2022, author = {Gorynina, Olga and Legoll, Frederic and Lelievre, Tony and Perez, Danny}, howpublished = {arXiv:2212.10508v1 [math.NA]}, title = {Combining machine-learned and empirical force fields with the parareal algorithm: application to the diffusion of atomistic defects}, url = {http://arxiv.org/abs/2212.10508v1}, year = {2022} }
Abstract for BibTeX entry GoryninaEtAl2022

We numerically investigate an adaptive version of the parareal algorithm in the context of molecular dynamics. This adaptive variant has been originally introduced in [F. Legoll, T. Lelievre and U. Sharma, SISC 2022]. We focus here on test cases of physical interest where the dynamics of the system is modelled by the Langevin equation and is simulated using the molecular dynamics software LAMMPS. In this work, the parareal algorithm uses a family of machine-learning spectral neighbor analysis potentials (SNAP) as fine, reference, potentials and embedded-atom method potentials (EAM) as coarse potentials. We consider a self-interstitial atom in a tungsten lattice and compute the average residence time of the system in metastable states. Our numerical results demonstrate significant computational gains using the adaptive parareal algorithm in comparison to a sequential integration of the Langevin dynamics. We also identify a large regime of numerical parameters for which statistical accuracy is reached without being a consequence of trajectorial accuracy.

J. Hahne, B. S. Southworth, and S. Friedhoff, “Asynchronous Truncated Multigrid-Reduction-in-Time,” SIAM Journal on Scientific Computing, pp. S281–S306, Nov. 2022, doi: 10.1137/21m1433149. [Online]. Available at: https://doi.org/10.1137/21m1433149

G. He, “Time Parallel Denoising Algorithm Based on P-M Equation for Real Image,” Wireless Communications and Mobile Computing, vol. 2022, pp. 1–9, Aug. 2022, doi: 10.1155/2022/8008912. [Online]. Available at: https://doi.org/10.1155/2022/8008912

Y. He and J. Liu, “A Vanka-type multigrid solver for complex-shifted Laplacian systems from diagonalization-based parallel-in-time algorithms,” Applied Mathematics Letters, vol. 132, p. 108125, Oct. 2022, doi: 10.1016/j.aml.2022.108125. [Online]. Available at: https://doi.org/10.1016/j.aml.2022.108125

K. Herb and P. Welter, “Parallel time integration using Batched BLAS (Basic Linear Algebra Subprograms) routines,” Computer Physics Communications, vol. 270, p. 108181, Jan. 2022, doi: 10.1016/j.cpc.2021.108181. [Online]. Available at: https://doi.org/10.1016/j.cpc.2021.108181

Y. Jiang and J. Liu, “Fast Parallel-in-Time Quasi-Boundary Value Methods for Backward Heat Conduction Problems,” Applied Numerical Mathematics, Oct. 2022, doi: 10.1016/j.apnum.2022.10.006. [Online]. Available at: https://doi.org/10.1016/j.apnum.2022.10.006

E. Kazakov, D. Efremenko, V. Zemlyakov, and J. Gao, “A Time-Parallel Ordinary Differential Equation Solver with an Adaptive Step Size: Performance Assessment,” in Lecture Notes in Computer Science, Springer International Publishing, 2022, pp. 3–17 [Online]. Available at: https://doi.org/10.1007/978-3-031-22941-1_1

KressnerEtAl2022

D. Kressner, S. Massei, and J. Zhu, “Improved parallel-in-time integration via low-rank updates and interpolation,” arXiv:2204.03073v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2204.03073v1
BibTeX entry KressnerEtAl2022

@unpublished{KressnerEtAl2022, author = {Kressner, Daniel and Massei, Stefano and Zhu, Junli}, howpublished = {arXiv:2204.03073v1 [math.NA]}, title = {Improved parallel-in-time integration via low-rank updates and interpolation}, url = {http://arxiv.org/abs/2204.03073v1}, year = {2022} }
Abstract for BibTeX entry KressnerEtAl2022

This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the context of parallel-in-time integration leading to a class of algorithms called ParaDiag. We develop and analyze two novel approaches for the numerical solution of such equations. Our first approach is based on the observation that the modification of these equations performed by ParaDiag in order to solve them in parallel has low rank. Building upon previous work on low-rank updates of matrix equations, this allows us to make use of tensorized Krylov subspace methods to account for the modification. Our second approach is based on interpolating the solution of the matrix equation from the solutions of several modifications. Both approaches avoid the use of iterative refinement needed by ParaDiag and related space-time approaches in order to attain good accuracy. In turn, our new approaches have the potential to outperform, sometimes significantly, existing approaches. This potential is demonstrated for several different types of PDEs.

Y. Lee, J. Park, and C.-O. Lee, “Parareal Neural Networks Emulating a Parallel-in-Time Algorithm,” IEEE Transactions on Neural Networks and Learning Systems, pp. 1–12, 2022, doi: 10.1109/tnnls.2022.3206797. [Online]. Available at: https://doi.org/10.1109/tnnls.2022.3206797

C.-O. Lee, Y. Lee, and J. Park, “A Parareal Architecture for Very Deep Convolutional Neural Networks,” in Lecture Notes in Computational Science and Engineering, Springer International Publishing, 2022, pp. 407–415 [Online]. Available at: http://dx.doi.org/10.1007/978-3-030-95025-5_43

F. Legoll, T. Lelièvre, and U. Sharma, “An Adaptive Parareal Algorithm: Application to the Simulation of Molecular Dynamics Trajectories,” SIAM Journal on Scientific Computing, vol. 44, no. 1, pp. B146–B176, Jan. 2022, doi: 10.1137/21m1412979. [Online]. Available at: https://doi.org/10.1137/21m1412979

S. Li, L. Xie, and L. Zhou, “Convergence analysis of space-time domain decomposition method for parabolic equations,” Computers &\mathsemicolon Mathematics with Applications, Aug. 2022, doi: 10.1016/j.camwa.2022.08.012. [Online]. Available at: https://doi.org/10.1016/j.camwa.2022.08.012

J. Liu and Z. Wang, “A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems in unsteady convection-diffusion PDEs,” vol. 416, p. 126750, Mar. 2022, doi: 10.1016/j.amc.2021.126750. [Online]. Available at: https://doi.org/10.1016/j.amc.2021.126750

J. Liu, X.-S. Wang, S.-L. Wu, and T. Zhou, “A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations,” Advances in Computational Mathematics, vol. 48, no. 3, Apr. 2022, doi: 10.1007/s10444-022-09928-4. [Online]. Available at: https://doi.org/10.1007%2Fs10444-022-09928-4

C. Lohmann, J. Dünnebacke, and S. Turek, “Fourier analysis of a time-simultaneous two-grid algorithm using a damped Jacobi waveform relaxation smoother for the one-dimensional heat equation,” Journal of Numerical Mathematics, vol. 0, no. 0, Jun. 2022, doi: 10.1515/jnma-2021-0045. [Online]. Available at: https://doi.org/10.1515/jnma-2021-0045

MoonEtAl2022

G. E. Moon and E. C. Cyr, “Parallel Training of GRU Networks with a Multi-Grid Solver for Long Sequences,” arXiv:2203.04738v1 [cs.CV], 2022 [Online]. Available at: http://arxiv.org/abs/2203.04738v1
BibTeX entry MoonEtAl2022

@unpublished{MoonEtAl2022, author = {Moon, Gordon Euhyun and Cyr, Eric C.}, howpublished = {arXiv:2203.04738v1 [cs.CV]}, title = {Parallel Training of GRU Networks with a Multi-Grid Solver for Long Sequences}, url = {http://arxiv.org/abs/2203.04738v1}, year = {2022} }
Abstract for BibTeX entry MoonEtAl2022

Parallelizing Gated Recurrent Unit (GRU) networks is a challenging task, as the training procedure of GRU is inherently sequential. Prior efforts to parallelize GRU have largely focused on conventional parallelization strategies such as data-parallel and model-parallel training algorithms. However, when the given sequences are very long, existing approaches are still inevitably performance limited in terms of training time. In this paper, we present a novel parallel training scheme (called parallel-in-time) for GRU based on a multigrid reduction in time (MGRIT) solver. MGRIT partitions a sequence into multiple shorter sub-sequences and trains the sub-sequences on different processors in parallel. The key to achieving speedup is a hierarchical correction of the hidden state to accelerate end-to-end communication in both the forward and backward propagation phases of gradient descent. Experimental results on the HMDB51 dataset, where each video is an image sequence, demonstrate that the new parallel training scheme achieves up to 6.5\times speedup over a serial approach. As efficiency of our new parallelization strategy is associated with the sequence length, our parallel GRU algorithm achieves significant performance improvement as the sequence length increases.

K. Pentland, M. Tamborrino, D. Samaddar, and L. C. Appel, “Stochastic Parareal: An Application of Probabilistic Methods to Time-Parallelization,” SIAM Journal on Scientific Computing, pp. S82–S102, Jul. 2022, doi: 10.1137/21m1414231. [Online]. Available at: https://doi.org/10.1137%2F21m1414231

PentlandEtAl2022c

K. Pentland, M. Tamborrino, and T. J. Sullivan, “Error bound analysis of the stochastic parareal algorithm,” arXiv:2211.05496v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2211.05496v1
BibTeX entry PentlandEtAl2022c

@unpublished{PentlandEtAl2022c, author = {Pentland, Kamran and Tamborrino, Massimiliano and Sullivan, T. J.}, howpublished = {arXiv:2211.05496v1 [math.NA]}, title = {Error bound analysis of the stochastic parareal algorithm}, url = {http://arxiv.org/abs/2211.05496v1}, year = {2022} }
Abstract for BibTeX entry PentlandEtAl2022c

Stochastic parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as parareal. Similarly to parareal, it combines fine- and coarse-grained solutions to an ordinary differential equation (ODE) using a predictor-corrector (PC) scheme. The key difference is that carefully chosen random perturbations are added to the PC to try to accelerate the location of a stochastic solution to the ODE. In this paper, we derive superlinear and linear mean-square error bounds for SParareal applied to nonlinear systems of ODEs using different types of perturbations. We illustrate these bounds numerically on a linear system of ODEs and a scalar nonlinear ODE, showing a good match between theory and numerics.

K. Pentland, M. Tamborrino, T. J. Sullivan, J. Buchanan, and L. C. Appel, “GParareal: a time-parallel ODE solver using Gaussian process emulation,” Statistics and Computing, vol. 33, no. 1, Dec. 2022, doi: 10.1007/s11222-022-10195-y. [Online]. Available at: https://doi.org/10.1007%2Fs11222-022-10195-y

PhilippiEtAl2022

B. Philippi and T. Slawig, “The Parareal Algorithm Applied to the FESOM 2 Ocean Circulation Model,” arXiv:2208.07598v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2208.07598v1
BibTeX entry PhilippiEtAl2022

@unpublished{PhilippiEtAl2022, author = {Philippi, Benedict and Slawig, Thomas}, howpublished = {arXiv:2208.07598v1 [math.NA]}, title = {The Parareal Algorithm Applied to the FESOM 2 Ocean Circulation Model}, url = {http://arxiv.org/abs/2208.07598v1}, year = {2022} }
Abstract for BibTeX entry PhilippiEtAl2022

In this work the parallel-in-time algorithm Parareal was applied to the ocean-circulation and sea-ice model FESOM2 developed by the Alfred-Wegener Institut (AWI). The climate model provides one time integration method and hence, the coarse and fine propagators were defined by time step width. The coarse method was executed at the CFL condition limit, while fine step-sizes where gradually refined. As a first assessment of the performance of Parareal a low-resolution test mesh was used with default settings provided by the AWI. An introduction to FESOM2 and the straightforward implementation of Parareal on the DKRZ cluster is given. The evaluation of numerical results for different simulation intervals and fine propagator configurations shows strong dependence on the simulation time and time step-size of the fine propagator. Increasing the latter leads to stagnation and eventually divergence of the Parareal algorithm.

M. K. Riahi, “PiTSBiCG: Parallel in time Stable Bi-Conjugate gradient algorithm,” Applied Numerical Mathematics, vol. 181, pp. 225–233, Nov. 2022, doi: 10.1016/j.apnum.2022.06.004. [Online]. Available at: https://doi.org/10.1016%2Fj.apnum.2022.06.004

RiffoEtAl2022

S. Riffo, F. Kwok, and J. Salomon, “Time-parallelization of sequential data assimilation problems,” arXiv:2212.02377v1 [math.OC], 2022 [Online]. Available at: http://arxiv.org/abs/2212.02377v1
BibTeX entry RiffoEtAl2022

@unpublished{RiffoEtAl2022, author = {Riffo, Sebastián and Kwok, Félix and Salomon, Julien}, howpublished = {arXiv:2212.02377v1 [math.OC]}, title = {Time-parallelization of sequential data assimilation problems}, url = {http://arxiv.org/abs/2212.02377v1}, year = {2022} }
Abstract for BibTeX entry RiffoEtAl2022

This paper is devoted to the problem of time parallelization of assimilation methods applying on unbounded time domain. In this way, we present a general procedure to couple the Luenberger observer with time parallelization algorithm. Our approach is based on a posteriori error estimates of the latter and preserves the rate of the non-parallelized observer. We then focus on the case where the Parareal algorithm is used as time parallelization algorithm, and derive a bound of the efficiency of our procedure. A variant devoted to the case a large number of processors is also proposed. We illustrate the performance of our approach with numerical experiments.
RosemeierEtAl2022

J. Rosemeier, T. Haut, and B. Wingate, “Multi-level Parareal algorithm with Averaging,” arXiv:2211.17239v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2211.17239v1
BibTeX entry RosemeierEtAl2022

@unpublished{RosemeierEtAl2022, author = {Rosemeier, Juliane and Haut, Terry and Wingate, Beth}, howpublished = {arXiv:2211.17239v1 [math.NA]}, title = {Multi-level Parareal algorithm with Averaging}, url = {http://arxiv.org/abs/2211.17239v1}, year = {2022} }
Abstract for BibTeX entry RosemeierEtAl2022

The present study is an extension of the work done in [16] and [10], where a two-level Parareal method with averaging was examined. The method proposed in this paper is a multi-level Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for multi-scale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level.
SaerkkaeEtAl2022

S. Särkkä and Á. F. García-Fernández, “Temporal Parallelisation of the HJB Equation and Continuous-Time Linear Quadratic Control,” arXiv:2212.11744v1 [math.OC], 2022 [Online]. Available at: http://arxiv.org/abs/2212.11744v1
BibTeX entry SaerkkaeEtAl2022

@unpublished{SaerkkaeEtAl2022, author = {Särkkä, Simo and García-Fernández, Ángel F.}, howpublished = {arXiv:2212.11744v1 [math.OC]}, title = {Temporal Parallelisation of the HJB Equation and Continuous-Time Linear Quadratic Control}, url = {http://arxiv.org/abs/2212.11744v1}, year = {2022} }
Abstract for BibTeX entry SaerkkaeEtAl2022

This paper presents a mathematical formulation to perform temporal parallelisation of continuous-time optimal control problems, which are solved via the Hamilton–Jacobi–Bellman (HJB) equation. We divide the time interval of the control problem into sub-intervals, and define a control problem in each sub-interval, conditioned on the start and end states, leading to conditional value functions for the sub-intervals. By defining an associative operator as the minimisation of the sum of conditional value functions, we obtain the elements and associative operators for a parallel associative scan operation. This allows for solving the optimal control problem on the whole time interval in parallel in logarithmic time complexity in the number of sub-intervals. We derive the HJB-type of backward and forward equations for the conditional value functions and solve them in closed form for linear quadratic problems. We also discuss other numerical methods for computing the conditional value functions and present closed form solutions for selected special cases. The computational advantages of the proposed parallel methods are demonstrated via simulations run on a multi-core central processing unit and a graphics processing unit.
SterckEtAl2022

H. D. Sterck, R. D. Falgout, and O. A. Krzysik, “Fast multigrid reduction-in-time for advection via modified semi-Lagrangian coarse-grid operators,” arXiv:2203.13382v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2203.13382v1
BibTeX entry SterckEtAl2022

@unpublished{SterckEtAl2022, author = {Sterck, H. De and Falgout, R. D. and Krzysik, O. A.}, howpublished = {arXiv:2203.13382v1 [math.NA]}, title = {Fast multigrid reduction-in-time for advection via modified semi-Lagrangian coarse-grid operators}, url = {http://arxiv.org/abs/2203.13382v1}, year = {2022} }
Abstract for BibTeX entry SterckEtAl2022

Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs), but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. The key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-Lagrangian-based coarse-grid operator that leads to fast and scalable multilevel time integration of linear advection PDEs. The coarse-grid operator is composed of a semi-Lagrangian discretization followed by a correction term, with the correction designed so that the leading-order truncation error of the composite operator is approximately equal to that of the ideal coarse-grid operator. Parallel results show substantial speed-ups over sequential time integration for variable-wave-speed advection problems in one and two spatial dimensions, and using high-order discretizations up to order five. The proposed approach establishes the first practical method that provides small and scalable MGRIT iteration counts for advection problems.
SterckEtAl2022b

H. D. Sterck, S. Friedhoff, O. A. Krzysik, and S. P. MacLachlan, “Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective,” arXiv:2208.01526v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2208.01526v1
BibTeX entry SterckEtAl2022b

@unpublished{SterckEtAl2022b, author = {Sterck, H. De and Friedhoff, S. and Krzysik, O. A. and MacLachlan, Scott P.}, howpublished = {arXiv:2208.01526v1 [math.NA]}, title = {Multigrid reduction-in-time convergence for advection problems: A Fourier analysis perspective}, url = {http://arxiv.org/abs/2208.01526v1}, year = {2022} }
Abstract for BibTeX entry SterckEtAl2022b

A long-standing issue in the parallel-in-time community is the poor convergence of standard iterative parallel-in-time methods for hyperbolic partial differential equations (PDEs), and for advection-dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two-level variant of the iterative parallel-in-time method of multigrid reduction-in-time (MGRIT). This closed-form theory allows for new insights into the poor convergence of MGRIT for advection-dominated PDEs when using the standard approach of rediscretizing the fine-grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse-grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady-state advection-dominated PDEs. We apply this convergence theory to show that, for certain semi-Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse-grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re-purposed in the MGRIT context to develop more robust parallel-in-time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.
SterckEtAl2022c

H. D. Sterck, R. D. Falgout, O. A. Krzysik, and J. B. Schroder, “Efficient multigrid reduction-in-time for method-of-lines discretizations of linear advection,” arXiv:2209.06916v1 [math.NA], 2022 [Online]. Available at: http://arxiv.org/abs/2209.06916v1
BibTeX entry SterckEtAl2022c

@unpublished{SterckEtAl2022c, author = {Sterck, H. De and Falgout, R. D. and Krzysik, O. A. and Schroder, J. B.}, howpublished = {arXiv:2209.06916v1 [math.NA]}, title = {Efficient multigrid reduction-in-time for method-of-lines discretizations of linear advection}, url = {http://arxiv.org/abs/2209.06916v1}, year = {2022} }
Abstract for BibTeX entry SterckEtAl2022c

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes known as characteristic components.We propose an alternative coarse-grid that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order.

J. Strake, D. Döhring, and A. Benigni, “MGRIT-Based Multi-Level Parallel-in-Time Electromagnetic Transient Simulation,” Energies, vol. 15, no. 21, p. 7874, Oct. 2022, doi: 10.3390/en15217874. [Online]. Available at: https://doi.org/10.3390/en15217874

M. Sugiyama, J. B. Schroder, B. S. Southworth, and S. Friedhoff, “Weighted relaxation for multigrid reduction in time,” Numerical Linear Algebra with Applications, Sep. 2022, doi: 10.1002/nla.2465. [Online]. Available at: https://doi.org/10.1002%2Fnla.2465

M. A. Sultanov, V. E. Misilov, and Y. Nurlanuly, “Efficient Parareal algorithm for solving time-fractional diffusion equation,” Dal nevostochnyi Matematicheskii Zhurnal, vol. 22, no. 2, pp. 245–251, 2022, doi: 10.47910/femj202233. [Online]. Available at: https://doi.org/10.47910/femj202233

Y. Takahashi, K. Fujiwara, and T. Iwashita, “Parallel-in-space-and-time finite-element analysis of electric machines using time step overlapping in a massively parallel computing environment,” COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Jul. 2022, doi: 10.1108/compel-04-2022-0161. [Online]. Available at: https://doi.org/10.1108/compel-04-2022-0161

R. Tielen, M. Möller, and C. Vuik, “Combining p-multigrid and Multigrid Reduction in Time methods to obtain a scalable solver for Isogeometric Analysis,” SN Applied Sciences, vol. 4, no. 6, May 2022, doi: 10.1007/s42452-022-05043-7. [Online]. Available at: https://doi.org/10.1007%2Fs42452-022-05043-7

Utkarsh, C. Elrod, Y. Ma, K. Althaus, and C. Rackauckas, “Parallelizing Explicit and Implicit Extrapolation Methods for Ordinary Differential Equations,” in 2022 IEEE High Performance Extreme Computing Conference (HPEC), 2022, doi: 10.1109/hpec55821.2022.9926357 [Online]. Available at: https://doi.org/10.1109%2Fhpec55821.2022.9926357

C.-Y. Wang, Y.-L. Jiang, and Z. Miao, “Time domain decomposition of parabolic control problems based on discontinuous Galerkin semi-discretization,” Applied Numerical Mathematics, Feb. 2022, doi: 10.1016/j.apnum.2022.02.016. [Online]. Available at: https://doi.org/10.1016/j.apnum.2022.02.016

R. Watschinger, M. Merta, G. Of, and J. Zapletal, “A Parallel Fast Multipole Method for a Space-Time Boundary Element Method for the Heat Equation,” SIAM Journal on Scientific Computing, vol. 44, no. 4, pp. C320–C345, Aug. 2022, doi: 10.1137/21m1430157. [Online]. Available at: https://doi.org/10.1137%2F21m1430157

J. Yang, Z. Yuan, and Z. Zhou, “Robust Convergence of Parareal Algorithms with Arbitrarily High-Order Fine Propagators,” SSRN Electronic Journal, 2022, doi: 10.2139/ssrn.4097528. [Online]. Available at: https://doi.org/10.2139%2Fssrn.4097528

L. Yang and H. Li, “A hybrid algorithm based on parareal and Schwarz waveform relaxation,” Electronic Research Archive, vol. 30, no. 11, pp. 4086–4107, 2022, doi: 10.3934/era.2022207. [Online]. Available at: https://doi.org/10.3934/era.2022207

R. Yoda, M. Bolten, K. Nakajima, and A. Fujii, “Assignment of idle processors to spatial redistributed domains on coarse levels in multigrid reduction in time,” in International Conference on High Performance Computing in Asia-Pacific Region, 2022, doi: 10.1145/3492805.3492810 [Online]. Available at: https://doi.org/10.1145/3492805.3492810

R. Yoda, M. Bolten, K. Nakajima, and A. Fujii, “Acceleration of Optimized Coarse-Grid Operators by Spatial Redistribution for Multigrid Reduction in Time,” in Computational Science – ICCS 2022, Springer International Publishing, 2022, pp. 214–221 [Online]. Available at: https://doi.org/10.1007/978-3-031-08754-7_29

R.-H. Zhang, Y.-L. Jiang, J. Li, and B. Song, “Analysis of the parareal algorithm for linear parametric differential equations,” International Journal of Computer Mathematics, pp. 1–0, Nov. 2022, doi: 10.1080/00207160.2022.2153225. [Online]. Available at: https://doi.org/10.1080/00207160.2022.2153225

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2021

AngelEtAl2021

J. Angel, S. Götschel, and D. Ruprecht, “Impact of spatial coarsening on Parareal convergence,” arXiv:2111.10228v1 [math.NA], 2021 [Online]. Available at: http://arxiv.org/abs/2111.10228v1
BibTeX entry AngelEtAl2021

@unpublished{AngelEtAl2021, author = {Angel, Judith and Götschel, Sebastian and Ruprecht, Daniel}, howpublished = {arXiv:2111.10228v1 [math.NA]}, title = {Impact of spatial coarsening on Parareal convergence}, url = {http://arxiv.org/abs/2111.10228v1}, year = {2021} }
Abstract for BibTeX entry AngelEtAl2021

We study the impact of spatial coarsening on the convergence of the Parareal algorithm, both theoretically and numerically. For initial value problems with a normal system matrix, we prove a lower bound for the Euclidean norm of the iteration matrix. When there is no physical or numerical diffusion, an immediate consequence is that the norm of the iteration matrix cannot be smaller than unoty as soon as the coarse problem has fewer degrees-of-freedom than the fine. This prevents a theoretical guarantee for monotonic convergence, which is necessary to obtain meaningful speedups. For diffusive problems, in the worst-case where the iteration error contracts only as fast as the powers of the iteration matrix norm, making Parareal as accurate as the fine method will take about as many iterations as there are processors, making meaningful speedup impossible. Numerical examples with a non-normal system matrix show that for diffusive problems good speedup is possible, but that for non-diffusive problems the negative impact of spatial coarsening on convergence is big.

P. Benedusi, M. L. Minion, and R. Krause, “An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem,” Computers & Mathematics with Applications, vol. 99, pp. 162–170, Oct. 2021, doi: 10.1016/j.camwa.2021.07.008. [Online]. Available at: https://doi.org/10.1016%2Fj.camwa.2021.07.008

S. Blanes, “Novel parallel in time integrators for ODEs,” Applied Mathematics Letters, p. 107542, Jul. 2021, doi: 10.1016/j.aml.2021.107542. [Online]. Available at: https://doi.org/10.1016/j.aml.2021.107542

A. L. Blumers, M. Yin, H. Nakajima, Y. Hasegawa, Z. Li, and G. E. Karniadakis, “Multiscale parareal algorithm for long-time mesoscopic simulations of microvascular blood flow in zebrafish,” Computational Mechanics, Aug. 2021, doi: 10.1007/s00466-021-02062-w. [Online]. Available at: https://doi.org/10.1007%2Fs00466-021-02062-w

T. Buvoli and M. Minion, “IMEX Runge-Kutta Parareal for Non-diffusive Equations,” in Springer Proceedings in Mathematics &\mathsemicolon Statistics, Springer International Publishing, 2021, pp. 95–127 [Online]. Available at: https://doi.org/10.1007%2F978-3-030-75933-9_5

M. Cai, J. Mahseredjian, I. Kocar, X. Fu, and A. Haddadi, “A parallelization-in-time approach for accelerating EMT simulations,” Electric Power Systems Research, vol. 197, p. 107346, Aug. 2021, doi: 10.1016/j.epsr.2021.107346. [Online]. Available at: https://doi.org/10.1016/j.epsr.2021.107346

CaldasEtAl2021

J. G. Caldas Steinstraesser, “Coupling large and small scale shallow water models with porosity in the presence of anisotropy,” PhD thesis, Université de Montpellier, 2021 [Online]. Available at: https://www.theses.fr/2021MONTS040
BibTeX entry CaldasEtAl2021

@phdthesis{CaldasEtAl2021, author = {Caldas Steinstraesser, Jo\~{a}o Guilherme}, school = {Universit\'{e} de Montpellier}, title = {Coupling large and small scale shallow water models with porosity in the presence of anisotropy}, url = {https://www.theses.fr/2021MONTS040}, year = {2021} }
Abstract for BibTeX entry CaldasEtAl2021

This PhD thesis aims to study the coupling of nonlinear shallow water models at different scales, with application to the numerical simulation of urban floods. Accurate simulations in this domain are usually prohibitively expensive due to the small mesh sizes necessary for the spatial discretization of the urban geometry and the associated small time steps constrained by stability conditions. Porosity-based shallow water models have been proposed in the past two decades as an alternative approach, consisting of upscaled models using larger mesh sizes and time steps and being able to provide good global approximations for the solution of the fine shallow water equations, with much smaller computational times. However, small-scale phenomena are not captured by this type of model. Therefore, we seek to formulate a numerical model coupling the fine and upscaled ones, in order to obtain more accurate solutions inside the urban zone, always with reduced computational costs relatively to the simulation of the fine model. The guideline for this objective lays on the use of predictor-corrector iterative parallel-in-time numerical methods, which naturally fit to this fine/coarse formulation. We focus on the parareal, one of the most popular parallel-in-time methods. As a main challenge, temporal parallelization suffers from instabilities and/or slow convergence when applied to hyperbolic or advection-dominated problems, such as the shallow water equations. Therefore, we consider a variant of the method using reduced-order models (ROMs) formulated on-the-fly along parareal iterations, using Proper Orthogonal Decomposition (POD) and the Empirical Interpolation Method (EIM), being able to improve the stability and convergence of the parareal method for solving nonlinear hyperbolic problems. We investigate the limitations of this ROM-based parareal method and we propose a number of modifications that provide further stability and convergence improvements: enrichment of the input snapshot sets used for the model reduction procedure; formulation of local-in-time ROMs; and incorporation of an adaptive parareal approach recently presented in the literature. The original and ROM-based parareal methods, including the proposed improvements, are compared and evaluated in terms of stability, convergence towards the fine solution and numerical speedup obtained in a parallel implementation. In a first part, the methods are formulated, studied and implemented considering a set of numerical simulations coupling the classical shallow water equations (without the porosity concept) at different scales. After this initial study, we implement them for coupling the classical and the porosity-based shallow water models, for the simulation of urban floods.
CaldasEtAl2021b

J. G. Caldas Steinstraesser, V. Guinot, and A. Rousseau, “Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes,” The SMAI journal of computational mathematics, vol. 7, pp. 159–184, 2021, doi: 10.5802/smai-jcm.75. [Online]. Available at: https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.75/
BibTeX entry CaldasEtAl2021b

@article{CaldasEtAl2021b, author = {Caldas Steinstraesser, Jo\~{a}o Guilherme and Guinot, Vincent and Rousseau, Antoine}, doi = {10.5802/smai-jcm.75}, journal = {The SMAI journal of computational mathematics}, language = {en}, pages = {159--184}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, title = {Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.75/}, volume = {7}, year = {2021} }
Abstract for BibTeX entry CaldasEtAl2021b

In this work, the POD-DEIM-based parareal method introduced in [8] is implemented for solving the two-dimensional nonlinear shallow water equations using a finite volume scheme. This method is a variant of the traditional parareal method, first introduced by [22], that improves the stability and convergence for nonlinear hyperbolic problems, and uses reduced-order models constructed via the Proper Orthogonal Decomposition - Discrete Empirical Interpolation Method (POD-DEIM) applied to snapshots of the solution of the parareal iterations. We propose a modification of this parareal method for further stability and convergence improvements. It consists in enriching the snapshots set for the POD-DEIM procedure with extra snapshots whose computation does not require any additional computational cost. The performances of the classical parareal method, the POD-DEIM-based parareal method and our proposed modification are compared using numerical tests with increasing complexity. Our modified method shows a more stable behaviour and converges in fewer iterations than the other two methods.

M. Caliari, L. Einkemmer, A. Moriggl, and A. Ostermann, “An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs,” Journal of Computational Physics, vol. 437, p. 110289, Jul. 2021, doi: 10.1016/j.jcp.2021.110289. [Online]. Available at: https://doi.org/10.1016%2Fj.jcp.2021.110289

ChaudhryEtAl2021

J. Chaudhry, D. Estep, and S. Tavener, “A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations,” arXiv:2111.00606v1 [math.NA], 2021 [Online]. Available at: http://arxiv.org/abs/2111.00606v1
BibTeX entry ChaudhryEtAl2021

@unpublished{ChaudhryEtAl2021, author = {Chaudhry, Jehanzeb and Estep, Donald and Tavener, Simon}, howpublished = {arXiv:2111.00606v1 [math.NA]}, title = {A posteriori error analysis for a space-time parallel discretization of parabolic partial differential equations}, url = {http://arxiv.org/abs/2111.00606v1}, year = {2021} }
Abstract for BibTeX entry ChaudhryEtAl2021

We construct a space-time parallel method for solving parabolic partial differential equations by coupling the Parareal algorithm in time with overlapping domain decomposition in space. Reformulating the original Parareal algorithm as a variational method and implementing a finite element discretization in space enables an adjoint-based a posteriori error analysis to be performed. Through an appropriate choice of adjoint problems and residuals the error analysis distinguishes between errors arising due to the temporal and spatial discretizations, as well as between the errors arising due to incomplete Parareal iterations and incomplete iterations of the domain decomposition solver. We first develop an error analysis for the Parareal method applied to parabolic partial differential equations, and then refine this analysis to the case where the associated spatial problems are solved using overlapping domain decomposition. These constitute our Time Parallel Algorithm (TPA) and Space-Time Parallel Algorithm (STPA) respectively. Numerical experiments demonstrate the accuracy of the estimator for both algorithms and the iterations between distinct components of the error.

Y.-C. Chen and K. Nakajima, “Optimized Cascadic Multigrid Parareal Method for Explicit Time-Marching Schemes,” in 2021 12th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems (ScalA), 2021, doi: 10.1109/scala54577.2021.00007 [Online]. Available at: https://doi.org/10.1109/scala54577.2021.00007

T. Cheng, N. Lin, T. Liang, and V. Dinavahi, “Parallel-in-time-and-space electromagnetic transient simulation of multi-terminal DC grids with device-level switch modelling,” IET Generation, Transmission & Distribution, Sep. 2021, doi: 10.1049/gtd2.12285. [Online]. Available at: https://doi.org/10.1049/gtd2.12285

F. Danieli and A. J. Wathen, “All-at-once solution of linear wave equations,” Numerical Linear Algebra with Applications, May 2021, doi: 10.1002/nla.2386. [Online]. Available at: https://doi.org/10.1002/nla.2386

V. Dinavahi and N. Lin, “Parallel-in-Time EMT and Transient Stability Simulation of AC-DC Grids,” in Parallel Dynamic and Transient Simulation of Large-Scale Power Systems, Springer International Publishing, 2021, pp. 313–357 [Online]. Available at: https://doi.org/10.1007/978-3-030-86782-9_7

M. Donatelli, R. Krause, M. Mazza, and K. Trotti, “All-at-once multigrid approaches for one-dimensional space-fractional diffusion equations,” vol. 58, no. 4, Oct. 2021, doi: 10.1007/s10092-021-00436-3. [Online]. Available at: https://doi.org/10.1007/s10092-021-00436-3

A. C. Ellison and B. Fornberg, “A parallel-in-time approach for wave-type PDEs,” Numerische Mathematik, Apr. 2021, doi: 10.1007/s00211-021-01197-5. [Online]. Available at: https://doi.org/10.1007/s00211-021-01197-5

R. D. Falgout, T. A. Manteuffel, B. O’Neill, and J. B. Schroder, “Multigrid reduction in time with Richardson extrapolation,” ETNA - Electronic Transactions on Numerical Analysis, vol. 54, pp. 210–233, 2021, doi: 10.1553/etna_vol54s210. [Online]. Available at: https://doi.org/10.1553%2Fetna_vol54s210

L. Fang, S. Vandewalle, and J. Meyers, “A parallel-in-time multiple shooting algorithm for large-scale PDE-constrained optimal control problems,” Journal of Computational Physics, p. 110926, Dec. 2021, doi: 10.1016/j.jcp.2021.110926. [Online]. Available at: https://doi.org/10.1016/j.jcp.2021.110926

GanderEtAl2021

M. J. Gander, J. Liu, S.-L. Wu, X. Yue, and T. Zhou, “ParaDiag: parallel-in-time algorithms based on the diagonalization technique,” arXiv preprint arXiv:2005.09158, 2021 [Online]. Available at: http://arxiv.org/abs/2005.09158
BibTeX entry GanderEtAl2021

@unpublished{GanderEtAl2021, author = {Gander, Martin J and Liu, Jun and Wu, Shu-Lin and Yue, Xiaoqiang and Zhou, Tao}, howpublished = {arXiv preprint arXiv:2005.09158}, title = {ParaDiag: parallel-in-time algorithms based on the diagonalization technique}, url = {http://arxiv.org/abs/2005.09158}, year = {2021} }
Abstract for BibTeX entry GanderEtAl2021

In 2008, Maday and Rønquist introduced an interesting new approach for the direct parallel-in-time (PinT) solution of time-dependent PDEs. The idea is to diagonalize the time stepping matrix, keeping the matrices for the space discretization unchanged, and then to solve all time steps in parallel. Since then, several variants appeared, and we call these closely related algorithms \em ParaDiag algorithms. ParaDiag algorithms in the literature can be classified into two groups: \beginitemize \item ParaDiag-I: direct standalone solvers, \item ParaDiag-II: iterative solvers. \enditemize We will explain the basic features of each group in this note. To have concrete examples, we will introduce ParaDiag-I and ParaDiag-II for the advection-diffusion equation. We will also introduce ParaDiag-II for the wave equation and an optimal control problem for the wave equation. We could have used the advection-diffusion equation as well to illustrate ParaDiag-II, but wave equations are known to cause problems for certain PinT algorithms and thus constitute an especially interesting example for which ParaDiag algorithms were tested. We show the main known theoretical results in each case, and also provide Matlab codes for testing. The goal of the Matlab codes is to help the interested reader understand the key features of the ParaDiag algorithms, without intention to be highly tuned for efficiency and/or low memory use. We also provide speedup measurements of ParaDiag algorithms for a 2D linear advection-diffusion equation. These results are obtained on the Tianhe-1 supercomputer in China and the SIUE Campus Cluster in the US, which is a multi-array, configurable and cooperative parallel system, and we compare these results to the performance of parareal and MGRiT, two widely used PinT algorithms. In a forthcoming update of this note, we will provide more material on ParaDiag algorithms, in particular further Matlab codes and parallel computing results, also for more realistic applications.

S. Götschel, M. Minion, D. Ruprecht, and R. Speck, “Twelve Ways to Fool the Masses When Giving Parallel-in-Time Results,” in Springer Proceedings in Mathematics & Statistics, Springer International Publishing, 2021, pp. 81–94 [Online]. Available at: https://doi.org/10.1007/978-3-030-75933-9_4

L. Grigori, S. A. Hirstoaga, V.-T. Nguyen, and J. Salomon, “Reduced model-based parareal simulations of oscillatory singularly perturbed ordinary differential equations,” Journal of Computational Physics, vol. 436, p. 110282, Jul. 2021, doi: 10.1016/j.jcp.2021.110282. [Online]. Available at: https://doi.org/10.1016/j.jcp.2021.110282

R. Kumari, S. Prasad, A. Kumari, and A. Kumar, “Parallel in Time Simulation of Automatic Generation Control System for Near Real-Time Transient Stability Analysis,” in 2021 International Conference on Control, Automation, Power and Signal Processing (CAPS), 2021, doi: 10.1109/caps52117.2021.9730716 [Online]. Available at: https://doi.org/10.1109/caps52117.2021.9730716

S. Lakshmiranganatha and S. S. Muknahallipatna, “Performance Analysis of Accelerator Architectures and Programming Models for Parareal Algorithm Solutions of Ordinary Differential Equations,” Journal of Computer and Communications, vol. 09, no. 02, pp. 29–56, 2021, doi: 10.4236/jcc.2021.92003. [Online]. Available at: https://doi.org/10.4236/jcc.2021.92003

U. Langer and M. Zank, “Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces,” SIAM Journal on Scientific Computing, vol. 43, no. 4, pp. A2714–A2736, Jan. 2021, doi: 10.1137/20m1358128. [Online]. Available at: https://doi.org/10.1137%2F20m1358128

S. Li, X. Shao, and R. Chen, “Multilevel space-time multiplicative Schwarz preconditioner for parabolic equations,” Numerical Linear Algebra with Applications, May 2021, doi: 10.1002/nla.2390. [Online]. Available at: https://doi.org/10.1002/nla.2390

X. Li and Y. Su, “A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations,” Journal of Algorithms & Computational Technology, vol. 15, p. 174830262110084, Jan. 2021, doi: 10.1177/17483026211008409. [Online]. Available at: https://doi.org/10.1177/17483026211008409

J. Li, P. Liu, and V. Dinavahi, “Space-Time-Parallel 3-D Finite Element Transformer Model with Adaptive TLM and Parareal Techniques for Electromagnetic Transient Analysis,” IEEE Open Journal of Industry Applications, pp. 1–1, 2021, doi: 10.1109/ojia.2021.3091557. [Online]. Available at: https://doi.org/10.1109/ojia.2021.3091557

J. Li, Y.-L. Jiang, and Z. Miao, “Analysis of the parareal approach based on discontinuous Galerkin method for time-dependent Stokes equations,” Numerical Methods for Partial Differential Equations, Jul. 2021, doi: 10.1002/num.22782. [Online]. Available at: https://doi.org/10.1002/num.22782

G. Li and J. Hu, “Wavelet-based Edge Multiscale Parareal Algorithm for Parabolic Equations with Heterogeneous Coefficients and Rough Initial Data,” Journal of Computational Physics, p. 110572, Jul. 2021, doi: 10.1016/j.jcp.2021.110572. [Online]. Available at: https://doi.org/10.1016/j.jcp.2021.110572

J. Li and Y.-L. Jiang, “The study of parareal algorithm for the linear switched systems,” p. 107763, Oct. 2021, doi: 10.1016/j.aml.2021.107763. [Online]. Available at: https://doi.org/10.1016/j.aml.2021.107763

X.-lei Lin, M. K. Ng, and Y. Zhi, “A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel,” Journal of Computational Physics, vol. 434, p. 110221, Jun. 2021, doi: 10.1016/j.jcp.2021.110221. [Online]. Available at: https://doi.org/10.1016%2Fj.jcp.2021.110221

LinEtAl2021b

X.-L. Lin and Z. Zhang, “A Parallel-in-Time Preconditioner for the Schur Complement of Parabolic Optimal Control Problems,” arXiv:2109.12524v2 [math.NA], 2021 [Online]. Available at: http://arxiv.org/abs/2109.12524v2
BibTeX entry LinEtAl2021b

@unpublished{LinEtAl2021b, author = {Lin, Xue-Lei and Zhang, Zhimin}, howpublished = {arXiv:2109.12524v2 [math.NA]}, title = {A Parallel-in-Time Preconditioner for the Schur Complement of Parabolic Optimal Control Problems}, url = {http://arxiv.org/abs/2109.12524v2}, year = {2021} }
Abstract for BibTeX entry LinEtAl2021b

For optimal control problems constrained by a initial-valued parabolic PDE, we have to solve a large scale saddle point algebraic system consisting of considering the discrete space and time points all together. A popular strategy to handle such a system is the Krylov subspace method, for which an efficient preconditioner plays a crucial role. The matching-Schur-complement preconditioner has been extensively studied in literature and the implementation of this preconditioner lies in solving the underlying PDEs twice, sequentially in time. In this paper, we propose a new preconditioner for the Schur complement, which can be used parallel-in-time (PinT) via the so called diagonalization technique. We show that the eigenvalues of the preconditioned matrix are low and upper bounded by positive constants independent of matrix size and the regularization parameter. The uniform boundedness of the eigenvalues leads to an optimal linear convergence rate of conjugate gradient solver for the preconditioned Schur complement system. To the best of our knowledge, it is the first time to have an optimal convergence analysis for a PinT preconditioning technique of the optimal control problem. Numerical results are reported to show that the performance of the proposed preconditioner is robust with respect to the discretization step-sizes and the regularization parameter.
LiuEtAl2021

J. Liu and S.-L. Wu, “Parallel-in-time preconditioners for the Sinc-Nyström method,” arXiv:2108.01700v1 [math.NA], 2021 [Online]. Available at: http://arxiv.org/abs/2108.01700v1
BibTeX entry LiuEtAl2021

@unpublished{LiuEtAl2021, author = {Liu, Jun and Wu, Shu-Lin}, howpublished = {arXiv:2108.01700v1 [math.NA]}, title = {Parallel-in-time preconditioners for the Sinc-Nyström method}, url = {http://arxiv.org/abs/2108.01700v1}, year = {2021} }
Abstract for BibTeX entry LiuEtAl2021

The Sinc-Nyström method in time is a high-order spectral method for solving evolutionary differential equations and it has wide applications in scientific computation. But in this method we have to solve all the time steps implicitly at one-shot, which may results in a large-scale nonsymmetric dense system that is expensive to solve. In this paper, we propose and analyze a parallel-in-time (PinT) preconditioner for solving such Sinc-Nyström systems, where both the parabolic and hyperbolic PDEs are investigated. Attributed to the special Toeplitz-like structure of the Sinc-Nyström systems, the proposed PinT preconditioner is indeed a low-rank perturbation of the system matrix and we show that the spectrum of the preconditioned system is highly clustered around one, especially when the time step size is refined. Such a clustered spectrum distribution matches very well with the numerically observed mesh-independent GMRES convergence rates in various examples. Several linear and nonlinear ODE and PDE examples are presented to illustrate the convergence performance of our proposed PinT preconditioners, where the achieved exponential order of accuracy are especially attractive to those applications in need of high accuracy.

C. Lyu, N. Lin, and V. Dinavahi, “Device-Level Parallel-in-time Simulation of MMC-Based Energy System for Electric Vehicles,” IEEE Transactions on Vehicular Technology, pp. 1–1, 2021, doi: 10.1109/tvt.2021.3081534. [Online]. Available at: https://doi.org/10.1109/tvt.2021.3081534

N. Margenberg and T. Richter, “Parallel time-stepping for fluid–structure interactions,” Mathematical Modelling of Natural Phenomena, vol. 16, p. 20, 2021, doi: 10.1051/mmnp/2021005. [Online]. Available at: https://doi.org/10.1051/mmnp/2021005

B. Park, K. Sun, A. Dimitrovski, Y. Liu, and S. Simunovic, “Examination of Semi-Analytical Solution Methods in the Coarse Operator of Parareal Algorithm for Power System Simulation,” IEEE Transactions on Power Systems, pp. 1–1, 2021, doi: 10.1109/tpwrs.2021.3069136. [Online]. Available at: https://doi.org/10.1109/tpwrs.2021.3069136

M. Patil and A. Datta, “Time-Parallel Scalable Solution of Periodic Rotor Dynamics for Large-Scale 3D Structures,” in AIAA Scitech 2021 Forum, 2021, doi: 10.2514/6.2021-1079 [Online]. Available at: https://doi.org/10.2514/6.2021-1079

A. Pels, I. Kulchytska-Ruchka, and S. Schöps, “Parallel-in-Time Simulation of Power Converters Using Multirate PDEs,” in Scientific Computing in Electrical Engineering, Springer International Publishing, 2021, pp. 33–41 [Online]. Available at: https://doi.org/10.1007%2F978-3-030-84238-3_4

J. Schütz, D. C. Seal, and J. Zeifang, “Parallel-in-Time High-Order Multiderivative IMEX Solvers,” Journal of Scientific Computing, vol. 90, no. 1, Dec. 2021, doi: 10.1007/s10915-021-01733-3. [Online]. Available at: https://doi.org/10.1007%2Fs10915-021-01733-3

A. A. Sivas, B. S. Southworth, and S. Rhebergen, “AIR Algebraic Multigrid for a Space-Time Hybridizable Discontinuous Galerkin Discretization of Advection(-Diffusion),” vol. 43, no. 5, pp. A3393–A3416, Jan. 2021, doi: 10.1137/20m1375103. [Online]. Available at: https://doi.org/10.1137%2F20m1375103

C. S. Skene, M. F. Eggl, and P. J. Schmid, “A parallel-in-time approach for accelerating direct-adjoint studies,” Journal of Computational Physics, vol. 429, p. 110033, Mar. 2021, doi: 10.1016/j.jcp.2020.110033. [Online]. Available at: https://doi.org/10.1016/j.jcp.2020.110033

Y. Sun, S.-L. Wu, and Y. Xu, “A Parallel-in-Time Implementation of the Numerov Method For Wave Equations,” Journal of Scientific Computing, vol. 90, no. 1, Nov. 2021, doi: 10.1007/s10915-021-01701-x. [Online]. Available at: https://doi.org/10.1007/s10915-021-01701-x

Y. Takahashi, K. Fujiwara, T. Iwashita, and H. Nakashima, “Comparison of Parallel-in-Space-and-Time Finite-Element Methods for Magnetic Field Analysis of Electric Machines,” IEEE Transactions on Magnetics, pp. 1–1, 2021, doi: 10.1109/tmag.2021.3064320. [Online]. Available at: https://doi.org/10.1109/tmag.2021.3064320

R. van Venetië and J. Westerdiep, “A Parallel Algorithm for Solving Linear Parabolic Evolution Equations,” Springer International Publishing, 2021, pp. 33–50 [Online]. Available at: https://doi.org/10.1007/978-3-030-75933-9_2

A. S. Walker and K. E. Niemeyer, “Applying the Swept Rule for Solving Two-Dimensional Partial Differential Equations on Heterogeneous Architectures,” Mathematical and Computational Applications, vol. 26, no. 3, p. 52, Jul. 2021, doi: 10.3390/mca26030052. [Online]. Available at: https://doi.org/10.3390/mca26030052

S.-L. Wu and T. Zhou, “Parallel implementation for the two-stage SDIRK methods via diagonalization,” Journal of Computational Physics, vol. 428, p. 110076, Mar. 2021, doi: 10.1016/j.jcp.2020.110076. [Online]. Available at: https://doi.org/10.1016/j.jcp.2020.110076

WuEtAl2021b

S. Wu, T. Zhou, and Z. Zhou, “Stability implies robust convergence of a class of preconditioned parallel-in-time iterative algorithms,” arXiv:2102.04646v2 [math.NA], 2021 [Online]. Available at: https://arxiv.org/abs/2102.04646v2
BibTeX entry WuEtAl2021b

@unpublished{WuEtAl2021b, author = {Wu, Shulin and Zhou, Tao and Zhou, Zhi}, howpublished = {arXiv:2102.04646v2 [math.NA]}, title = {Stability implies robust convergence of a class of preconditioned parallel-in-time iterative algorithms}, url = {https://arxiv.org/abs/2102.04646v2}, year = {2021} }
Abstract for BibTeX entry WuEtAl2021b

Solving evolutionary equations in a parallel-in-time manner is an attractive topic and many algorithms are proposed in recent two decades. The algorithm based on the block α-circulant preconditioning technique has shown promising advantages, especially for wave propagation problems. By fast Fourier transform for factorizing the involved circulant matrices, the preconditioned iteration can be computed efficiently via the so-called diagonalization technique, which yields a direct parallel implementation across all time levels. In recent years, considerable efforts have been devoted to exploring the convergence of the preconditioned iteration by studying the spectral radius of the iteration matrix, and this leads to many case-by-case studies depending on the used time-integrator. In this paper, we propose a unified convergence analysis for the algorithm applied to u’+Au=f, where σ(A)⊂\mathbbC^+ with σ(A) being the spectrum of A∈\mathbbC^m\times m. For any one-step method (such as the Runge-Kutta methods) with stability function \mathcalR(z), we prove that the decay rate of the global error is bounded by α/(1-α), provided the method is stable, i.e., \max_λ∈σ(A)|\mathcalR(∆tλ)|\leq1. For any linear multistep method, such a bound becomes cα/(1-cα), where c\geq1 is a constant specified by the multistep method itself. Our proof only relies on the stability of the time-integrator and the estimate is independent of the step size ∆t and the spectrum σ(A).

S. Wu and Z. Zhou, “A Parallel-in-Time Algorithm for High-Order BDF Methods for Diffusion and Subdiffusion Equations,” vol. 43, no. 6, pp. A3627–A3656, Jan. 2021, doi: 10.1137/20m1355690. [Online]. Available at: https://doi.org/10.1137/20m1355690

D. Xue, Y. Hou, and Y. Li, “Analysis of the local and parallel space-time algorithm for the heat equation,” Computers & Mathematics with Applications, vol. 100, pp. 167–181, Oct. 2021, doi: 10.1016/j.camwa.2021.09.008. [Online]. Available at: https://doi.org/10.1016/j.camwa.2021.09.008

X. Yue, K. Pan, J. Zhou, Z. Weng, S. Shu, and J. Tang, “A multigrid-reduction-in-time solver with a new two-level convergence for unsteady fractional Laplacian problems,” Computers & Mathematics with Applications, vol. 89, pp. 57–67, May 2021, doi: 10.1016/j.camwa.2021.02.020. [Online]. Available at: https://doi.org/10.1016/j.camwa.2021.02.020

Y. Zeng, Y. Duan, and B.-S. Liu, “Solving 2D parabolic equations by using time parareal coupling with meshless collocation RBFs methods,” Engineering Analysis with Boundary Elements, vol. 127, pp. 102–112, Jun. 2021, doi: 10.1016/j.enganabound.2021.03.008. [Online]. Available at: https://doi.org/10.1016/j.enganabound.2021.03.008

Y.-L. Zhao, X.-M. Gu, and A. Ostermann, “A Preconditioning Technique for an All-at-once System from Volterra Subdiffusion Equations with Graded Time Steps,” Journal of Scientific Computing, vol. 88, no. 1, May 2021, doi: 10.1007/s10915-021-01527-7. [Online]. Available at: https://doi.org/10.1007/s10915-021-01527-7

ZhaoEtAl2021b

Y.-L. Zhao, J. Wu, and X.-M. Gu, “On the bilateral preconditioning for a L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations,” arXiv:2109.06510v1 [math.NA], 2021 [Online]. Available at: http://arxiv.org/abs/2109.06510v1
BibTeX entry ZhaoEtAl2021b

@unpublished{ZhaoEtAl2021b, author = {Zhao, Yong-Liang and Wu, Jing and Gu, Xian-Ming}, howpublished = {arXiv:2109.06510v1 [math.NA]}, title = {On the bilateral preconditioning for a L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations}, url = {http://arxiv.org/abs/2109.06510v1}, year = {2021} }
Abstract for BibTeX entry ZhaoEtAl2021b

Time-space fractional Bloch-Torrey equations are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order scheme for this equation by employing the recently proposed L2-type formula [A. A. Alikhanov, C. Huang, Appl. Math. Comput. (2021) 126545]. Then, we prove the stability and the convergence of this scheme. Based on such the numerical scheme, a L2-type all-at-once system is derived. In order to solve this system in a parallel-in-time pattern, a bilateral preconditioning technique is designed according to the special structure of the system. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant for the time fractional order α∈(0,0.3624). Numerical results are reported to show the efficiency of our method.

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2020

AgbohEtAl2020

W. Agboh, O. Grainger, D. Ruprecht, and M. Dogar, “Parareal with a Learned Coarse Model for Robotic Manipulation,” Computing and Visualization in Science, vol. 23, no. 8, 2020 [Online]. Available at: https://doi.org/10.1007/s00791-020-00327-0
BibTeX entry AgbohEtAl2020

@article{AgbohEtAl2020, author = {Agboh, Wisdom and Grainger, Oliver and Ruprecht, Daniel and Dogar, Mehmet}, journal = {Computing and Visualization in Science}, number = {8}, title = {Parareal with a Learned Coarse Model for Robotic Manipulation}, url = {https://doi.org/10.1007/s00791-020-00327-0}, volume = {23}, year = {2020} }
Abstract for BibTeX entry AgbohEtAl2020

A key component of many robotics model-based planning and control algorithms is physics predictions, that is, forecasting a sequence of states given an initial state and a sequence of controls. This process is slow and a major computational bottleneck for robotics planning algorithms. Parallel-in-time integration methods can help to leverage parallel computing to accelerate physics predictions and thus planning. The Parareal algorithm iterates between a coarse serial integrator and a fine parallel integrator. A key challenge is to devise a coarse level model that is computationally cheap but accurate enough for Parareal to converge quickly. Here, we investigate the use of a deep neural network physics model as a coarse model for Parareal in the context of robotic manipulation. In simulated experiments using the physics engine Mujoco as fine propagator we show that the learned coarse model leads to faster Parareal convergence than a coarse physics-based model. We further show that the learned coarse model allows to apply Parareal to scenarios with multiple objects, where the physics-based coarse model is not applicable. Finally, We conduct experiments on a real robot and show that Parareal predictions are close to real-world physics predictions for robotic pushing of multiple objects. Some real robot manipulation plans using Parareal can be found at https://www.youtube.com/watch?v=wCh2o1rf-gA .

D. Bast, I. Kulchytska-Ruchka, S. Schoeps, and O. Rain, “Accelerated Steady-State Torque Computation for Induction Machines Using Parallel-In-Time Algorithms,” IEEE Transactions on Magnetics, pp. 1–1, 2020, doi: 10.1109/tmag.2019.2945510. [Online]. Available at: https://doi.org/10.1109/tmag.2019.2945510

C.-E. Brehier and X. Wang, “On Parareal Algorithms for Semilinear Parabolic Stochastic PDEs,” SIAM Journal on Numerical Analysis, vol. 58, no. 1, pp. 254–278, Jan. 2020, doi: 10.1137/19m1251011. [Online]. Available at: https://doi.org/10.1137/19m1251011

Buvoli2020

T. Buvoli, “Exponential Polynomial Time Integrators,” arXiv:2011.00670v1 [math.NA], 2020 [Online]. Available at: http://arxiv.org/abs/2011.00670v1
BibTeX entry Buvoli2020

@unpublished{Buvoli2020, author = {Buvoli, Tommaso}, howpublished = {arXiv:2011.00670v1 [math.NA]}, title = {Exponential Polynomial Time Integrators}, url = {http://arxiv.org/abs/2011.00670v1}, year = {2020} }
Abstract for BibTeX entry Buvoli2020

In this paper we extend the polynomial time integration framework to include exponential integration for both partitioned and unpartitioned initial value problems. We then demonstrate its utility by constructing a new class of parallel exponential block methods based on the Legendre points. These new integrators can be constructed at arbitrary orders of accuracy, have improved stability compared to existing polynomial based exponential linear multistep methods, and can offer significant computational savings compared to current state-of-the-art methods.
BuvoliEtAl2020

T. Buvoli and M. L. Minion, “IMEX Parareal Integrators,” arXiv:2011.01604v1 [math.NA], 2020 [Online]. Available at: http://arxiv.org/abs/2011.01604v1
BibTeX entry BuvoliEtAl2020

@unpublished{BuvoliEtAl2020, author = {Buvoli, Tommaso and Minion, Michael L.}, howpublished = {arXiv:2011.01604v1 [math.NA]}, title = {IMEX Parareal Integrators}, url = {http://arxiv.org/abs/2011.01604v1}, year = {2020} }
Abstract for BibTeX entry BuvoliEtAl2020

Parareal is a widely studied parallel-in-time method that can achieve meaningful speedup on certain problems. However, it is well known that the method typically performs poorly on dispersive equations. This paper analyzes linear stability and convergence for IMEX Runge-Kutta parareal methods on dispersive equations. By combining standard linear stability analysis with a simple convergence analysis, we find that certain parareal configurations can achieve parallel speedup on dispersive equations. These stable configurations all posses low iteration counts, large block sizes, and a large number of processors.

T. Cheng, T. Duan, and V. Dinavahi, “Parallel-in-Time Object-Oriented Electromagnetic Transient Simulation of Power Systems,” IEEE Open Access Journal of Power and Energy, pp. 1–1, 2020, doi: 10.1109/oajpe.2020.3012636. [Online]. Available at: https://doi.org/10.1109/oajpe.2020.3012636

C.-K. Cheng, C.-T. Ho, C. Jia, X. Wang, Z. Zen, and X. Zha, “A Parallel-in-Time Circuit Simulator for Power Delivery Networks with Nonlinear Load Models,” in 2020 IEEE 29th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS), 2020, doi: 10.1109/epeps48591.2020.9231406 [Online]. Available at: https://doi.org/10.1109/epeps48591.2020.9231406

J. Christopher, R. D. Falgout, J. B. Schroder, S. M. Guzik, and X. Gao, “A space-time parallel algorithm with adaptive mesh refinement for computational fluid dynamics,” Computing and Visualization in Science, vol. 23, no. 1-4, Sep. 2020, doi: 10.1007/s00791-020-00334-1. [Online]. Available at: https://doi.org/10.1007/s00791-020-00334-1

ClarkeEtAl2020a

A. T. Clarke, C. J. Davies, D. Ruprecht, and S. M. Tobias, “Parallel-in-time integration of Kinematic Dynamos,” Journal of Computational Physics: X, vol. 7, p. 100057, 2020, doi: 10.1016/j.jcpx.2020.100057. [Online]. Available at: https://doi.org/10.1016/j.jcpx.2020.100057
BibTeX entry ClarkeEtAl2020a

@article{ClarkeEtAl2020a, author = {Clarke, Andrew T. and Davies, Christopher J. and Ruprecht, Daniel and Tobias, Steven M.}, doi = {10.1016/j.jcpx.2020.100057}, journal = {Journal of Computational Physics: X}, pages = {100057}, title = {Parallel-in-time integration of Kinematic Dynamos}, url = {https://doi.org/10.1016/j.jcpx.2020.100057}, volume = {7}, year = {2020} }
Abstract for BibTeX entry ClarkeEtAl2020a

The precise mechanisms responsible for the natural dynamos in the Earth and Sun are still not fully understood. Numerical simulations of natural dynamos are extremely computationally intensive, and are carried out in parameter regimes many orders of magnitude away from real conditions. Parallelization in space is a common strategy to speed up simulations on high performance computers, but eventually hits a scaling limit. Additional directions of parallelization are desirable to utilise the high number of processor cores now available. Parallel-in-time methods can deliver speed up in addition to that offered by spatial partitioning but have not yet been applied to dynamo simulations. This paper investigates the feasibility of using the parallel-in-time algorithm Parareal to speed up initial value problem simulations of the kinematic dynamo, using the open source Dedalus spectral solver. Both the time independent Roberts and time dependent Galloway-Proctor 2.5D dynamos are investigated over a range of magnetic Reynolds numbers. Speedups beyond those possible from spatial parallelisation are found in both cases. Results for the Galloway-Proctor flow are promising, with Parareal efficiency found to be close to 0.3. Roberts flow results are less efficient, but Parareal still shows some speed up over spatial parallelisation alone. Parallel in space and time speed ups of ∼300 were found for 1600 cores for the Galloway-Proctor flow, with total parallel efficiency of ∼0.16.
ClarkeEtAl2020b

A. Clarke, C. Davies, D. Ruprecht, S. Tobias, and J. S. Oishi, “Performance of parallel-in-time integration for Rayleigh Bénard Convection,” Computing and Visualization in Science, vol. 23, no. 10, 2020 [Online]. Available at: https://doi.org/10.1007/s00791-020-00332-3
BibTeX entry ClarkeEtAl2020b

@article{ClarkeEtAl2020b, author = {Clarke, Andrew and Davies, Chris and Ruprecht, Daniel and Tobias, Steven and Oishi, Jeffrey S.}, journal = {Computing and Visualization in Science}, number = {10}, title = {Performance of parallel-in-time integration for {R}ayleigh {B}énard Convection}, url = {https://doi.org/10.1007/s00791-020-00332-3}, volume = {23}, year = {2020} }
Abstract for BibTeX entry ClarkeEtAl2020b

Rayleigh-Bénard convection (RBC) is a fundamental problem of fluid dynamics, with many applications to geophysical, astrophysical, and industrial flows. Understanding RBC at parameter regimes of interest requires complex physical or numerical experiments. Numerical simulations require large amounts of computational resources; in order to more efficiently use the large numbers of processors now available in large high performance computing clusters, novel parallelisation strategies are required. To this end, we investigate the performance of the parallel-in-time algorithm Parareal when used in numerical simulations of RBC. We present the first parallel-in-time speedups for RBC simulations at finite Prandtl number. We also investigate the problem of convergence of Parareal with respect to to statistical numerical quantities, such as the Nusselt number, and discuss the importance of reliable online stopping criteria in these cases.

L. D’Amore and R. Cacciapuoti, “Model Reduction in Space and Time for the ab initio decomposition of 4D Variational Data Assimilation Problems,” Applied Numerical Mathematics, Oct. 2020, doi: 10.1016/j.apnum.2020.10.003. [Online]. Available at: https://doi.org/10.1016/j.apnum.2020.10.003

FlamantEtAl2020

C. Flamant, P. Protopapas, and D. Sondak, “Solving Differential Equations Using Neural Network Solution Bundles,” arXiv:2006.14372v1 [cs.LG], 2020 [Online]. Available at: http://arxiv.org/abs/2006.14372v1
BibTeX entry FlamantEtAl2020

@unpublished{FlamantEtAl2020, author = {Flamant, Cedric and Protopapas, Pavlos and Sondak, David}, howpublished = {arXiv:2006.14372v1 [cs.LG]}, title = {Solving Differential Equations Using Neural Network Solution Bundles}, url = {http://arxiv.org/abs/2006.14372v1}, year = {2020} }
Abstract for BibTeX entry FlamantEtAl2020

The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution whose values are computed at discrete times. When many varied solutions with different initial conditions to the ODE are required, the computational cost can become significant. We propose that a neural network be used as a solution bundle, a collection of solutions to an ODE for various initial states and system parameters. The neural network solution bundle is trained with an unsupervised loss that does not require any prior knowledge of the sought solutions, and the resulting object is differentiable in initial conditions and system parameters. The solution bundle exhibits fast, parallelizable evaluation of the system state, facilitating the use of Bayesian inference for parameter estimation in real dynamical systems.

M. J. Gander and T. Lunet, “ParaStieltjes: Parallel computation of Gauss quadrature rules using a Parareal-like approach for the Stieltjes procedure,” Numerical Linear Algebra with Applications, Jun. 2020, doi: 10.1002/nla.2314. [Online]. Available at: https://doi.org/10.1002/nla.2314

M. J. Gander, F. Kwok, and J. Salomon, “PARAOPT: A Parareal Algorithm for Optimality Systems,” SIAM Journal on Scientific Computing, vol. 42, no. 5, pp. A2773–A2802, Jan. 2020, doi: 10.1137/19m1292291. [Online]. Available at: https://doi.org/10.1137/19m1292291

M. J. Gander and S.-L. Wu, “A Diagonalization-Based Parareal Algorithm for Dissipative and Wave Propagation Problems,” SIAM Journal on Numerical Analysis, vol. 58, no. 5, pp. 2981–3009, Jan. 2020, doi: 10.1137/19m1271683. [Online]. Available at: https://doi.org/10.1137/19m1271683

M. J. Gander, I. Kulchytska-Ruchka, and S. Schöps, “A New Parareal Algorithm for Time-Periodic Problems with Discontinuous Inputs,” in Lecture Notes in Computational Science and Engineering, Springer International Publishing, 2020, pp. 243–250 [Online]. Available at: https://doi.org/10.1007/978-3-030-56750-7_27

I. C. Garcia, I. Kulchytska-Ruchka, M. Clemens, and S. Schops, “Parallel-in-Time Solution of Eddy Current Problems Using Implicit and Explicit Time-stepping Methods,” in 2020 IEEE 19th Biennial Conference on Electromagnetic Field Computation (CEFC), 2020, doi: 10.1109/cefc46938.2020.9451465 [Online]. Available at: https://doi.org/10.1109%2Fcefc46938.2020.9451465

A. Garmon and D. Perez, “Exploiting Model Uncertainty to Improve the Scalability of Long-Time Simulations using Parallel Trajectory Splicing,” Modelling and Simulation in Materials Science and Engineering, Jul. 2020, doi: 10.1088/1361-651x/aba511. [Online]. Available at: https://doi.org/10.1088/1361-651x/aba511

X.-M. Gu and S.-L. Wu, “A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel,” Journal of Computational Physics, vol. 417, p. 109576, Sep. 2020, doi: 10.1016/j.jcp.2020.109576. [Online]. Available at: https://doi.org/10.1016/j.jcp.2020.109576

S. Günther, L. Ruthotto, J. B. Schroder, E. C. Cyr, and N. R. Gauger, “Layer-Parallel Training of Deep Residual Neural Networks,” SIAM Journal on Mathematics of Data Science, vol. 2, no. 1, pp. 1–23, Jan. 2020, doi: 10.1137/19m1247620. [Online]. Available at: https://doi.org/10.1137/19m1247620

HahneEtAl2020

J. Hahne, S. Friedhoff, and M. Bolten, “PyMGRIT: A Python Package for the parallel-in-time method MGRIT,” arXiv:2008.05172v1 [cs.MS], 2020 [Online]. Available at: http://arxiv.org/abs/2008.05172v1
BibTeX entry HahneEtAl2020

@unpublished{HahneEtAl2020, author = {Hahne, Jens and Friedhoff, Stephanie and Bolten, Matthias}, howpublished = {arXiv:2008.05172v1 [cs.MS]}, title = {PyMGRIT: A Python Package for the parallel-in-time method MGRIT}, url = {http://arxiv.org/abs/2008.05172v1}, year = {2020} }
Abstract for BibTeX entry HahneEtAl2020

In this paper, we introduce the Python framework PyMGRIT, which implements the multigrid-reduction-in-time (MGRIT) algorithm for solving the (non-)linear systems arising from the discretization of time-dependent problems. The MGRIT algorithm is a reduction-based iterative method that allows parallel-in-time simulations, i. e., calculating multiple time steps simultaneously in a simulation, by using a time-grid hierarchy. The PyMGRIT framework features many different variants of the MGRIT algorithm, ranging from different multigrid cycle types and relaxation schemes, as well as various coarsening strategies, including time-only and space-time coarsening, to using different time integrators on different levels in the multigrid hierachy. PyMGRIT allows serial runs for prototyping and testing of new approaches, as well as parallel runs using the Message Passing Interface (MPI). Here, we describe the implementation of the MGRIT algorithm in PyMGRIT and present the usage from both user and developer point of views. Three examples illustrate different aspects of the package, including pure time parallelism as well as space-time parallelism by coupling PyMGRIT with PETSc or Firedrake, which enable spatial parallelism through MPI.

F. P. Hamon, M. Schreiber, and M. L. Minion, “Parallel-in-time multi-level integration of the shallow-water equations on the rotating sphere,” Journal of Computational Physics, vol. 407, p. 109210, Apr. 2020, doi: 10.1016/j.jcp.2019.109210. [Online]. Available at: https://doi.org/10.1016/j.jcp.2019.109210

A. Hessenthaler, B. S. Southworth, D. Nordsletten, O. Röhrle, R. D. Falgout, and J. B. Schroder, “Multilevel Convergence Analysis of Multigrid-Reduction-in-Time,” SIAM Journal on Scientific Computing, vol. 42, no. 2, pp. A771–A796, Jan. 2020, doi: 10.1137/19m1238812. [Online]. Available at: https://doi.org/10.1137/19m1238812

N. E. Hodge, “Towards Improved Speed and Accuracy of Laser Powder Bed FusionSimulations via Representation of Multiple Time Scales,” Additive Manufacturing, p. 101600, Oct. 2020, doi: 10.1016/j.addma.2020.101600. [Online]. Available at: https://doi.org/10.1016/j.addma.2020.101600

X. Hu, C. Rodrigo, and F. J. Gaspar, “Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation,” Journal of Computational Physics, p. 109540, May 2020, doi: 10.1016/j.jcp.2020.109540. [Online]. Available at: https://doi.org/10.1016/j.jcp.2020.109540

K. Jałowiecki, A. Więckowski, P. Gawron, and B. Gardas, “Parallel in time dynamics with quantum annealers,” Scientific Reports, vol. 10, no. 1, Aug. 2020, doi: 10.1038/s41598-020-70017-x. [Online]. Available at: https://doi.org/10.1038/s41598-020-70017-x

A. Kirby, S. Samsi, M. Jones, A. Reuther, J. Kepner, and V. Gadepally, “Layer-Parallel Training with GPU Concurrency of Deep Residual Neural Networks via Nonlinear Multigrid,” in 2020 IEEE High Performance Extreme Computing Conference (HPEC), 2020, doi: 10.1109/hpec43674.2020.9286180 [Online]. Available at: https://doi.org/10.1109/hpec43674.2020.9286180

S. Lakshmiranganatha and S. S. Muknahallipatna, “Graphical Processing Unit Based Time-Parallel Numerical Method for Ordinary Differential Equations,” Journal of Computer and Communications, vol. 08, no. 02, pp. 39–63, 2020, doi: 10.4236/jcc.2020.82004. [Online]. Available at: https://doi.org/10.4236/jcc.2020.82004

F. Legoll, T. Lelièvre, K. Myerscough, and G. Samaey, “Parareal computation of stochastic differential equations with time-scale separation: a numerical convergence study,” Computing and Visualization in Science, vol. 23, no. 1-4, Sep. 2020, doi: 10.1007/s00791-020-00329-y. [Online]. Available at: https://doi.org/10.1007/s00791-020-00329-y

H. Liu, A. Cheng, and H. Wang, “A Parareal Finite Volume Method for Variable-Order Time-Fractional Diffusion Equations,” Journal of Scientific Computing, vol. 85, no. 1, Oct. 2020, doi: 10.1007/s10915-020-01321-x. [Online]. Available at: https://doi.org/10.1007/s10915-020-01321-x

LiuEtAl2020b

J. Liu and Z. Wang, “A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems from evolutionary PDEs,” arXiv:2012.09148v1 [math.NA], 2020 [Online]. Available at: http://arxiv.org/abs/2012.09148v1
BibTeX entry LiuEtAl2020b

@unpublished{LiuEtAl2020b, author = {Liu, Jun and Wang, Zhu}, howpublished = {arXiv:2012.09148v1 [math.NA]}, title = {A ROM-accelerated parallel-in-time preconditioner for solving all-at-once systems from evolutionary PDEs}, url = {http://arxiv.org/abs/2012.09148v1}, year = {2020} }
Abstract for BibTeX entry LiuEtAl2020b

In this paper we propose to use model reduction techniques for speeding up the diagonalization-based parallel-in-time (ParaDIAG) preconditioner, for iteratively solving all-at-once systems from evolutionary PDEs. In particular, we use the reduced basis method to seek a low-dimensional approximation to the sequence of complex-shifted systems arising from Step-(b) of the ParaDIAG preconditioning procedure. Different from the standard reduced order modeling that uses the separation of offline and online stages, we have to build the reduced order model (ROM) online for the considered systems at each iteration. Therefore, several heuristic acceleration techniques are introduced in the greedy basis generation algorithm, that is built upon a residual-based error indicator, to further boost up its computational efficiency. Several numerical experiments are conducted, which illustrate the favorable computational efficiency of our proposed ROM-accelerated ParaDIAG preconditioner, in comparison with the state of the art multigrid-based ParaDIAG preconditioner.

J. Liu and S.-L. Wu, “A Fast Block \textdollar}alpha\textdollar-Circulant Preconditoner for All-at-Once Systems From Wave Equations,” SIAM Journal on Matrix Analysis and Applications, vol. 41, no. 4, pp. 1912–1943, Jan. 2020, doi: 10.1137/19m1309869. [Online]. Available at: https://doi.org/10.1137/19m1309869

E. Lorin, “Derivation and analysis of parallel-in-time neural ordinary differential equations,” Annals of Mathematics and Artificial Intelligence, Jul. 2020, doi: 10.1007/s10472-020-09702-6. [Online]. Available at: https://doi.org/10.1007/s10472-020-09702-6

Y. Maday and O. Mula, “An adaptive parareal algorithm,” Journal of Computational and Applied Mathematics, vol. 377, p. 112915, Oct. 2020, doi: 10.1016/j.cam.2020.112915. [Online]. Available at: https://doi.org/10.1016/j.cam.2020.112915

X. Meng, Z. Li, D. Zhang, and G. E. Karniadakis, “PPINN: Parareal physics-informed neural network for time-dependent PDEs,” Computer Methods in Applied Mechanics and Engineering, vol. 370, p. 113250, Oct. 2020, doi: 10.1016/j.cma.2020.113250. [Online]. Available at: https://doi.org/10.1016/j.cma.2020.113250

NguyenTsai2020

H. Nguyen and R. Tsai, “A stable parareal-like method for the second order wave equation,” Journal of Computational Physics, vol. 405, p. 109156, 2020, doi: https://doi.org/10.1016/j.jcp.2019.109156. [Online]. Available at: http://www.sciencedirect.com/science/article/pii/S0021999119308617
BibTeX entry NguyenTsai2020

@article{NguyenTsai2020, author = {Nguyen, Hieu and Tsai, Richard}, doi = {https://doi.org/10.1016/j.jcp.2019.109156}, issn = {0021-9991}, journal = {Journal of Computational Physics}, keywords = {Parallel-in-time, Wave equation, Procrustes problem}, pages = {109156}, title = {A stable parareal-like method for the second order wave equation}, url = {http://www.sciencedirect.com/science/article/pii/S0021999119308617}, volume = {405}, year = {2020} }
Abstract for BibTeX entry NguyenTsai2020

A new parallel-in-time iterative method is proposed for solving the homogeneous second-order wave equation. The new method involves a coarse scale propagator, allowing for larger time steps, and a fine scale propagator which fully resolves the medium using finer spatial grid and uses shorter time steps. The fine scale propagator is run in parallel for short time intervals. The two propagators are coupled in an iterative way that resembles the standard parareal method [24]. We present a data-driven strategy in which the computed data gathered from each iteration are re-used to stabilize the coupling by minimizing the wave energy residual of the fine and coarse propagated solutions. Several examples, including a wave speed with discontinuities, are provided to demonstrate the effectiveness of the proposed method.

B. W. Ong and J. B. Schroder, “Applications of time parallelization,” Computing and Visualization in Science, vol. 23, no. 1-4, Sep. 2020, doi: 10.1007/s00791-020-00331-4. [Online]. Available at: https://doi.org/10.1007/s00791-020-00331-4

B. Park et al., “Performance and Feature Improvements in Parareal-based Power System Dynamic Simulation,” in 2020 IEEE International Conference on Power Systems Technology (POWERCON), 2020, doi: 10.1109/powercon48463.2020.9230544 [Online]. Available at: https://doi.org/10.1109/powercon48463.2020.9230544

H. Rittich and R. Speck, “Time-parallel simulation of the Schrödinger Equation,” Computer Physics Communications, vol. 255, p. 107363, Oct. 2020, doi: 10.1016/j.cpc.2020.107363. [Online]. Available at: https://doi.org/10.1016/j.cpc.2020.107363

R. Schöbel and R. Speck, “PFASST-ER: combining the parallel full approximation scheme in space and time with parallelization across the method,” Computing and Visualization in Science, vol. 23, no. 1-4, Sep. 2020, doi: 10.1007/s00791-020-00330-5. [Online]. Available at: https://doi.org/10.1007/s00791-020-00330-5

L. Z. sci, “Convergence Analysis of Parareal Algorithm Based on Milstein Scheme for Stochastic Differential Equations,” Journal of Computational Mathematics, vol. 38, no. 3, pp. 487–501, Jun. 2020, doi: 10.4208/jcm.1901-m2018-0085. [Online]. Available at: https://doi.org/10.4208/jcm.1901-m2018-0085

B. Song, Y.-L. Jiang, and X. Wang, “Analysis of two new parareal algorithms based on the Dirichlet-Neumann/Neumann-Neumann waveform relaxation method for the heat equation,” Numerical Algorithms, Jun. 2020, doi: 10.1007/s11075-020-00949-y. [Online]. Available at: https://doi.org/10.1007/s11075-020-00949-y

B. Stump and A. Plotkowski, “Spatiotemporal parallelization of an analytical heat conduction model for additive manufacturing via a hybrid OpenMP \mathplus MPI approach,” Computational Materials Science, vol. 184, p. 109861, Nov. 2020, doi: 10.1016/j.commatsci.2020.109861. [Online]. Available at: https://doi.org/10.1016/j.commatsci.2020.109861

S.-L. Wu and J. Liu, “A Parallel-In-Time Block-Circulant Preconditioner for Optimal Control of Wave Equations,” SIAM Journal on Scientific Computing, vol. 42, no. 3, pp. A1510–A1540, Jan. 2020, doi: 10.1137/19m1289613. [Online]. Available at: https://doi.org/10.1137/19m1289613

WuEtAl2020b

S. Wu and Z. Zhou, “Parallel-in-time high-order BDF schemes for diffusion and subdiffusion equations,” arXiv:2007.13125v1 [math.NA], 2020 [Online]. Available at: http://arxiv.org/abs/2007.13125v1
BibTeX entry WuEtAl2020b

@unpublished{WuEtAl2020b, author = {Wu, Shuonan and Zhou, Zhi}, howpublished = {arXiv:2007.13125v1 [math.NA]}, title = {Parallel-in-time high-order BDF schemes for diffusion and subdiffusion equations}, url = {http://arxiv.org/abs/2007.13125v1}, year = {2020} }
Abstract for BibTeX entry WuEtAl2020b

In this paper, we propose a parallel-in-time algorithm for approximately solving parabolic equations. In particular, we apply the k-step backward differentiation formula, and then develop an iterative solver by using the waveform relaxation technique. Each resulting iterate represents a periodic-like system, which could be further solved in parallel by using the diagonalization technique. The convergence of the waveform relaxation iteration is theoretically examined by using the generating function method. The approach we established in this paper extends the existing argument of single-step methods in Gander and Wu [Numer. Math., 143 (2019), pp. 489–527] to general BDF methods up to order 6. The argument could be further applied to the time-fractional subdiffusion equation, whose discretization shares common properties of the standard BDF methods, because of the nonlocality of the fractional differential operator. Illustrative numerical results are presented to complement the theoretical analysis.

S.-L. Wu and T. Zhou, “Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 26, p. 88, 2020, doi: 10.1051/cocv/2020012. [Online]. Available at: https://doi.org/10.1051/cocv/2020012

Y.-L. Zhao, X.-M. Gu, M. Li, and H.-Y. Jian, “Preconditioners for all-at-once system from the fractional mobile/immobile advection–diffusion model,” Journal of Applied Mathematics and Computing, Jul. 2020, doi: 10.1007/s12190-020-01410-y. [Online]. Available at: https://doi.org/10.1007/s12190-020-01410-y

ZhaoEtAl2020b

Y.-L. Zhao, X.-M. Gu, and A. Ostermann, “A parallel preconditioning technique for an all-at-once system from subdiffusion equations with variable time steps,” arXiv:2007.14636v1 [math.NA], 2020 [Online]. Available at: http://arxiv.org/abs/2007.14636v1
BibTeX entry ZhaoEtAl2020b

@unpublished{ZhaoEtAl2020b, author = {Zhao, Yong-Liang and Gu, Xian-Ming and Ostermann, Alexander}, howpublished = {arXiv:2007.14636v1 [math.NA]}, title = {A parallel preconditioning technique for an all-at-once system from subdiffusion equations with variable time steps}, url = {http://arxiv.org/abs/2007.14636v1}, year = {2020} }
Abstract for BibTeX entry ZhaoEtAl2020b

Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well.The graded L1 scheme is often chosen to discretize such problems since it can handle the singularity of the solution near t = 0. In this paper, we propose a modification. We first split the time interval [0, T] into [0, T_0] and [T_0, T], where T_0 (0 < T_0 < T) is reasonably small. Then, the graded L1 scheme is applied in [0, T_0], while the uniform one is used in [T_0, T]. Our all-at-once system is derived based on this strategy. In order to solve the arising system efficiently, we split it into two subproblems and design two preconditioners. Some properties of these two preconditioners are also investigated. Moreover, we extend our method to solve semilinear subdiffusion problems. Numerical results are reported to show the efficiency of our method.

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2019

BlumersEtAl2019

A. L. Blumers, Z. Li, and G. E. Karniadakis, “Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics,” Journal of Computational Physics, vol. 393, pp. 214–228, 2019, doi: 10.1016/j.jcp.2019.05.016. [Online]. Available at: https://doi.org/10.1016/j.jcp.2019.05.016
BibTeX entry BlumersEtAl2019

@article{BlumersEtAl2019, author = {Blumers, Ansel L. and Li, Zhen and Karniadakis, George Em}, doi = {10.1016/j.jcp.2019.05.016}, journal = {Journal of Computational Physics}, pages = {214 - 228}, title = {Supervised parallel-in-time algorithm for long-time Lagrangian simulations of stochastic dynamics: Application to hydrodynamics}, url = {https://doi.org/10.1016/j.jcp.2019.05.016}, volume = {393}, year = {2019} }
Abstract for BibTeX entry BlumersEtAl2019

Lagrangian particle methods based on detailed atomic and molecular models are powerful computational tools for studying the dynamics of microscale and nanoscale systems. However, the maximum time step is limited by the smallest oscillation period of the fastest atomic motion, rendering long-time simulations very expensive. To resolve this bottleneck, we propose a supervised parallel-in-time algorithm for stochastic dynamics (SPASD) to accelerate long-time Lagrangian particle simulations. Our method is inspired by bottom-up coarse-graining projections that yield mean-field hydrodynamic behavior in the continuum limit. Here as an example, we use the dissipative particle dynamics (DPD) as the Lagrangian particle simulator that is supervised by its macroscopic counterpart, i.e., the Navier-Stokes simulator. The low-dimensional macroscopic system (here, the Navier-Stokes solver) serves as a predictor to supervise the high-dimensional Lagrangian simulator, in a predictor-corrector type algorithm. The results of the Lagrangian simulation then correct the mean-field prediction and provide the proper microscopic details (e.g., consistent fluctuations, correlations, etc.). The unique feature that sets SPASD apart from other multiscale methods is the use of a low-fidelity macroscopic model as a predictor. The macro-model can be approximate and even inconsistent with the microscale description, but SPASD anticipates the deviation and corrects it internally to recover the true dynamics. We first present the algorithm and analyze its theoretical speedup, and subsequently we present the accuracy and convergence of the algorithm for the time-dependent plane Poiseuille flow, demonstrating that SPASD converges exponentially fast over iterations, irrespective of the accuracy of the predictor. Moreover, the fluctuating characteristics of the stochastic dynamics are identical to the unsupervised (serial in time) DPD simulation. We also compare the performance of SPASD to the conventional spatial decomposition method, which is one of the most parallel-efficient methods for particle simulations. We find that the parallel efficiency of SPASD and the conventional spatial decomposition method are similar for a small number of computing cores, but for a large number of cores the performance of SPASD is superior. Furthermore, SPASD can be used in conjunction with spatial decomposition for enhanced performance. Lastly, we simulate a two-dimensional cavity flow that requires more iterations to converge compared to the Poiseuille flow, and we observe that SPASD converges to the correct solution. Although a DPD solver is used to demonstrate the results, SPASD is a general framework and can be readily applied to other Lagrangian approaches including molecular dynamics and Langevin dynamics.

K. Carlberg, L. Brencher, B. Haasdonk, and A. Barth, “Data-Driven Time Parallelism via Forecasting,” SIAM Journal on Scientific Computing, vol. 41, no. 3, pp. B466–B496, Jan. 2019, doi: 10.1137/18m1174362. [Online]. Available at: https://doi.org/10.1137/18m1174362

DohrEtAl2019

S. Dohr, J. Zapletal, G. Of, M. Merta, and M. Kravčenko, “A parallel space–time boundary element method for the heat equation,” Computers & Mathematics with Applications, 2019, doi: https://doi.org/10.1016/j.camwa.2018.12.031. [Online]. Available at: http://www.sciencedirect.com/science/article/pii/S0898122118307296
BibTeX entry DohrEtAl2019

@article{DohrEtAl2019, author = {Dohr, Stefan and Zapletal, Jan and Of, Günther and Merta, Michal and Kravčenko, Michal}, doi = {https://doi.org/10.1016/j.camwa.2018.12.031}, journal = {Computers \& Mathematics with Applications}, title = {A parallel space–time boundary element method for the heat equation}, url = {http://www.sciencedirect.com/science/article/pii/S0898122118307296}, year = {2019} }
Abstract for BibTeX entry DohrEtAl2019

In this paper we introduce a new parallel solver for the weakly singular space–time boundary integral equation for the heat equation. The space–time boundary mesh is decomposed into a given number of submeshes. Pairs of the submeshes represent dense blocks in the system matrices, which are distributed among computational nodes by an algorithm based on a cyclic decomposition of complete graphs ensuring load balance. In addition, we employ vectorization and threading in shared memory to ensure intra-node efficiency. We present scalability experiments on different CPU architectures to evaluate the performance of the proposed parallelization techniques. All levels of parallelism allow us to tackle large problems and lead to an almost optimal speedup.

S. Friedhoff, J. Hahne, I. Kulchytska-Ruchka, and S. Schöps, “Exploring Parallel-in-Time Approaches for Eddy Current Problems,” in Progress in Industrial Mathematics at ECMI 2018, Springer International Publishing, 2019, pp. 373–379 [Online]. Available at: https://doi.org/10.1007/978-3-030-27550-1_47

S. Friedhoff, J. Hahne, and S. Schöps, “Multigrid-reduction-in-time for Eddy Current problems,” PAMM, vol. 19, no. 1, Nov. 2019, doi: 10.1002/pamm.201900262. [Online]. Available at: https://doi.org/10.1002/pamm.201900262

FriedhoffSouthworth2019

S. Friedhoff and B. S. Southworth, “On ‘Optimal’ h-Independent Convergence of Parareal and MGRIT Using Runge-Kutta Time Integration,” arXiv:1906.06672 [math.NA], 2019 [Online]. Available at: https://arxiv.org/abs/1906.06672
BibTeX entry FriedhoffSouthworth2019

@unpublished{FriedhoffSouthworth2019, author = {Friedhoff, Stephanie and Southworth, Ben S.}, howpublished = {arXiv:1906.06672 [math.NA]}, title = {On ``{O}ptimal'' h-{I}ndependent {C}onvergence of {P}arareal and {MGRIT} {U}sing {R}unge-{K}utta {T}ime {I}ntegration}, url = {https://arxiv.org/abs/1906.06672}, year = {2019} }
Abstract for BibTeX entry FriedhoffSouthworth2019

Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as two-level integrators, with fine and coarse schemes, and have been designed to achieve a ‘parallel in real time’ implementation. In this work, the fine integrator is given by the exponential Euler scheme. Two choices for the coarse integrator are considered: the linear implicit Euler scheme, and the exponential Euler scheme. The influence on the performance of the parareal algorithm, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations, is investigated, with theoretical analysis results and with extensive numerical experiments.
GanderEtAl2019

M. Gander, L. Halpern, J. Rannou, and J. Ryan, “A Direct Time Parallel Solver by Diagonalization for the Wave Equation,” SIAM Journal on Scientific Computing, vol. 41, no. 1, pp. A220–A245, 2019, doi: 10.1137/17M1148347. [Online]. Available at: https://doi.org/10.1137/17M1148347
BibTeX entry GanderEtAl2019

@article{GanderEtAl2019, author = {Gander, M. and Halpern, L. and Rannou, J. and Ryan, J.}, doi = {10.1137/17M1148347}, journal = {SIAM Journal on Scientific Computing}, number = {1}, pages = {A220--A245}, title = {A Direct Time Parallel Solver by Diagonalization for the Wave Equation}, url = {https://doi.org/10.1137/17M1148347}, volume = {41}, year = {2019} }
Abstract for BibTeX entry GanderEtAl2019

With the advent of very large scale parallel computers, it has become more and more important to also use the time direction for parallelization when solving evolution problems. While there are many successful algorithms for diffusive problems, only some of them are also effective for hyperbolic problems. We present here a mathematical analysis of a new method based on the diagonalization of the time stepping matrix proposed by Maday and Rønquist in 2007. Like many time-parallelization methods, at first this does not seem to be a very promising approach: the matrix is essentially triangular, or, for equidistant time steps, actually a Jordan block, and thus not diagonalizable. If one chooses however different time steps, diagonalization is possible, and one has to trade off between the accuracy due to necessarily having different time steps, and numerical errors in the diagonalization process of these almost nondiagonalizable matrices. We present for the first time such a diagonalization technique for the Newmark scheme for solving wave equations, and derive a mathematically rigorous optimization strategy for the choice of the parameters in the special case when the Newmark scheme becomes Crank–Nicolson. Our analysis shows that small to medium scale time parallelization is possible with this approach. We illustrate our results with numerical experiments for model wave equations in various dimensions and also an industrial test case for the elasticity equations with variable coefficients.
GanderEtAl2019b

M. Gander, Y. Jiang, and B. Song, “A Superlinear Convergence Estimate for the Parareal Schwarz Waveform Relaxation Algorithm,” SIAM Journal on Scientific Computing, vol. 41, no. 2, pp. A1148–A1169, 2019, doi: 10.1137/18M1177226. [Online]. Available at: https://doi.org/10.1137/18M1177226
BibTeX entry GanderEtAl2019b

@article{GanderEtAl2019b, author = {Gander, M. and Jiang, Y. and Song, B.}, doi = {10.1137/18M1177226}, journal = {SIAM Journal on Scientific Computing}, number = {2}, pages = {A1148--A1169}, title = {A Superlinear Convergence Estimate for the Parareal Schwarz Waveform Relaxation Algorithm}, url = {https://doi.org/10.1137/18M1177226}, volume = {41}, year = {2019} }
Abstract for BibTeX entry GanderEtAl2019b

The parareal Schwarz waveform relaxation algorithm is a new space-time parallel algorithm for the solution of evolution partial differential equations. It is based on a decomposition of the entire space-time domain both in space and in time into smaller space-time subdomains, and then computes by an iteration in parallel on all these small space-time subdomains a better and better approximation of the overall solution in space-time. The initial conditions in the space-time subdomains are updated using a parareal mechanism, while the boundary conditions are updated using Schwarz waveform relaxation techniques. A first precursor of this algorithm was presented 15 years ago, and while the method works well in practice, the convergence of the algorithm is not yet understood, and to analyze it is technically difficult. We present in this paper for the first time an accurate superlinear convergence estimate when the algorithm is applied to the heat equation. We illustrate our analysis with numerical experiments including cases not covered by the analysis, which opens up many further research directions.
GanderEtAl2019c

M. J. Gander, I. Kulchytska-Ruchka, I. Niyonzima, and S. Schöps, “A New Parareal Algorithm for Problems with Discontinuous Sources,” SIAM Journal on Scientific Computing, vol. 41, no. 2, pp. B375–B395, 2019, doi: 10.1137/18M1175653. [Online]. Available at: https://doi.org/10.1137/18M1175653
BibTeX entry GanderEtAl2019c

@article{GanderEtAl2019c, author = {Gander, Martin J. and Kulchytska-Ruchka, Iryna and Niyonzima, Innocent and Schöps, Sebastian}, doi = {10.1137/18M1175653}, journal = {SIAM Journal on Scientific Computing}, number = {2}, pages = {B375--B395}, title = {A New Parareal Algorithm for Problems with Discontinuous Sources}, url = {https://doi.org/10.1137/18M1175653}, volume = {41}, year = {2019} }
Abstract for BibTeX entry GanderEtAl2019c

The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal algorithm for ordinary differential equations which involve discontinuous right-hand sides. Such situations occur in various applications, e.g., when an electric device is supplied with a pulse-width-modulated signal. Our new Parareal algorithm uses a smooth input for the coarse problem with reduced dynamics. We derive error estimates that show how the input reduction influences the overall convergence rate of the algorithm. We support our theoretical results by numerical experiments, and also test our new Parareal algorithm in an eddy current simulation of an induction machine.

M. J. Gander and S.-L. Wu, “Convergence analysis of a periodic-like waveform relaxation method for initial-value problems via the diagonalization technique,” Numerische Mathematik, vol. 143, no. 2, pp. 489–527, Jun. 2019, doi: 10.1007/s00211-019-01060-8. [Online]. Available at: https://doi.org/10.1007/s00211-019-01060-8

S. Götschel and M. L. Minion, “An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs,” SIAM Journal on Scientific Computing, vol. 41, no. 6, pp. C603–C626, Jan. 2019, doi: 10.1137/19m1239313. [Online]. Available at: https://doi.org/10.1137/19m1239313

HedinLelievre2019

F. Hédin and T. Lelièvre, “gen.parRep: A first implementation of the Generalized Parallel Replica dynamics for the long time simulation of metastable biochemical systems,” Computer Physics Communications, 2019, doi: 10.1016/j.cpc.2019.01.005. [Online]. Available at: https://doi.org/10.1016/j.cpc.2019.01.005
BibTeX entry HedinLelievre2019

@article{HedinLelievre2019, author = {Hédin, Florent and Lelièvre, Tony}, doi = {10.1016/j.cpc.2019.01.005}, journal = {Computer Physics Communications}, title = {gen.parRep: A first implementation of the Generalized Parallel Replica dynamics for the long time simulation of metastable biochemical systems}, url = {https://doi.org/10.1016/j.cpc.2019.01.005}, year = {2019} }
Abstract for BibTeX entry HedinLelievre2019

Metastability is one of the major encountered obstacles when performing long molecular dynamics simulations, and many methods were developed to address this challenge. The “Parallel Replica”(ParRep) dynamics is known for allowing to simulate very long trajectories of metastable Langevin dynamics in the materials science community, but it relies on assumptions that can hardly be transposed to the world of biochemical simulations. The later developed “Generalized ParRep” variant solves those issues, but it was not applied to significant systems of interest so far. In this article, we present the program gen.parRep, the first publicly available implementation of the Generalized Parallel Replica method (BSD 3-Clause license), targeting frequently encountered metastable biochemical systems, such as conformational equilibria or dissociation of protein–ligand complexes. It will be shown that the resulting C++ implementation exhibits a strong linear scalability, providing up to 70% of the maximum possible speedup on several hundreds of CPUs.

J. Hong, X. Wang, and L. Zhang, “Parareal Exponential \textdollar}theta\textdollar-Scheme for Longtime Simulation of Stochastic Schrödinger Equations with Weak Damping,” SIAM Journal on Scientific Computing, vol. 41, no. 6, pp. B1155–B1177, Jan. 2019, doi: 10.1137/18m1176749. [Online]. Available at: https://doi.org/10.1137/18m1176749

HowseEtAl2019

A. Howse, H. Sterck, R. Falgout, S. MacLachlan, and J. Schroder, “Parallel-In-Time Multigrid with Adaptive Spatial Coarsening for The Linear Advection and Inviscid Burgers Equations,” SIAM Journal on Scientific Computing, vol. 41, no. 1, pp. A538–A565, 2019, doi: 10.1137/17M1144982. [Online]. Available at: https://dx.doi.org/10.1137/17M1144982
BibTeX entry HowseEtAl2019

@article{HowseEtAl2019, author = {Howse, A. and Sterck, H. and Falgout, R. and MacLachlan, S. and Schroder, J.}, doi = {10.1137/17M1144982}, journal = {SIAM Journal on Scientific Computing}, number = {1}, pages = {A538--A565}, title = {Parallel-In-Time Multigrid with Adaptive Spatial Coarsening for The Linear Advection and Inviscid Burgers Equations}, url = {https://dx.doi.org/10.1137/17M1144982}, volume = {41}, year = {2019} }
Abstract for BibTeX entry HowseEtAl2019

We apply a multigrid reduction-in-time (MGRIT) algorithm to hyperbolic partial differential equations in one spatial dimension. This study is motivated by the observation that sequential time-stepping is a computational bottleneck when attempting to implement highly concurrent algorithms, thus parallel-in-time methods are desirable. MGRIT adds parallelism by using a hierarchy of successively coarser temporal levels to accelerate the solution on the finest level. In the case of explicit time-stepping, spatial coarsening is a suitable approach to ensure that stability conditions are satisfied on all levels, and it may be useful for implicit time-stepping by producing cheaper multigrid cycles. Unfortunately, uniform spatial coarsening results in extremely slow convergence when the wave speed is near zero, even if only locally. We present an adaptive spatial coarsening strategy that addresses this issue for the variable coefficient linear advection equation and the inviscid Burgers equation using first-order explicit or implicit time-stepping methods. Serial numerical results show this method offers significant improvements over uniform coarsening and is convergent for the inviscid Burgers equation with and without shocks. Parallel scaling tests on up to 128K cores indicate that run-time improvements over serial time-stepping strategies are possible when spatial parallelism alone saturates, and that scalability is robust for oscillatory solutions which change on the scale of the grid spacing.
KrzysikEtAl2019

O. A. Krzysik, H. D. Sterck, S. P. MacLachlan, and S. Friedhoff, “On selecting coarse-grid operators for Parareal and MGRIT applied to linear advection,” arXiv:1902.07757 [math.NA], 2019 [Online]. Available at: https://arxiv.org/abs/1902.07757
BibTeX entry KrzysikEtAl2019

@unpublished{KrzysikEtAl2019, author = {Krzysik, Oliver A. and Sterck, Hans De and MacLachlan, Scott P. and Friedhoff, Stephanie}, howpublished = {arXiv:1902.07757 [math.NA]}, title = {On selecting coarse-grid operators for Parareal and MGRIT applied to linear advection}, url = {https://arxiv.org/abs/1902.07757}, year = {2019} }
Abstract for BibTeX entry KrzysikEtAl2019

We consider the parallel time integration of the linear advection equation with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms. Our aim is to develop a better understanding of the convergence behaviour of these algorithms for this problem, which is known to be poor relative to the diffusion equation, its model parabolic counterpart. Using Fourier analysis, we derive new convergence estimates for these algorithms which, in conjunction with existing convergence theory, provide insight into the origins of this poor performance. We then use this theory to explore improved coarse-grid time-stepping operators. For several high-order discretizations of the advection equation, we demonstrate that there exist non-standard coarse-grid time stepping operators that yield significant improvements over the standard choice of rediscretization.
KwokOng2019

F. Kwok and B. Ong, “Schwarz Waveform Relaxation with Adaptive Pipelining,” SIAM Journal on Scientific Computing, vol. 41, no. 1, pp. A339–A364, 2019, doi: 10.1137/17M115311X. [Online]. Available at: https://doi.org/10.1137/17M115311X
BibTeX entry KwokOng2019

@article{KwokOng2019, author = {Kwok, F. and Ong, B.}, doi = {10.1137/17M115311X}, journal = {SIAM Journal on Scientific Computing}, number = {1}, pages = {A339--A364}, title = {Schwarz Waveform Relaxation with Adaptive Pipelining}, url = {https://doi.org/10.1137/17M115311X}, volume = {41}, year = {2019} }
Abstract for BibTeX entry KwokOng2019

Schwarz waveform relaxation (SWR) methods have been developed to solve a wide range of diffusion-dominated and reaction-dominated equations. The appeal of these methods stems primarily from their ability to use nonconforming space-time discretizations; SWR methods are consequently well-adapted for coupling models with highly varying spatial and time scales. The efficacy of SWR methods is questionable, however, since in each iteration, one propagates an error across the entire time interval. In this manuscript, we introduce an adaptive pipeline approach wherein one subdivides the computational domain into space-time blocks, and adaptively selects the waveform iterates which should be updated given a fixed number of computational workers. Our method is complementary to existing space and time parallel methods, and can be used to obtain additional speedup when the saturation point is reached for other types of parallelism. We analyze these waveform relaxation with adaptive pipelining (WRAP) methods to show convergence and the theoretical speedup that can be expected. Numerical experiments on solutions to the linear heat equation, the advection-diffusion equation, and a reaction-diffusion equation illustrate features and efficacy of WRAP methods for various transmission conditions.
LiEtAl2019

S. Li, R. Chen, and X. Shao, “Parallel two-level space–time hybrid Schwarz method for solving linear parabolic equations,” Applied Numerical Mathematics, vol. 139, pp. 120–135, 2019, doi: 10.1016/j.apnum.2019.01.016. [Online]. Available at: https://doi.org/10.1016/j.apnum.2019.01.016
BibTeX entry LiEtAl2019

@article{LiEtAl2019, author = {Li, Shishun and Chen, Rongliang and Shao, Xinping}, doi = {10.1016/j.apnum.2019.01.016}, journal = {Applied Numerical Mathematics}, pages = {120--135}, title = {Parallel two-level space–time hybrid Schwarz method for solving linear parabolic equations}, url = {https://doi.org/10.1016/j.apnum.2019.01.016}, volume = {139}, year = {2019} }
Abstract for BibTeX entry LiEtAl2019

In this paper, we present a two-level space–time hybrid Schwarz preconditioner for GMRES based on the classical hybrid Schwarz method and the second-order backward differentiation formula. In the proposed method, the parabolic equations are solved in parallel on both of the space and time directions. Under some reasonable assumptions, the optimal convergence theory is developed for the proposed space–time method, i.e., the convergence rate is independent of the mesh parameters, the number of subdomains and the window size. Some numerical results are given to confirm the theory very well in terms of the convergence rate and accuracy. And the strong/weak scalability obtained with 4096 processors is also reported to show the efficiency of the proposed method.
LiEtAl2019b

S. Li, X. Shao, and X.-C. Cai, “Highly parallel space-time domain decomposition methods for parabolic problems,” CCF Transactions on High Performance Computing, 2019, doi: 10.1007/s42514-019-00003-x. [Online]. Available at: https://doi.org/10.1007/s42514-019-00003-x
BibTeX entry LiEtAl2019b

@article{LiEtAl2019b, author = {Li, Shishun and Shao, Xinping and Cai, Xiao-Chuan}, doi = {10.1007/s42514-019-00003-x}, journal = {CCF Transactions on High Performance Computing}, title = {Highly parallel space-time domain decomposition methods for parabolic problems}, url = {https://doi.org/10.1007/s42514-019-00003-x}, year = {2019} }
Abstract for BibTeX entry LiEtAl2019b

In the past few years, the number of processor cores of top ranked supercomputers has increased drastically. It is challenging to design efficient parallel algorithms that offer such a high degree of parallelism, especially for certain time-dependent problems because of the sequential nature of “time”. To increase the degree of parallelization, some parallel-in-time algorithms have been developed. In this paper, we give an overview of some recently introduced parallel-in-time methods, and present in detail the class of space-time Schwarz methods, including the standard and the restricted versions, for solving parabolic partial differential equations. Some numerical experiments carried out on a parallel computer with a large number of processor cores for three-dimensional problems are given to show the parallel scalability of the methods. In the end of the paper, we provide a comparison of the parallel-in-time algorithms with a traditional algorithm that is parallelized only in space.
MeleEtAl2019

V. Mele, D. Romano, E. M. Constantinescu, L. Carracciuolo, and L. D’Amore, “Performance Evaluation for a PETSc Parallel-in-Time Solver Based on the MGRIT Algorithm,” in Euro-Par 2018: Parallel Processing Workshops, 2019, pp. 716–728, doi: 10.1002/cpe.4928 [Online]. Available at: https://doi.org/10.1002/cpe.4928
BibTeX entry MeleEtAl2019

@inproceedings{MeleEtAl2019, author = {Mele, Valeria and Romano, Diego and Constantinescu, Emil M. and Carracciuolo, Luisa and D'Amore, Luisa}, booktitle = {Euro-Par 2018: Parallel Processing Workshops}, doi = {10.1002/cpe.4928}, editor = {Mencagli, Gabriele and B. Heras, Dora and Cardellini, Valeria and Casalicchio, Emiliano and Jeannot, Emmanuel and Wolf, Felix and Salis, Antonio and Schifanella, Claudio and Manumachu, Ravi Reddy and Ricci, Laura and Beccuti, Marco and Antonelli, Laura and Garcia Sanchez, Jos{\'e} Daniel and Scott, Stephen L.}, pages = {716--728}, publisher = {Springer International Publishing}, title = {Performance Evaluation for a PETSc Parallel-in-Time Solver Based on the MGRIT Algorithm}, url = {https://doi.org/10.1002/cpe.4928}, year = {2019} }
Abstract for BibTeX entry MeleEtAl2019

We herein describe the performance evaluation of a modular implementation of the MGRIT (MultiGrid-In-Time) algorithm within the context of the PETSc (the Portable, Extensible Toolkit for Scientific computing) library. Our aim is to give the PETSc users the opportunity of testing the MGRIT parallel-in-time approach as an alternative to the Time Stepping integrator (TS), when solving their problems arising from the discretization of linear evolutionary models. To this end, we analyzed the performance parameters of the algorithm in order to underline the relationship between the configuration factors and problem characteristics, intentionally overlooking any accuracy issue and spacial parallelism.
NeumuellerSmears2019

M. Neumüller and I. Smears, “Time-Parallel Iterative Solvers for Parabolic Evolution Equations,” SIAM Journal on Scientific Computing, vol. 41, no. 1, pp. C28–C51, 2019, doi: 10.1137/18M1172466. [Online]. Available at: https://doi.org/10.1137/18M1172466
BibTeX entry NeumuellerSmears2019

@article{NeumuellerSmears2019, author = {Neum{\"u}ller, M. and Smears, I.}, doi = {10.1137/18M1172466}, journal = {SIAM Journal on Scientific Computing}, number = {1}, pages = {C28--C51}, title = {Time-Parallel Iterative Solvers for Parabolic Evolution Equations}, url = {https://doi.org/10.1137/18M1172466}, volume = {41}, year = {2019} }
Abstract for BibTeX entry NeumuellerSmears2019

We present original time-parallel algorithms for the solution of the implicit Euler discretization of general linear parabolic evolution equations with time-dependent self-adjoint spatial operators. Motivated by the inf-sup theory of parabolic problems, we show that the standard nonsymmetric time-global system can be equivalently reformulated as an original symmetric saddle-point system that remains inf-sup stable with respect to the same natural parabolic norms. We then propose and analyse an efficient and readily implementable parallel-in-time preconditioner to be used with an inexact Uzawa method. The proposed preconditioner is non-intrusive and easy to implement in practice, and also features the key theoretical advantages of robust spectral bounds, leading to convergence rates that are independent of the number of time-steps, final time, or spatial mesh sizes, and also a theoretical parallel complexity that grows only logarithmically with respect to the number of time-steps. Numerical experiments with large-scale parallel computations show the effectiveness of the method, along with its good weak and strong scaling properties.

A. G. Peddle, T. Haut, and B. Wingate, “Parareal Convergence for Oscillatory PDEłowercases with Finite Time-Scale Separation,” SIAM Journal on Scientific Computing, vol. 41, no. 6, pp. A3476–A3497, Jan. 2019, doi: 10.1137/17m1131611. [Online]. Available at: https://doi.org/10.1137/17m1131611

Rosa-Raı́ces Jorge L., B. Zhang, and T. F. Miller, “Path-accelerated stochastic molecular dynamics: Parallel-in-time integration using path integrals,” The Journal of Chemical Physics, vol. 151, no. 16, p. 164120, Oct. 2019, doi: 10.1063/1.5125455. [Online]. Available at: https://doi.org/10.1063/1.5125455

SamaddarEtAl2019

D. Samaddar, D. P. Coster, X. Bonnin, L. A. Berry, W. R. Elwasif, and D. B. Batchelor, “Application of the parareal algorithm to simulations of ELMs in ITER plasma,” Computer Physics Communications, vol. 235, pp. 246–257, 2019, doi: 10.1016/j.cpc.2018.08.007. [Online]. Available at: https://doi.org/10.1016/j.cpc.2018.08.007
BibTeX entry SamaddarEtAl2019

@article{SamaddarEtAl2019, author = {Samaddar, D. and Coster, D.P. and Bonnin, X. and Berry, L.A. and Elwasif, W.R. and Batchelor, D.B.}, doi = {10.1016/j.cpc.2018.08.007}, journal = {Computer Physics Communications}, pages = {246--257}, title = {Application of the parareal algorithm to simulations of {ELM}s in {ITER} plasma}, url = {https://doi.org/10.1016/j.cpc.2018.08.007}, volume = {235}, year = {2019} }
Abstract for BibTeX entry SamaddarEtAl2019

This paper explores the application of the parareal algorithm to simulations of ELMs in ITER plasma. The primary focus of this research is identifying the parameters that lead to optimum performance. Since the plasma dynamics vary extremely fast during an ELM cycle, a straightforward application of the algorithm is not possible and a modification to the standard parareal correction is implemented. The size of the time chunks also have an impact on the performance and needs to be optimized. A computational gain of 7.8 is obtained with 48 processors to illustrate that the parareal algorithm can be successfully applied to ELM plasma.
SchreiberLoft2019

M. Schreiber, N. Schaeffer, and R. Loft, “Exponential Integrators with Parallel-in-Time Rational Approximations for Shallow-Water Equations on the Rotating Sphere,” Parallel Computing, 2019, doi: 10.1016/j.parco.2019.01.005. [Online]. Available at: https://dx.doi.org/10.1016/j.parco.2019.01.005
BibTeX entry SchreiberLoft2019

@article{SchreiberLoft2019, author = {Schreiber, M. and Schaeffer, N. and Loft, R.}, doi = {10.1016/j.parco.2019.01.005}, journal = {Parallel Computing}, title = {Exponential Integrators with Parallel-in-Time Rational Approximations for Shallow-Water Equations on the Rotating Sphere}, url = {https://dx.doi.org/10.1016/j.parco.2019.01.005}, year = {2019} }
Abstract for BibTeX entry SchreiberLoft2019

High-performance computing trends towards many-core systems are expected to continue over the next decade. As a result, parallel-in-time methods, mathematical formulations which exploit additional degrees of parallelism in the time dimension, have gained increasing interest in recent years. In this work we study a massively parallel rational approximation of exponential integrators (REXI). This method replaces a time integration of stiff linear oscillatory and diffusive systems by the sum of the solutions of many decoupled systems, which can be solved in parallel. Previous numerical studies showed that this reformulation allows taking arbitrarily long time steps for the linear oscillatory parts. The present work studies the non-linear shallow-water equations on the rotating sphere, a simplified system of equations used to study properties of space and time discretization methods in the context of atmospheric simulations. After introducing time integrators, we first compare the time step sizes to the errors in the simulation, discussing pros and cons of different formulations of REXI. Here, REXI already shows superior properties compared to explicit and implicit time stepping methods. Additionally, we present wallclock-time-to-error results revealing the sweet spots of REXI obtaining either an over 6 ×  higher accuracy within the same time frame or an about 3 ×  reduced time-to-solution for a similar error threshold. Our results motivate further explorations of REXI for operational weather/climate systems.

M. Schreiber and R. Loft, “A parallel time integrator for solving the linearized shallow water equations on the rotating sphere,” Numerical Linear Algebra with Applications, vol. 26, no. 2, p. e2220, 2019, doi: 10.1002/nla.2220. [Online]. Available at: https://onlinelibrary.wiley.com/doi/abs/10.1002/nla.2220

Southworth2019

B. S. Southworth, “Necessary Conditions and Tight Two-level Convergence Bounds for Parareal and Multigrid Reduction in Time,” SIAM J. Matrix Anal. Appl., vol. 40, no. 2, pp. 564–608, 2019, doi: https://doi.org/10.1137/18M1226208.
BibTeX entry Southworth2019

@article{Southworth2019, author = {Southworth, Ben S.}, doi = {https://doi.org/10.1137/18M1226208}, journal = {SIAM J. Matrix Anal. Appl.}, number = {2}, pages = {564--608}, title = {Necessary {C}onditions and {T}ight {T}wo-level {C}onvergence {B}ounds for {P}arareal and {M}ultigrid {R}eduction in {T}ime}, volume = {40}, year = {2019} }
Abstract for BibTeX entry Southworth2019

Parareal and multigrid reduction in time (MGRiT) are two of the most popular parallel-in-time methods. The basic idea is to treat time integration in a parallel context by using a multigrid method in time. If Φis the (fine-grid) time-stepping scheme of interest, such as any Runge–Kutta scheme, then let Ψdenote a “coarse-grid" time-stepping scheme chosen to approximate k steps of Φ, where k≥1. In particular, Ψdefines the coarse-grid correction, and evaluating Ψshould be (significantly) cheaper than evaluating Φ^k. Parareal is a two-level method with a fixed relaxation scheme, and MGRiT is a generalization to the multilevel setting, with the additional option of a modified, stronger relaxation scheme. A number of papers have studied the convergence of Parareal and MGRiT. However, general conditions on the convergence of Parareal or MGRiT that answer the following simple questions have yet to be developed: (i) For a given Φand k, what is the best Ψ? (ii) Can Parareal/MGRiT converge for my problem? This work derives necessary and sufficient conditions for the convergence of Parareal and MGRiT applied to linear problems, along with tight two-level convergence bounds, under minimal additional assumptions on Φand Ψ. Results all rest on the introduction of a temporal approximation property (TAP) that indicates how Φ^k must approximate the action of Ψon different vectors. Loosely, for unitarily diagonalizable operators, the TAP indicates that the fine-grid and coarse-grid time integration schemes must integrate geometrically smooth spatial components similarly, and less so for geometrically high frequency. In the (nonunitarily) diagonalizable setting, the conditioning of each eigenvector, v_i, must also be reflected in how well Ψv_i ∼Φ^kv_i. In general, worst-case convergence bounds are exactly given by \min \varphi < 1 such that an inequality along the lines of \|(Ψ-Φ^k)v\| ≤\varphi \|(I - Ψ)v\| holds for all v. Such inequalities are formalized as different realizations of the TAP in section 2 and form the basis for convergence of MGRiT and Parareal applied to linear problems.

R. Speck, “Algorithm 997: pySDC - Prototyping Spectral Deferred Corrections,” ACM Transactions on Mathematical Software, vol. 45, no. 3, pp. 1–23, Aug. 2019, doi: 10.1145/3310410. [Online]. Available at: https://doi.org/10.1145/3310410

SpeckEtAl2019

R. Speck, M. Knobloch, A. Gocht, and S. Lührs, “Using performance analysis tools for parallel-in-time integrators – Does my time-parallel code do what I think it does?,” arXiv:1911.13027v1 [cs.PF], 2019 [Online]. Available at: http://arxiv.org/abs/1911.13027v1
BibTeX entry SpeckEtAl2019

@unpublished{SpeckEtAl2019, author = {Speck, Robert and Knobloch, Michael and Gocht, Andreas and Lührs, Sebastian}, howpublished = {arXiv:1911.13027v1 [cs.PF]}, title = {Using performance analysis tools for parallel-in-time integrators -- Does my time-parallel code do what I think it does?}, url = {http://arxiv.org/abs/1911.13027v1}, year = {2019} }
Abstract for BibTeX entry SpeckEtAl2019

While many ideas and proofs of concept for parallel-in-time integration methods exists, the number of large-scale, accessible time-parallel codes is rather small. This is often due to the apparent or subtle complexity of the algorithms and the many pitfalls awaiting developers of parallel numerical software. One example of such a time-parallel code is pySDC, which implements, among others, the parallel full approximation scheme in space and time (PFASST). Inspired by nonlinear multigrid ideas, PFASST allows to integrate multiple time-steps simultaneously using a space-time hierarchy of spectral deferred corrections. In this paper we demonstrate the application of performance analysis tools to the PFASST implementation pySDC. We aim to answer whether the code works as intended and whether the time-parallelization is as efficient as expected. Tracing the path we took for this work, we highlight the obstacles encountered, describe remedies and explain the sometimes surprising findings made possible by the tools. Although focusing only on a single implementation of a particular parallel-in-time integrator, we hope that our results and in particular the way we obtained them are a blueprint for other time-parallel codes.

S. Wang, Y. Shao, and Z. Peng, “A Parallel-in-Space-and-Time Method for Transient Electromagnetic Problems,” IEEE Transactions on Antennas and Propagation, vol. 67, no. 6, pp. 3961–3973, 2019, doi: 10.1109/TAP.2019.2909937. [Online]. Available at: https://doi.org/10.1109/TAP.2019.2909937

S.-L. Wu and T. Zhou, “Acceleration of the Two-Level MGRIT Algorithm via the Diagonalization Technique,” SIAM Journal on Scientific Computing, vol. 41, no. 5, pp. A3421–A3448, Jan. 2019, doi: 10.1137/18m1207697. [Online]. Available at: https://doi.org/10.1137/18m1207697

L. Zhang, W. Zhou, and L. Ji, “Parareal algorithms applied to stochastic differential equations with conserved quantities,” Journal of Computational Mathematics, vol. 37, no. 1, pp. 48–60, 2019, doi: 10.4208/jcm.1708-m2017-0089. [Online]. Available at: https://doi.org/10.4208/jcm.1708-m2017-0089

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2018

BadiaEtAl2018

S. Badia and M. Olm, “Nonlinear parallel-in-time Schur complement solvers for ordinary differential equations,” Journal of Computational and Applied Mathematics, vol. 344, pp. 794–806, 2018, doi: 10.1016/j.cam.2017.09.033. [Online]. Available at: https://doi.org/10.1016/j.cam.2017.09.033
BibTeX entry BadiaEtAl2018

@article{BadiaEtAl2018, author = {Badia, Santiago and Olm, Marc}, doi = {10.1016/j.cam.2017.09.033}, journal = {Journal of Computational and Applied Mathematics}, pages = {794--806}, title = {Nonlinear parallel-in-time Schur complement solvers for ordinary differential equations}, url = {https://doi.org/10.1016/j.cam.2017.09.033}, volume = {344}, year = {2018} }
Abstract for BibTeX entry BadiaEtAl2018

Abstract In this work, we propose a parallel-in-time solver for linear and nonlinear ordinary differential equations. The approach is based on an efficient multilevel solver of the Schur complement related to a multilevel time partition. For linear problems, the scheme leads to a fast direct method. Next, two different strategies for solving nonlinear ODEs are proposed. First, we consider a Newton method over the global nonlinear ODE, using the multilevel Schur complement solver at every nonlinear iteration. Second, we state the global nonlinear problem in terms of the nonlinear Schur complement (at an arbitrary level), and perform nonlinear iterations over it. Numerical experiments show that the proposed schemes are weakly scalable, i.e., we can efficiently exploit increasing computational resources to solve for more time steps the same problem.

P. Benedusi, C. Garoni, R. Krause, X. Li, and S. Serra-Capizzano, “Space-Time FE-DG Discretization of the Anisotropic Diffusion Equation in Any Dimension: The Spectral Symbol,” SIAM Journal on Matrix Analysis and Applications, vol. 39, no. 3, pp. 1383–1420, 2018, doi: 10.1137/17M113527X. [Online]. Available at: https://doi.org/10.1137/17M113527X

BoltenEtAl2018

M. Bolten, D. Moser, and R. Speck, “Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems,” Numerical Linear Algebra with Applications, vol. 25, no. 6, p. e2208, 2018, doi: 10.1002/nla.2208. [Online]. Available at: https://onlinelibrary.wiley.com/doi/abs/10.1002/nla.2208
BibTeX entry BoltenEtAl2018

@article{BoltenEtAl2018, author = {Bolten, Matthias and Moser, Dieter and Speck, Robert}, doi = {10.1002/nla.2208}, journal = {Numerical Linear Algebra with Applications}, number = {6}, pages = {e2208}, title = {Asymptotic convergence of the parallel full approximation scheme in space and time for linear problems}, url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nla.2208}, volume = {25}, year = {2018} }
Abstract for BibTeX entry BoltenEtAl2018

For time-dependent partial differential equations, parallel-in-time integration using the “parallel full approximation scheme in space and time” (PFASST) is a promising way to accelerate existing space-parallel approaches beyond their scaling limits. Inspired by the classical Parareal method and multigrid ideas, PFASST allows to integrate multiple time steps simultaneously using a space–time hierarchy of spectral deferred correction sweeps. While many use cases and benchmarks exist, a solid and reliable mathematical foundation is still missing. Very recently, however, PFASST for linear problems has been identified as a multigrid method. In this paper, we will use this multigrid formulation and, in particular, PFASST’s iteration matrix to show that, in the nonstiff and stiff limit, PFASST indeed is a convergent iterative method. We will provide upper bounds for the spectral radius of the iteration matrix and investigate how PFASST performs for increasing numbers of parallel time steps. Finally, we will demonstrate that the results obtained here indeed relate to actual PFASST runs.
BorregalesEtAl2018

Manuel Borregales and Kundan Kumar and Florin Adrian Radu and Carmen Rodrigo and Francisco José Gaspar, “A partially parallel-in-time fixed-stress splitting method for Biot’s consolidation model,” Computers & Mathematics with Applications, 2018, doi: 10.1016/j.camwa.2018.09.005. [Online]. Available at: https://doi.org/10.1016/j.camwa.2018.09.005
BibTeX entry BorregalesEtAl2018

@article{BorregalesEtAl2018, author = {{Manuel Borregales and Kundan Kumar and Florin Adrian Radu and Carmen Rodrigo and Francisco Jos{\'e} Gaspar}}, doi = {10.1016/j.camwa.2018.09.005}, journal = {Computers \& Mathematics with Applications}, title = {{A partially parallel-in-time fixed-stress splitting method for Biot’s consolidation model}}, url = {https://doi.org/10.1016/j.camwa.2018.09.005}, year = {2018} }
Abstract for BibTeX entry BorregalesEtAl2018

In this work, we study the parallel-in-time iterative solution of coupled flow and geomechanics in porous media, modelled by a two-field formulation of Biot’s equations. In particular, we propose a new version of the fixed-stress splitting method, which has been widely used as solution method of these problems. This new approach forgets about the sequential nature of the temporal variable and considers the time direction as a further direction for parallelization. The method is partially parallel-in-time. We present a rigorous convergence analysis of the method and numerical experiments to demonstrate the robust behaviour of the algorithm.
Bu2018

S. Bu, “Time parallelization scheme with an adaptive time step size for solving stiff initial value problems,” Open Mathematics, vol. 16, no. 1, pp. 210–218, 2018, doi: 10.1515/math-2018-0022. [Online]. Available at: https://doi.org/10.1515/math-2018-0022
BibTeX entry Bu2018

@article{Bu2018, author = {Bu, Sunyoung}, doi = {10.1515/math-2018-0022}, issue = {1}, journal = {Open Mathematics}, pages = {210--218}, title = {Time parallelization scheme with an adaptive time step size for solving stiff initial value problems}, url = {https://doi.org/10.1515/math-2018-0022}, volume = {16}, year = {2018} }
Abstract for BibTeX entry Bu2018

In this paper, we introduce a practical strategy to select an adaptive time step size suitable for the parareal algorithm designed to parallelize a numerical scheme for solving stiff initial value problems. For the adaptive time step size, a technique to detect stiffness of a given system is first considered since the step size will be chosen according to the extent of stiffness. Finally, the stiffness detection technique is applied to an initial prediction step of the parareal algorithm, and select an adaptive step size to each time interval according to the stiffness. Several numerical experiments demonstrate the efficiency of the proposed method.
DamoreEtAl2018

L. D’Amore and R. Cacciapuoti, “DD-DA PinT-based model: A Domain Decomposition approach in space and time, based on Parareal, for solving the 4D-Var Data Assimilation model,” arXiv:1807.07107 [math.NA], 2018 [Online]. Available at: https://arxiv.org/abs/1807.07107
BibTeX entry DamoreEtAl2018

@unpublished{DamoreEtAl2018, author = {D'Amore, Luisa and Cacciapuoti, Rosalba}, howpublished = {arXiv:1807.07107 [math.NA]}, title = {DD-DA PinT-based model: A Domain Decomposition approach in space and time, based on Parareal, for solving the 4D-Var Data Assimilation model}, url = {https://arxiv.org/abs/1807.07107}, year = {2018} }
Abstract for BibTeX entry DamoreEtAl2018

We present the mathematical framework of a Domain Decomposition (DD) aproach based on Parallel-in-Time methods (PinT-based approach) for solving the 4D-Var Data Assimilation (DA) model. The main outcome of the proposed DD PinT-based approach is: 1. DA acts as coarse/predictor for the local PDE-based forecasting model, increasing the accuracy of the local solution. 2. The fine and coarse solvers can be used in parallel, increasing the efficiency of the algorithm.3. Data locality is preserved and data movement is reduced, increasing the software scalability. We provide the mathematical framework including convergence analysis and error propagation.
DuanEtAl2018

N. Duan, S. Simunovic, A. Dimitrovski, and K. Sun, “Improving the Convergence Rate of Parareal-in-time Power System Simulation using the Krylov Subspace,” in 2018 IEEE Power Energy Society General Meeting (PESGM), 2018, pp. 1–5, doi: 10.1109/PESGM.2018.8586354 [Online]. Available at: https://dx.doi.org/10.1109/PESGM.2018.8586354
BibTeX entry DuanEtAl2018

@inproceedings{DuanEtAl2018, author = {{Duan}, N. and {Simunovic}, S. and {Dimitrovski}, A. and {Sun}, K.}, booktitle = {2018 IEEE Power Energy Society General Meeting (PESGM)}, doi = {10.1109/PESGM.2018.8586354}, pages = {1--5}, title = {Improving the Convergence Rate of Parareal-in-time Power System Simulation using the {K}rylov Subspace}, url = {https://dx.doi.org/10.1109/PESGM.2018.8586354}, year = {2018} }
Abstract for BibTeX entry DuanEtAl2018

The performance of parareal-in-time algorithms is determined on the number of sequential, coarse step iterations. A common tradeoff in designing an efficient parareal-in-time algorithm is between accuracy of the coarse solver and the number of iterations. Traditional parareal implementation for the power system simulation can also have difficulties handling complex power systems. In this paper, we propose a Krylov subspace enhanced parareal algorithm to reduce the number of coarse iterations. The proposed approach is demonstrated on a single-machine-infinite-bus system and the IEEE 10-machine 39-bus system. Noticeable decrease of number of iterations is observed in both cases.
DyjaEtal2018

R. Dyja, B. Ganapathysubramanian, and K. G. van der Zee, “Parallel-In-Space-Time, Adaptive Finite Element Framework for Nonlinear Parabolic Equations,” SIAM Journal on Scientific Computing, vol. 40, no. 3, pp. C283–C304, 2018, doi: 10.1137/16M108985X. [Online]. Available at: https://doi.org/10.1137/16M108985X
BibTeX entry DyjaEtal2018

@article{DyjaEtal2018, author = {Dyja, Robert and Ganapathysubramanian, Baskar and van der Zee, Kristoffer G.}, doi = {10.1137/16M108985X}, journal = {SIAM Journal on Scientific Computing}, number = {3}, pages = {C283--C304}, title = {Parallel-In-Space-Time, Adaptive Finite Element Framework for Nonlinear Parabolic Equations}, url = {https://doi.org/10.1137/16M108985X}, volume = {40}, year = {2018} }
Abstract for BibTeX entry DyjaEtal2018

We present an adaptive methodology for the solution of (linear and) nonlinear time dependent problems that is especially tailored for massively parallel computations. The basic concept is to solve for large blocks of space-time unknowns instead of marching sequentially in time. The methodology is a combination of a computationally efficient implementation of a parallel-in-space-time finite element solver coupled with a posteriori space-time error estimates and a parallel mesh generator. While we focus on spatial adaptivity in this work, the methodology enables simultaneous adaptivity in both space and time domains. We explore this basic concept in the context of a variety of time steppers including Θ-schemes and backward difference formulas. We specifically illustrate this framework with applications involving time dependent linear, quasi-linear, and semilinear diffusion equations. We focus on investigating how the coupled space-time refinement indicators for this class of problems affect spatial adaptivity. Finally, we show good scaling behavior up to 150,000 processors on the NCSA Blue Waters machine. This conceptually simple methodology enables scaling on next generation multicore machines by simultaneously solving for a large number of timesteps, and reducing computational overhead by locally refining spatial blocks that can track localized features. This methodology also opens up the possibility of efficiently incorporating adjoint equations for error estimators and inverse design problems, since blocks of space-time are simultaneously solved and stored in memory.
FischerEtAl2018

L. Fischer, S. Götschel, and M. Weiser, “Lossy data compression reduces communication time in hybrid time-parallel integrators,” Computing and Visualization in Science, vol. 19, no. 1, pp. 19–30, 2018, doi: 10.1007/s00791-018-0293-2. [Online]. Available at: https://doi.org/10.1007/s00791-018-0293-2
BibTeX entry FischerEtAl2018

@article{FischerEtAl2018, author = {Fischer, L. and G\"otschel, S. and Weiser, M.}, doi = {10.1007/s00791-018-0293-2}, journal = {Computing and Visualization in Science}, number = {1}, pages = {19--30}, title = {Lossy data compression reduces communication time in hybrid time-parallel integrators}, url = {https://doi.org/10.1007/s00791-018-0293-2}, volume = {19}, year = {2018} }
Abstract for BibTeX entry FischerEtAl2018

Parallel in time methods for solving initial value problems are a means to increase the parallelism of numerical simulations. Hybrid parareal schemes interleaving the parallel in time iteration with an iterative solution of the individual time steps are among the most efficient methods for general nonlinear problems. Despite the hiding of communication time behind computation, communication has in certain situations a significant impact on the total runtime. Here we present strict, yet no sharp, error bounds for hybrid parareal methods with inexact communication due to lossy data compression, and derive theoretical estimates of the impact of compression on parallel efficiency of the algorithms. These and some computational experiments suggest that compression is a viable method to make hybrid parareal schemes robust with respect to low bandwidth setups.
FrancoEtAl2018

S. R. Franco, F. J. Gaspar, M. A. V. Pinto, and C. Rodrigo, “Multigrid method based on a space-time approach with standard coarsening for parabolic problems,” Applied Mathematics and Computation, vol. 317, no. Supplement C, pp. 25–34, 2018, doi: 10.1016/j.amc.2017.08.043. [Online]. Available at: https://doi.org/10.1016/j.amc.2017.08.043
BibTeX entry FrancoEtAl2018

@article{FrancoEtAl2018, author = {Franco, Sebasti\~{a}o Romero and Gaspar, Francisco Jos\'{e} and Pinto, Marcio Augusto Villela and Rodrigo, Carmen}, doi = {10.1016/j.amc.2017.08.043}, journal = {Applied Mathematics and Computation}, number = {Supplement C}, pages = {25--34}, title = {Multigrid method based on a space-time approach with standard coarsening for parabolic problems}, url = {https://doi.org/10.1016/j.amc.2017.08.043}, volume = {317}, year = {2018} }
Abstract for BibTeX entry FrancoEtAl2018

In this work, a space-time multigrid method which uses standard coarsening in both temporal and spatial domains and combines the use of different smoothers is proposed for the solution of the heat equation in one and two space dimensions. In particular, an adaptive smoothing strategy, based on the degree of anisotropy of the discrete operator on each grid-level, is the basis of the proposed multigrid algorithm. Local Fourier analysis is used for the selection of the crucial parameter defining such an adaptive smoothing approach. Central differences are used to discretize the spatial derivatives and both implicit Euler and Crank–Nicolson schemes are considered for approximating the time derivative. For the solution of the second-order scheme, we apply a double discretization approach within the space-time multigrid method. The good performance of the method is illustrated through several numerical experiments.
FrancoEtAl2018a

S. R. Franco, C. Rodrigo, F. J. Gaspar, and M. A. V. Pinto, “A multigrid waveform relaxation method for solving the poroelasticity equations,” Computational and Applied Mathematics, pp. 1–16, 2018, doi: 10.1007/s40314-018-0603-9. [Online]. Available at: https://doi.org/10.1007/s40314-018-0603-9
BibTeX entry FrancoEtAl2018a

@article{FrancoEtAl2018a, author = {Franco, Sebasti\~{a}o Romero and Rodrigo, Carmen and Gaspar, Francisco Jos\'{e} and Pinto, Marcio Augusto Villela}, doi = {10.1007/s40314-018-0603-9}, journal = {Computational and Applied Mathematics}, pages = {1--16}, title = {A multigrid waveform relaxation method for solving the poroelasticity equations}, url = {https://doi.org/10.1007/s40314-018-0603-9}, year = {2018} }
Abstract for BibTeX entry FrancoEtAl2018a

In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.
FuWang2018

H. Fu and H. Wang, “A Preconditioned Fast Parareal Finite Difference Method for Space-Time Fractional Partial Differential Equation,” Journal of Scientific Computing, 2018, doi: 10.1007/s10915-018-0835-2. [Online]. Available at: https://doi.org/10.1007/s10915-018-0835-2
BibTeX entry FuWang2018

@article{FuWang2018, author = {Fu, Hongfei and Wang, Hong}, doi = {10.1007/s10915-018-0835-2}, journal = {Journal of Scientific Computing}, title = {A Preconditioned Fast Parareal Finite Difference Method for Space-Time Fractional Partial Differential Equation}, url = {https://doi.org/10.1007/s10915-018-0835-2}, year = {2018} }
Abstract for BibTeX entry FuWang2018

We develop a fast parareal finite difference method for space-time fractional partial differential equation. The method properly handles the heavy tail behavior in the numerical discretization, while retaining the numerical advantages of conventional parareal algorithm. At each time step, we explore the structure of the stiffness matrix to develop a matrix-free preconditioned fast Krylov subspace iterative solver for the finite difference method without resorting to any lossy compression. Consequently, the method has significantly reduced computational complexity and memory requirement. Numerical experiments show the strong potential of the method.

M. J. Gander, S. Güttel, and M. Petcu, “A Nonlinear ParaExp Algorithm,” in Lecture Notes in Computational Science and Engineering, Springer International Publishing, 2018, pp. 261–270 [Online]. Available at: https://doi.org/10.1007/978-3-319-93873-8_24

GanderEtAl2018_cvs

M. J. Gander, F. Kwok, and H. Zhang, “Multigrid interpretations of the parareal algorithm leading to an overlapping variant and MGRIT,” Computing and Visualization in Science, 2018, doi: 10.1007/s00791-018-0297-y. [Online]. Available at: https://doi.org/10.1007/s00791-018-0297-y
BibTeX entry GanderEtAl2018_cvs

@article{GanderEtAl2018_cvs, author = {Gander, Martin J. and Kwok, Felix and Zhang, Hui}, doi = {10.1007/s00791-018-0297-y}, journal = {Computing and Visualization in Science}, title = {Multigrid interpretations of the parareal algorithm leading to an overlapping variant and MGRIT}, url = {https://doi.org/10.1007/s00791-018-0297-y}, year = {2018} }
Abstract for BibTeX entry GanderEtAl2018_cvs

The parareal algorithm is by construction a two level method, and there are several ways to interpret the parareal algorithm to obtain multilevel versions. We first review the three main interpretations of the parareal algorithm as a two-level method, a direct one, one based on geometric multigrid and one based on algebraic multigrid. The algebraic multigrid interpretation leads to the MGRIT algorithm, when using instead of only an F-smoother, a so called FCF -smoother. We show that this can be interpreted at the level of the parareal algorithm as generous overlap in time. This allows us to generalize the method to even larger overlap, corresponding in MGRIT to F(CF)ν -smoothing, ν >=1 , and we prove two new convergence results for such algorithms in the fully non-linear setting: convergence in a finite number of steps, becoming smaller when ν increases, and a general superlinear convergence estimate for this generalized version of MGRIT. We illustrate our results with numerical experiments, both for linear and non-linear systems of ordinary and partial differential equations. Our results show that overlap only sometimes leads to faster algorithms.
GoddenWathen2018

A. Goddard and A. Wathen, “A note on parallel preconditioning for all-at-once evolutionary PDEs,” pp. 135–150, 2018, doi: 10.1553/etna_vol51s135. [Online]. Available at: https://dx.doi.org/10.1553/etna_vol51s135
BibTeX entry GoddenWathen2018

@article{GoddenWathen2018, address = {Wien}, author = {Goddard, Anthony and Wathen, Andy}, doi = {10.1553/etna_vol51s135}, editor = {Ronny Ramlau, Lothar Reichel (Hg.)}, pages = {135-150}, publisher = {Verlag der Österreichischen Akademie der Wissenschaften}, title = {A note on parallel preconditioning for all-at-once evolutionary PDEs}, url = {https://dx.doi.org/10.1553/etna_vol51s135}, year = {2018} }
Abstract for BibTeX entry GoddenWathen2018

McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40 (2018), pp. A1012–A1033] present a method for preconditioning time-dependent PDEs via an approximation by a nearby time-periodic problem, that is, they employ circulant-related matrices as preconditioners for the non-symmetric block Toeplitz matrices which arise from an all-at-once formulation. They suggest that such an approach might be efficiently implemented in parallel. In this short article, we present parallel numerical results for their preconditioner which exhibit strong scaling. We also extend their preconditioner via a Neumann series approach which also allows for efficient parallel execution. Results are shown for both parabolic and hyperbolic PDEs. Our simple implementation (in C++ and MPI) is available at the Git repository PARALAAOMPI.
GoetschelMinion2018

S. Götschel and M. L. Minion, “Parallel-in-Time for Parabolic Optimal Control Problems Using PFASST,” in Domain Decomposition Methods in Science and Engineering XXIV, 2018, pp. 363–371, doi: 10.1007/978-3-319-93873-8_34 [Online]. Available at: https://doi.org/10.1007/978-3-319-93873-8_34
BibTeX entry GoetschelMinion2018

@inproceedings{GoetschelMinion2018, author = {G\"otschel, Sebastian and Minion, Michael L.}, booktitle = {Domain Decomposition Methods in Science and Engineering {XXIV}}, doi = {10.1007/978-3-319-93873-8_34}, editor = {Bj{\o}rstad, Petter E. and Brenner, Susanne C. and Halpern, Lawrence and Kim, Hyea Hyun and Kornhuber, Ralf and Rahman, Talal and Widlund, Olof B.}, pages = {363--371}, publisher = {Springer International Publishing}, title = {Parallel-in-Time for Parabolic Optimal Control Problems Using {PFASST}}, url = {https://doi.org/10.1007/978-3-319-93873-8_34}, year = {2018} }
Abstract for BibTeX entry GoetschelMinion2018

In gradient-based methods for parabolic optimal control problems, it is necessary to solve both the state equation and a backward-in-time adjoint equation in each iteration of the optimization method. In order to facilitate fully parallel gradient-type and nonlinear conjugate gradient methods for the solution of such optimal control problems, we discuss the application of the parallel-in-time method PFASST to adjoint gradient computation. In addition to enabling time parallelism, PFASST provides high flexibility for handling nonlinear equations, as well as potential extra computational savings from reusing previous solutions in the optimization loop. The approach is demonstrated here for a model reaction-diffusion optimal control problem.
GuentherEtAl2018

S. Günther, N. R. Gauger, and J. B. Schroder, “A Non-Intrusive Parallel-in-Time Adjoint Solver with the XBraid Library,” Computing and Visualization in Science, 2018, doi: 10.1007/s00791-018-0300-7. [Online]. Available at: https://doi.org/10.1007/s00791-018-0300-7
BibTeX entry GuentherEtAl2018

@article{GuentherEtAl2018, author = {G\"unther, S. and Gauger, N. R. and Schroder, J. B.}, doi = {10.1007/s00791-018-0300-7}, journal = {Computing and Visualization in Science}, title = {A Non-Intrusive Parallel-in-Time Adjoint Solver with the {XBraid} Library}, url = {https://doi.org/10.1007/s00791-018-0300-7}, year = {2018} }
Abstract for BibTeX entry GuentherEtAl2018

In this paper, an adjoint solver for the multigrid in time software library XBraid is presented. XBraid provides a non-intrusive approach for simulating unsteady dynamics on multiple processors while parallelizing not only in space but also in the time domain. It applies an iterative multigrid reduction in time algorithm to existing spatially parallel classical time propagators and computes the unsteady solution parallel in time. Techniques from Automatic Differentiation are used to develop a consistent discrete adjoint solver which provides sensitivity information of output quantities with respect to design parameter changes. The adjoint code runs backwards through the primal XBraid actions and accumulates gradient information parallel in time. It is highly non-intrusive as existing adjoint time propagators can easily be integrated through the adjoint interface. The adjoint code is validated on advection-dominated flow with periodic upstream boundary condition. It provides similar strong scaling results as the primal XBraid solver and offers great potential for speeding up the overall computational costs for sensitivity analysis using multiple processors
HessenthalerEtAl2018

A. Hessenthaler, D. Nordsletten, O. Röhrle, J. B. Schroder, and R. D. Falgout, “Convergence of the multigrid reduction in time algorithm for the linear elasticity equations,” Numerical Linear Algebra with Applications, vol. 25, no. 3, p. e2155, 2018, doi: 10.1002/nla.2155. [Online]. Available at: https://dx.doi.org/10.1002/nla.2155
BibTeX entry HessenthalerEtAl2018

@article{HessenthalerEtAl2018, author = {Hessenthaler, A. and Nordsletten, D. and Röhrle, O. and Schroder, J. B. and Falgout, R. D.}, doi = {10.1002/nla.2155}, journal = {Numerical Linear Algebra with Applications}, number = {3}, pages = {e2155}, title = {Convergence of the multigrid reduction in time algorithm for the linear elasticity equations}, url = {https://dx.doi.org/10.1002/nla.2155}, volume = {25}, year = {2018} }
Abstract for BibTeX entry HessenthalerEtAl2018

This paper presents some recent advances for parallel‐in‐time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel‐in‐time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel‐in‐time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF‐relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel‐in‐time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.
HwangMunster2018

J. T. Hwang and D. Munster, “Solution of ordinary differential equations in gradient-based multidisciplinary design optimization,” in 2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, American Institute of Aeronautics and Astronautics, 2018 [Online]. Available at: https://doi.org/10.2514/6.2018-1646
BibTeX entry HwangMunster2018

@inbook{HwangMunster2018, author = {Hwang, John T. and Munster, Drayton}, booktitle = {2018 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference}, doi = {doi:10.2514/6.2018-1646}, publisher = {American Institute of Aeronautics and Astronautics}, title = {Solution of ordinary differential equations in gradient-based multidisciplinary design optimization}, url = {https://doi.org/10.2514/6.2018-1646}, year = {2018} }
Abstract for BibTeX entry HwangMunster2018

A gradient-based approach to multidisciplinary design optimization enables efficient scalability to large numbers of design variables. However, the need for derivatives imposes a difficult requirement when integrating ordinary differential equations in models. To simplify this, we propose the use of the general linear methods framework, which unifies all Runge–Kutta and linear multistep methods. This approach enables rapid implementation of integration methods without the need to differentiate each individual method, even in a gradient-based optimization context. We also develop a new parallel time integration algorithm that enables vectorization across time steps. We present a set of benchmarking results using a stiff ODE, a non-stiff nonlinear ODE, and an orbital dynamics ODE, and compare integration methods. In a modular gradient-based multidisciplinary design optimization context, we find that the new parallel time integration algorithm with high-order implicit methods, especially Gauss–Legendre collocation, is the best choice for a broad range of problems.
IizukaEtAl2018

M. Iizuka and K. Ono, “Influence of the phase accuracy of the coarse solver calculation on the convergence of the parareal method iteration for hyperbolic PDEs,” Computing and Visualization in Science, 2018, doi: 10.1007/s00791-018-0299-9. [Online]. Available at: https://doi.org/10.1007/s00791-018-0299-9
BibTeX entry IizukaEtAl2018

@article{IizukaEtAl2018, author = {Iizuka, Mikio and Ono, Kenji}, doi = {10.1007/s00791-018-0299-9}, journal = {Computing and Visualization in Science}, title = {Influence of the phase accuracy of the coarse solver calculation on the convergence of the parareal method iteration for hyperbolic PDEs}, url = {https://doi.org/10.1007/s00791-018-0299-9}, year = {2018} }
Abstract for BibTeX entry IizukaEtAl2018

Gander and Petcu (ESAIM Proc 25:114–129, 2008) reported that, theoretically, the convergence of the parareal method iteration for hyperbolic PDEs is strongly influenced by the phase (frequency) accuracy of the coarse solver calculation. However, no numerical study has clearly shown this. Therefore, through numerical tests, we investigate the influence of the phase accuracy of the coarse solver calculation on the convergence of the parareal method iteration for hyperbolic PDEs. First, we consider a simple harmonic motion and a multi-DOF mass-spring system (MDMSS) as examples of hyperbolic PDEs using the modified Newmark- β method (Mizuta et al. in J JSCE 268:15–21, 1977), which can provide the exact phase of the time integration of a simple harmonic motion. Based on the results of the numerical tests, we show that the convergence of the parareal method iteration for hyperbolic PDEs is approximately independent of the parameters of parallel-in-time integration (PinT) and instead is dependent primarily on the phase accuracy of the coarse solver calculation. In addition, we show that reducing the number of bases in the reduced basis method (RBM) (Chen et al., in: Rozza (ed) Reduced order methods for modeling and computational reduction, MS and a modeling, simulation and applications, vol 9, Springer, Berlin, pp 187–214, 2014) causes the saturation of a decrease in an error during the parareal iteration for the MDMSS using the mode analysis method. The RBM is expected to make available accurate phase calculation in the coarse solver by maintaining the time step width as same as that of the fine solver. Second, we investigate whether the same saturation appears for the linear advection–diffusion equation when we use the RBM. We use the time evolution basis method in the RBM for the linear advection–diffusion equation. As a result, we show that reducing the number of bases causes the saturation of the decrease in the error in the linear advection–diffusion equation. Based on the results of the present study, an increase in the phase accuracy of the coarse solver calculation is strongly required for better convergence of the parareal method iteration for hyperbolic PDEs. Moreover, the saturation of the decrease in the error during the parareal method iteration should be overcome when using the RBM.
KooijEtAl2018

G. L. Kooij, M. A. Botchev, and B. J. Geurts, “An Exponential Time Integrator for the Incompressible Navier–Stokes Equation,” SIAM Journal on Scientific Computing, vol. 40, no. 3, pp. B684–B705, 2018, doi: 10.1137/17M1121950. [Online]. Available at: https://doi.org/10.1137/17M1121950
BibTeX entry KooijEtAl2018

@article{KooijEtAl2018, author = {Kooij, Gijs L. and Botchev, Mike A. and Geurts, Bernard J.}, doi = {10.1137/17M1121950}, journal = {SIAM Journal on Scientific Computing}, number = {3}, pages = {B684--B705}, title = {An Exponential Time Integrator for the Incompressible Navier--Stokes Equation}, url = {https://doi.org/10.1137/17M1121950}, volume = {40}, year = {2018} }
Abstract for BibTeX entry KooijEtAl2018

We present an exponential time integration method for the incompressible Navier–Stokes equation. An essential step in our procedure is the treatment of the pressure by applying a divergence-free projection to the momentum equation. The differential-algebraic equation for the discrete velocity and pressure is then reduced to a conventional ordinary differential equation that can be solved with the proposed exponential integrator. A promising feature of exponential time integration is its potential time parallelism within the Paraexp algorithm. We demonstrate that our approach leads to parallel speedup assuming negligible parallel communication.
LedermanBilyeu2018

C. Lederman and D. Bilyeu, “An Approximate Time-Parallel Method for the Fast and Accurate Computation of Particle Trajectories in a Magnetic Field,” Journal of Applied Mathematics and Physics, vol. 6, pp. 498–519, 2018, doi: 10.4236/jamp.2018.63046 . [Online]. Available at: https://doi.org/10.4236/jamp.2018.63046
BibTeX entry LedermanBilyeu2018

@article{LedermanBilyeu2018, author = {Lederman, Carl and Bilyeu, David}, doi = {10.4236/jamp.2018.63046 }, journal = {Journal of Applied Mathematics and Physics}, pages = {498--519}, title = {An Approximate Time-Parallel Method for the Fast and Accurate Computation of Particle Trajectories in a Magnetic Field}, url = {https://doi.org/10.4236/jamp.2018.63046}, volume = {6}, year = {2018} }
Abstract for BibTeX entry LedermanBilyeu2018

A temporal multiscale hybridization method is presented that carefully couples coarse scale gyrokinetic models with exact charged particle solution trajectories (that is, with full phase information) in a magnetic field. The approach is based on the careful approximation of a sum, generally employed for time-parallel (TP) computing applications. While the hybridization method presented is highly parallelizable, a computational efficiency gain is seen from considering serial computations only. A complete numerical method is only presented for the aforementioned charged particle application, however, the general approach depicted likely has relevance to a wide swath of challenging multiscale/multiphysics problems. Additionally, the approach has obvious relevance to TP computing applications (such as variable selection on which to perform TP calculations and fine scale sampling strategies).
LiangLin2018

J. Liang and M. C. Lin, “Time-Domain Parallelization for Accelerating Cloth Simulation,” Computer Graphics Forum, vol. 37, no. 8, pp. 21–34, 2018, doi: 10.1111/cgf.13509. [Online]. Available at: https://dx.doi.org/10.1111/cgf.13509
BibTeX entry LiangLin2018

@article{LiangLin2018, author = {Liang, Junbang and Lin, Ming C.}, doi = {10.1111/cgf.13509}, journal = {Computer Graphics Forum}, number = {8}, pages = {21--34}, title = {Time-Domain Parallelization for Accelerating Cloth Simulation}, url = {https://dx.doi.org/10.1111/cgf.13509}, volume = {37}, year = {2018} }
Abstract for BibTeX entry LiangLin2018

Cloth simulations, widely used in computer animation and apparel design, can be computationally expensive for real-time applications. Some parallelization techniques have been proposed for visual simulation of cloth using CPU or GPU clusters and often rely on parallelization using spatial domain decomposition techniques that have a large communication overhead. In this paper, we propose a novel time-domain parallelization technique that makes use of the two-level mesh representation to resolve the time-dependency issue and develop a practical algorithm to smooth the state transition from the corresponding coarse to fine meshes. A load estimation and a load balancing technique used in online partitioning are also proposed to maximize the performance acceleration. Our method achieves a nearly linear performance scaling on manycore clusters and outperforms spatial-domain parallelization on a diverse set of benchmarks.
LunetEtAl2018

T. Lunet, J. Bodart, S. Gratton, and X. Vasseur, “Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening,” Computing and Visualization in Science, vol. 19, no. 1, pp. 31–44, 2018, doi: 10.1007/s00791-018-0295-0. [Online]. Available at: https://doi.org/10.1007/s00791-018-0295-0
BibTeX entry LunetEtAl2018

@article{LunetEtAl2018, author = {Lunet, Thibaut and Bodart, Julien and Gratton, Serge and Vasseur, Xavier}, doi = {10.1007/s00791-018-0295-0}, journal = {Computing and Visualization in Science}, number = {1}, pages = {31--44}, title = {Time-parallel simulation of the decay of homogeneous turbulence using Parareal with spatial coarsening}, url = {https://doi.org/10.1007/s00791-018-0295-0}, volume = {19}, year = {2018} }
Abstract for BibTeX entry LunetEtAl2018

Direct Numerical Simulation of turbulent flows is a computationally demanding problem that requires efficient parallel algorithms. We investigate the applicability of the time-parallel Parareal algorithm to an instructional case study related to the simulation of the decay of homogeneous isotropic turbulence in three dimensions. We combine a Parareal variant based on explicit time integrators and spatial coarsening with the space-parallel Hybrid Navier–Stokes solver. We analyse the performance of this space–time parallel solver with respect to speedup and quality of the solution. The results are compared with reference data obtained with a classical explicit integration, using an error analysis which relies on the energetic content of the solution. We show that a single Parareal iteration is able to reproduce with high fidelity the main statistical quantities characterizing the turbulent flow field.
MadayMula2018

Y. Maday and O. Mula, “A Scalable Adaptive Parareal Algorithm With Online Stopping Criterion,” hal-01781257, version 1, 2018 [Online]. Available at: https://hal.archives-ouvertes.fr/hal-01781257/
BibTeX entry MadayMula2018

@unpublished{MadayMula2018, author = {Maday, Yvon and Mula, Olga}, howpublished = {hal-01781257, version 1}, title = {A Scalable Adaptive Parareal Algorithm With Online Stopping Criterion}, url = {https://hal.archives-ouvertes.fr/hal-01781257/}, year = {2018} }
Abstract for BibTeX entry MadayMula2018

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems. Our main contribution is the significant improvement of the parallel efficiency of the parareal in time method, an iterative predictor-corrector algorithm. This is achieved by first reformulating the algorithm in a rigorous infinite dimensional functional space setting. We then formulate implementable versions where time dependent subproblems are solved at increasing accuracy across the parareal iterations (in opposition to the classical version where the subproblems are solved at a fixed high accuracy). Aside from the important improvement in parallel efficiency and as a natural by product, the new approach provides a rigourous online stopping criterion with a posteriori error estimators and the numerical cost to achieve a certain final accuracy is designed to be near-minimal. We illustrate the gain in efficiency of the new approach on simple numerical experiments. In addition to this, we discuss the potential benefits of reusing information from previous parareal iterations to enhance efficiency even more.
MagoulesEtAl2018

F. Magoulès, G. Gbikpi-Benissan, and Q. Zou, “Asynchronous Iterations of Parareal Algorithm for Option Pricing Models,” Mathematics, vol. 6, no. 4, 2018, doi: 10.3390/math6040045. [Online]. Available at: https://doi.org/10.3390/math6040045
BibTeX entry MagoulesEtAl2018

@article{MagoulesEtAl2018, author = {Magoul{\`e}s, Fr{\'e}d{\'e}ric and Gbikpi-Benissan, Guillaume and Zou, Qinmeng}, doi = {10.3390/math6040045}, journal = {Mathematics}, number = {4}, title = {Asynchronous Iterations of Parareal Algorithm for Option Pricing Models}, url = {https://doi.org/10.3390/math6040045}, volume = {6}, year = {2018} }
Abstract for BibTeX entry MagoulesEtAl2018

Spatial domain decomposition methods have been largely investigated in the last decades, while time domain decomposition seems to be contrary to intuition and so is not as popular as the former. However, many attractive methods have been proposed, especially the parareal algorithm, which showed both theoretical and experimental efficiency in the context of parallel computing. In this paper, we present an original model of asynchronous variant based on the parareal scheme, applied to the European option pricing problem. Some numerical experiments are given to illustrate the convergence performance and computational efficiency of such a method.

F. Magoulès and G. Gbikpi-Benissan, “Asynchronous Parareal Time Discretization For Partial Differential Equations,” SIAM Journal on Scientific Computing, vol. 40, no. 6, pp. C704–C725, Jan. 2018, doi: 10.1137/17m1149225. [Online]. Available at: https://doi.org/10.1137/17m1149225

T. Manteuffel, J. Ruge, and B. Southworth, “Nonsymmetric Algebraic Multigrid Based on Local Approximate Ideal Restriction,” SIAM Journal on Scientific Computing, vol. 40, no. 6, pp. A4105–A4130, 2018.

MeleEtAl2018

V. Mele, E. M. Constantinescu, L. Carracciuolo, and L. D’Amore, “A PETSc parallel-in-time solver based on MGRIT algorithm,” Concurrency and Computation: Practice and Experience, vol. 0, no. 0, p. e4928, 2018, doi: 10.1002/cpe.4928. [Online]. Available at: https://dx.doi.org/10.1002/cpe.4928
BibTeX entry MeleEtAl2018

@article{MeleEtAl2018, author = {Mele, Valeria and Constantinescu, Emil M. and Carracciuolo, Luisa and D'Amore, Luisa}, doi = {10.1002/cpe.4928}, journal = {Concurrency and Computation: Practice and Experience}, number = {0}, pages = {e4928}, title = {{A PETSc parallel-in-time solver based on MGRIT algorithm}}, url = {https://dx.doi.org/10.1002/cpe.4928}, volume = {0}, year = {2018} }
Abstract for BibTeX entry MeleEtAl2018

We address the development of a modular implementation of the MGRIT (MultiGrid-In-Time) algorithm to solve linear and nonlinear systems that arise from the discretization of evolutionary models with a parallel-in-time approach in the context of the PETSc (the Portable, Extensible Toolkit for Scientific computing) library. Our aim is to give the opportunity of predicting the performance gain achievable when using the MGRIT approach instead of the Time Stepping integrator (TS). To this end, we analyze the performance parameters of the algorithm that provide a-priori the best number of processing elements and grid levels to use to address the scaling of MGRIT, regarded as a parallel iterative algorithm proceeding along the time dimension.

V. Mele, D. Romano, E. M. Constantinescu, L. Carracciuolo, and L. D’Amore, “Performance Evaluation for a PETSc Parallel-in-Time Solver Based on the MGRIT Algorithm,” in Euro-Par 2018: Parallel Processing Workshops, Springer International Publishing, 2018, pp. 716–728 [Online]. Available at: http://dx.doi.org/10.1007/978-3-030-10549-5_56

MiaoEtAl2018

Z. Miao, Y.-L. Jiang, and Y.-B. Yang, “Convergence analysis of a parareal-in-time algorithm for the incompressible non-isothermal flows,” International Journal of Computer Mathematics, vol. 0, no. 0, pp. 1–18, 2018, doi: 10.1080/00207160.2018.1498484. [Online]. Available at: https://doi.org/10.1080/00207160.2018.1498484
BibTeX entry MiaoEtAl2018

@article{MiaoEtAl2018, author = {Miao, Zhen and Jiang, Yao-Lin and Yang, Yun-Bo}, doi = {10.1080/00207160.2018.1498484}, journal = {International Journal of Computer Mathematics}, number = {0}, pages = {1--18}, title = {Convergence analysis of a parareal-in-time algorithm for the incompressible non-isothermal flows}, url = {https://doi.org/10.1080/00207160.2018.1498484}, volume = {0}, year = {2018} }
Abstract for BibTeX entry MiaoEtAl2018

This paper presents and analyzes a parareal-in-time scheme for the incompressible non-isothermal Navier–Stokes equations with Boussinesq approximation. Standard finite element method is adopted for the spatial discretization.The proposed algorithm is proved to be unconditional stability. The convergence factor of iteration error for the velocity and temperature is given at time-continuous case. It theoretically demonstrates the superlinearly convergence of the parareal iteration combined with finite element method for incompressible non-isothermal flows. Finally, several numerical experiments that confirm feasibility and applicability of the algorithm perform well as expected.

S. A. Morton, “An Implicit BDF2 Time-Parallel Algorithm for Solving Convection Diffusion Equations,” in 2018 AIAA Aerospace Sciences Meeting, 2018 [Online]. Available at: https://doi.org/10.2514/6.2018-1046

NielsenEtAl2018

A. S. Nielsen, G. Brunner, and J. S. Hesthaven, “Communication-aware adaptive parareal with application to a nonlinear hyperbolic system of partial differential equations,” Journal of Computational Physics, 2018, doi: 10.1016/j.jcp.2018.04.056. [Online]. Available at: https://doi.org/10.1016/j.jcp.2018.04.056
BibTeX entry NielsenEtAl2018

@article{NielsenEtAl2018, author = {Nielsen, Allan S. and Brunner, Gilles and Hesthaven, Jan S.}, doi = {10.1016/j.jcp.2018.04.056}, journal = {Journal of Computational Physics}, title = {Communication-aware adaptive parareal with application to a nonlinear hyperbolic system of partial differential equations}, url = {https://doi.org/10.1016/j.jcp.2018.04.056}, year = {2018} }
Abstract for BibTeX entry NielsenEtAl2018

In the strong scaling limit, the performance of conventional spatial domain decomposition techniques for the parallel solution of PDEs saturates. When sub-domains become small, halo-communication and other overhead come to dominate. A potential path beyond this scaling limit is to introduce domain-decomposition in time, with one such popular approach being the Parareal algorithm which has received a lot of attention due to its generality and potential scalability. Low efficiency, particularly on convection dominated problems, has however limited the adoption of the method. In this paper we demonstrate trough large-scale numerical experiments that it is possible not only to obtain time-parallel speedup on the non-linear shallow water wave equation, but also that we may obtain parallel acceleration beyond what is possible using conventional spatial domain-decomposition techniques alone. Two factors were essential in achieving this. First, for Parareal to converge on the hyperbolic problem we used an approximate Riemann solver as the preconditioner. This preconditioner introduces only dissipative errors with respect to the 3rd order accurate WENO-RK discretization used to solve the PDE system. The preconditoner is comparatively expensive and convergence is slow unless time-subdomains are short. We therefore introduce a new scheduler that we denote Communication Aware Adaptive Parareal (CAAP). CAAP increases obtainable speed-up by minimizing the time-subdomain length without making communication of time-subdomains too costly whilst also adaptively overlapping consecutive cycles of Parareal so to mitigate the impact of a relatively expensive coarse operator.
OngMandal2018

B. W. Ong and B. C. Mandal, “Pipeline implementations of Neumann–Neumann and Dirichlet–Neumann waveform relaxation methods,” Numerical Algorithms, vol. 78, no. 1, pp. 1–20, May 2018, doi: 10.1007/s11075-017-0364-3. [Online]. Available at: https://doi.org/10.1007/s11075-017-0364-3
BibTeX entry OngMandal2018

@article{OngMandal2018, author = {Ong, Benjamin W. and Mandal, Bankim C.}, day = {01}, doi = {10.1007/s11075-017-0364-3}, journal = {Numerical Algorithms}, month = may, number = {1}, pages = {1--20}, title = {{Pipeline implementations of Neumann--Neumann and Dirichlet--Neumann waveform relaxation methods}}, url = {https://doi.org/10.1007/s11075-017-0364-3}, volume = {78}, year = {2018} }
Abstract for BibTeX entry OngMandal2018

This paper is concerned with the reformulation of Neumann–Neumann waveform relaxation (NNWR) methods and Dirichlet–Neumann waveform relaxation (DNWR) methods, a family of parallel space-time approaches to solving time-dependent PDEs. By changing the order of the operations, pipeline-parallel computation of the waveform iterates are possible, without changing the solution of each waveform iterate. The parallel efficiency of the pipeline implementation is analyzed, as well as the change in the communication pattern. Numerical studies are presented to show the effectiveness of the pipeline NNWR and DNWR algorithms.
PagesEtAl2018

G. Pagès, O. Pironneau, and G. Sall, “The Parareal Algorithm for American Options,” SIAM Journal on Financial Mathematics, vol. 9, no. 3, pp. 966–993, 2018, doi: 10.1137/17M1138832. [Online]. Available at: https://doi.org/10.1137/17M1138832
BibTeX entry PagesEtAl2018

@article{PagesEtAl2018, author = {Pagès, G. and Pironneau, O. and Sall, G.}, doi = {10.1137/17M1138832}, journal = {SIAM Journal on Financial Mathematics}, number = {3}, pages = {966--993}, title = {The Parareal Algorithm for American Options}, url = {https://doi.org/10.1137/17M1138832}, volume = {9}, year = {2018} }
Abstract for BibTeX entry PagesEtAl2018

With parallelism in mind we investigate the parareal method for American contracts, both theoretically and numerically. Least-square Monte Carlo (LSMC) and parareal time decomposition with two or more levels are used, leading to an efficient parallel implementation which scales linearly with the number of processors and is appropriate to any multiprocessor-memory architecture in its multilevel version. We prove L^2 superlinear convergence for an LSMC backward in time computation of American contracts, when the conditional expectations are known, i.e., before Monte Carlo discretization. The argument provides also a tool to analyze the multilevel parareal algorithm; in all cases the computing time is increased only by a constant factor, compared to the sequential algorithm on the finest grid, and speedup is guaranteed when the number of processors is larger than that constant. A numerical implementation confirms the theoretical error estimates.
Ruprecht2018

D. Ruprecht, “Wave propagation characteristics of Parareal,” Computing and Visualization in Science, vol. 19, no. 1, pp. 1–17, 2018, doi: 10.1007/s00791-018-0296-z. [Online]. Available at: https://doi.org/10.1007/s00791-018-0296-z
BibTeX entry Ruprecht2018

@article{Ruprecht2018, author = {Ruprecht, D.}, doi = {10.1007/s00791-018-0296-z}, journal = {Computing and Visualization in Science}, number = {1}, pages = {1--17}, title = {Wave propagation characteristics of Parareal}, url = {https://doi.org/10.1007/s00791-018-0296-z}, volume = {19}, year = {2018} }
Abstract for BibTeX entry Ruprecht2018

The paper derives and analyses the (semi-)discrete dispersion relation of the Parareal parallel-in-time integration method. It investigates Parareal’s wave propagation characteristics with the aim to better understand what causes the well documented stability problems for hyperbolic equations. The analysis shows that the instability is caused by convergence of the amplification factor to the exact value from above for medium to high wave numbers. Phase errors in the coarse propagator are identified as the culprit, which suggests that specifically tailored coarse level methods could provide a remedy.
SamaeyEtAl2018

G. Samaey and T. Slawig, “A micro/macro parallel-in-time (parareal) algorithm applied to a climate model with discontinuous non-monotone coefficients and oscillatory forcing,” arXiv:1806.04442 [math.NA], 2018 [Online]. Available at: https://arxiv.org/abs/1806.04442
BibTeX entry SamaeyEtAl2018

@unpublished{SamaeyEtAl2018, author = {Samaey, Giovanni and Slawig, Thomas}, howpublished = {arXiv:1806.04442 [math.NA]}, title = {A micro/macro parallel-in-time (parareal) algorithm applied to a climate model with discontinuous non-monotone coefficients and oscillatory forcing}, url = {https://arxiv.org/abs/1806.04442}, year = {2018} }
Abstract for BibTeX entry SamaeyEtAl2018

We present the application of a micro/macro parareal algorithm for a 1-D energy balance climate model with discontinuous and non-monotone coefficients and forcing terms. The micro/macro parareal method uses a coarse propagator, based on a (macroscopic) 0-D approximation of the underlying (microscopic) 1-D model. We compare the performance of the method using different versions of the macro model, as well as different numerical schemes for the micro propagator, namely an explicit Euler method with constant stepsize and an adaptive library routine. We study convergence of the method and the theoretical gain in computational time in a realization on parallel processors. We show that, in this example and for all settings, the micro/macro parareal method converges in fewer iterations than the number of used parareal subintervals, and that a theoretical gain in performance of up to 10 is possible.
SchmittEtAl2018

A. Schmitt, M. Schreiber, P. Peixoto, and M. Schäfer, “A numerical study of a semi-Lagrangian Parareal method applied to the viscous Burgers equation,” Computing and Visualization in Science, vol. 19, no. 1, pp. 45–57, 2018, doi: 10.1007/s00791-018-0294-1. [Online]. Available at: https://doi.org/10.1007/s00791-018-0294-1
BibTeX entry SchmittEtAl2018

@article{SchmittEtAl2018, author = {Schmitt, A. and Schreiber, M. and Peixoto, P. and Sch{\"a}fer, M.}, doi = {10.1007/s00791-018-0294-1}, journal = {Computing and Visualization in Science}, number = {1}, pages = {45--57}, title = {A numerical study of a semi-Lagrangian Parareal method applied to the viscous Burgers equation}, url = {https://doi.org/10.1007/s00791-018-0294-1}, volume = {19}, year = {2018} }
Abstract for BibTeX entry SchmittEtAl2018

This work focuses on the Parareal parallel-in-time method and its application to the viscous Burgers equation. A crucial component of Parareal is the coarse time stepping scheme, which strongly impacts the convergence of the parallel-in-time method. Three choices of coarse time stepping schemes are investigated in this work: explicit Runge–Kutta, implicit–explicit Runge–Kutta, and implicit Runge–Kutta with semi-Lagrangian advection. Manufactured solutions are used to conduct studies, which provide insight into the viability of each considered time stepping method for the coarse time step of Parareal. One of our main findings is the advantageous convergence behavior of the semi-Lagrangian scheme for advective flows.
SchoepsEtAl2018

S. Schöps, I. Niyonzima, and M. Clemens, “Parallel-In-Time Simulation of Eddy Current Problems Using Parareal,” IEEE Transactions on Magnetics, vol. 54, no. 3, pp. 1–4, 2018, doi: 10.1109/TMAG.2017.2763090. [Online]. Available at: https://dx.doi.org/10.1109/TMAG.2017.2763090
BibTeX entry SchoepsEtAl2018

@article{SchoepsEtAl2018, author = {Schöps, Sebastian and Niyonzima, Innocent and Clemens, Markus}, doi = {10.1109/TMAG.2017.2763090}, journal = {IEEE Transactions on Magnetics}, number = {3}, pages = {1--4}, title = {Parallel-In-Time Simulation of Eddy Current Problems Using Parareal}, url = {https://dx.doi.org/10.1109/TMAG.2017.2763090}, volume = {54}, year = {2018} }
Abstract for BibTeX entry SchoepsEtAl2018

In this paper, the usage of the Parareal method is proposed for the time-parallel solution of the eddy current problem. The method is adapted to the particular challenges of the problem that are related to the differential algebraic character due to non-conducting regions. It is shown how the necessary modification can be automatically incorporated by using a suitable time-stepping method. This paper closes with the first demonstration of a simulation of a realistic four-pole induction machine model using Parareal.
SchreiberEtAl2018

M. Schreiber, P. S. Peixoto, T. Haut, and B. Wingate, “Beyond spatial scalability limitations with a massively parallel method for linear oscillatory problems,” The International Journal of High Performance Computing Applications, vol. 32, no. 6, pp. 913–933, 2018, doi: 10.1177/1094342016687625. [Online]. Available at: https://doi.org/10.1177/1094342016687625
BibTeX entry SchreiberEtAl2018

@article{SchreiberEtAl2018, author = {Schreiber, Martin and Peixoto, Pedro S and Haut, Terry and Wingate, Beth}, doi = {10.1177/1094342016687625}, journal = {The International Journal of High Performance Computing Applications}, number = {6}, pages = {913--933}, title = {Beyond spatial scalability limitations with a massively parallel method for linear oscillatory problems}, url = {https://doi.org/10.1177/1094342016687625}, volume = {32}, year = {2018} }
Abstract for BibTeX entry SchreiberEtAl2018

This paper presents, discusses and analyses a massively parallel-in-time solver for linear oscillatory partial differential equations, which is a key numerical component for evolving weather, ocean, climate and seismic models. The time parallelization in this solver allows us to significantly exceed the computing resources used by parallelization-in-space methods and results in a correspondingly significantly reduced wall-clock time. One of the major difficulties of achieving Exascale performance for weather prediction is that the strong scaling limit – the parallel performance for a fixed problem size with an increasing number of processors – saturates. A main avenue to circumvent this problem is to introduce new numerical techniques that take advantage of time parallelism. In this paper, we use a time-parallel approximation that retains the frequency information of oscillatory problems. This approximation is based on (a) reformulating the original problem into a large set of independent terms and (b) solving each of these terms independently of each other which can now be accomplished on a large number of high-performance computing resources. Our results are conducted on up to 3586 cores for problem sizes with the parallelization-in-space scalability limited already on a single node. We gain significant reductions in the time-to-solution of 118.3× for spectral methods and 1503.0× for finite-difference methods with the parallelization-in-time approach. A developed and calibrated performance model gives the scalability limitations a priori for this new approach and allows us to extrapolate the performance of the method towards large-scale systems. This work has the potential to contribute as a basic building block of parallelization-in-time approaches, with possible major implications in applied areas modelling oscillatory dominated problems.
SchreiberLoft2018

M. Schreiber and R. Loft, “A parallel time integrator for solving the linearized shallow water equations on the rotating sphere,” Numerical Linear Algebra with Applications, 2018, doi: 10.1002/nla.2220. [Online]. Available at: https://doi.org/10.1002/nla.2220
BibTeX entry SchreiberLoft2018

@article{SchreiberLoft2018, author = {Schreiber, M. and Loft, R.}, doi = {10.1002/nla.2220}, journal = {Numerical Linear Algebra with Applications}, title = {A parallel time integrator for solving the linearized shallow water equations on the rotating sphere}, url = {https://doi.org/10.1002/nla.2220}, year = {2018} }
Abstract for BibTeX entry SchreiberLoft2018

With the stagnation of processor core performance, further reductions in the time to solution for geophysical fluid problems are becoming increasingly difficult with standard time integrators. Parallel‐in‐time exposes and exploits additional parallelism in the time dimension, which is inherently sequential in traditional methods. The rational approximation of exponential integrators (REXI) method allows taking arbitrarily long time steps based on a sum over a number of decoupled complex PDEs that can be solved independently massively parallel. Hence, REXI is assumed to be well suited for modern massively parallel super computers, which are currently trending. To date, the study and development of the REXI approach have been limited to linearized problems on the periodic two‐dimensional plane. This work extends the REXI time stepping method to the linear shallow‐water equations on the rotating sphere, thus moving the method one step closer to solving fully nonlinear fluid problems of geophysical interest on the sphere. The rotating sphere poses particular challenges for finding an efficient solver due to the zonal dependence of the Coriolis term. Here, we present an efficient REXI solver based on spherical harmonics, showing the results of a geostrophic balance test, a comparison with alternative time stepping methods, an analysis of dispersion relations indicating superior properties of REXI, and finally, a performance comparison on the Cheyenne supercomputer. Our results indicate that REXI not only can take larger time steps but also can be used to gain higher accuracy and significantly reduced time to solution compared with currently existing time stepping methods.

J. B. Schroder, R. D. Falgout, C. S. Woodward, P. Top, and M. Lecouvez, “Parallel-in-Time Solution of Power Systems with Scheduled Events,” in 2018 IEEE Power & Energy Society General Meeting (PESGM), 2018, pp. 1–5.

Speck2018

R. Speck, “Parallelizing spectral deferred corrections across the method,” Computing and Visualization in Science, 2018, doi: 10.1007/s00791-018-0298-x. [Online]. Available at: https://doi.org/10.1007/s00791-018-0298-x
BibTeX entry Speck2018

@article{Speck2018, author = {Speck, Robert}, doi = {10.1007/s00791-018-0298-x}, journal = {Computing and Visualization in Science}, title = {Parallelizing spectral deferred corrections across the method}, url = {https://doi.org/10.1007/s00791-018-0298-x}, year = {2018} }
Abstract for BibTeX entry Speck2018

In this paper we present two strategies to enable "parallelization across the method" for spectral deferred corrections (SDC). Using standard low-order time-stepping methods in an iterative fashion, SDC can be seen as preconditioned Picard iteration for the collocation problem. Typically, a serial Gauss-Seidel-like preconditioner is used, computing updates for each collocation node one by one. The goal of this paper is to show how this process can be parallelized, so that all collocation nodes are updated simultaneously. The first strategy aims at finding parallel preconditioners for the Picard iteration and we test three choices using four different test problems. For the second strategy we diagonalize the quadrature matrix of the collocation problem directly. In order to integrate non-linear problems we employ simplified and inexact Newton methods. Here, we estimate the speed of convergence depending on the time-step size and verify our results using a non-linear diffusion problem.
Subber2018

W. Subber and A. Sarkar, “A Parallel Time Integrator for Noisy Nonlinear Oscillatory Systems,” Journal of Computational Physics, 2018, doi: 10.1016/j.jcp.2018.01.019. [Online]. Available at: https://doi.org/10.1016/j.jcp.2018.01.019
BibTeX entry Subber2018

@article{Subber2018, author = {Subber, Waad and Sarkar, Abhijit}, doi = {10.1016/j.jcp.2018.01.019}, journal = {Journal of Computational Physics}, title = {A Parallel Time Integrator for Noisy Nonlinear Oscillatory Systems}, url = {https://doi.org/10.1016/j.jcp.2018.01.019}, year = {2018} }
Abstract for BibTeX entry Subber2018

In this paper, we adapt a parallel time integration scheme to track the trajectories of noisy non-linear dynamical systems. Specifically, we formulate a parallel algorithm to generate the sample path of nonlinear oscillator defined by stochastic differential equations (SDEs) using the so-called parareal method for ordinary differential equations (ODEs). The presence of Wiener process in {SDEs} causes difficulties in the direct application of any numerical integration techniques of {ODEs} including the parareal algorithm. The parallel implementation of the algorithm involves two {SDEs} solvers, namely a fine-level scheme to integrate the system in parallel and a coarse-level scheme to generate and correct the required initial conditions to start the fine-level integrators. For the numerical illustration, a randomly excited Duffing oscillator is investigated in order to study the performance of the stochastic parallel algorithm with respect to a range of system parameters. The distributed implementation of the algorithm exploits Massage Passing Interface (MPI).

A. T. Weaver, S. Gürol, J. Tshimanga, M. Chrust, and A. Piacentini, “‘Time’-Parallel diffusion-based correlation operators,” Quarterly Journal of the Royal Meteorological Society, vol. 144, no. 716, pp. 2067–2088, Oct. 2018, doi: 10.1002/qj.3302. [Online]. Available at: https://doi.org/10.1002/qj.3302

Wu2018

S. Wu, “Toward Parallel Coarse Grid Correction for the Parareal Algorithm,” SIAM Journal on Scientific Computing, vol. 40, no. 3, pp. A1446–A1472, 2018, doi: 10.1137/17M1141102. [Online]. Available at: https://doi.org/10.1137/17M1141102
BibTeX entry Wu2018

@article{Wu2018, author = {Wu, S.}, doi = {10.1137/17M1141102}, journal = {SIAM Journal on Scientific Computing}, number = {3}, pages = {A1446--A1472}, title = {Toward Parallel Coarse Grid Correction for the Parareal Algorithm}, url = {https://doi.org/10.1137/17M1141102}, volume = {40}, year = {2018} }
Abstract for BibTeX entry Wu2018

In this paper, we present an idea toward parallel coarse grid correction (CGC) for the parareal algorithm. It is well known that such a CGC procedure is often the bottleneck of speedup of the parareal algorithm. For an ODE system with initial-value condition u(0)=u_0 the idea can be explained as follows. First, we apply the \mathcalG-propagator to the same ODE system but with a special condition u(0)=αu(T), where α∈\mathbbR is a crux parameter. Second, in each iteration of the parareal algorithm the CGC procedure will be carried out by the so-called diagonalization technique established recently. The parameter αcontrols both the roundoff error arising from such a diagonalization technique and the convergence rate of the resulting parareal algorithm. We show that there exists some threshold α^* such that the parareal algorithm with diagonalization-based CGC possesses the same convergence rate as that of the parareal algorithm with classical CGC if |α|≤α^*. With |α|=α^*, we show that the condition number associated with the diagonalization technique is a moderate quantity of order \mathcalO(1) (and therefore the roundoff error is small) and is independent of the length of the time interval. Numerical results are given to support our findings.
WuZhou2018_JCP

S.-L. Wu and T. Zhou, “Parareal algorithms with local time-integrators for time fractional differential equations,” Journal of Computational Physics, vol. 358, pp. 135–149, 2018, doi: 10.1016/j.jcp.2017.12.029. [Online]. Available at: https://doi.org/10.1016/j.jcp.2017.12.029
BibTeX entry WuZhou2018_JCP

@article{WuZhou2018_JCP, author = {Wu, Shu-Lin and Zhou, Tao}, doi = {10.1016/j.jcp.2017.12.029}, journal = {Journal of Computational Physics}, pages = {135--149}, title = {Parareal algorithms with local time-integrators for time fractional differential equations}, url = {https://doi.org/10.1016/j.jcp.2017.12.029}, volume = {358}, year = {2018} }
Abstract for BibTeX entry WuZhou2018_JCP

It is challenge work to design parareal algorithms for time-fractional differential equations due to the historical effect of the fractional operator. A direct extension of the classical parareal methed to such equations will lead to unbalance computational time in each process. In this work, we present an efficient parareal iteration scheme to overcome this issue, by adopting two recently developed local time-integrators for time fractional operators. In both approaches, one introduces auxiliary variables to localized the fractional operator. To this end, we propose a new strategy to perform the coarse grid correction so that the auxiliary variables and the solution variable are corrected separately in a mixed pattern. It is shown that the proposed parareal algorithm admits robust rate of convergence. Numerical examples are presented to support our conclusions.
YallaEnquist2018

G. R. Yalla and B. Engquist, “Parallel in Time Algorithms for Multiscale Dynamical Systems Using Interpolation and Neural Networks,” in Proceedings of the High Performance Computing Symposium, 2018, pp. 9:1–9:12 [Online]. Available at: http://dl.acm.org/citation.cfm?id=3213069.3213078
BibTeX entry YallaEnquist2018

@inproceedings{YallaEnquist2018, articleno = {9}, author = {Yalla, Gopal R. and Engquist, Bjorn}, booktitle = {Proceedings of the High Performance Computing Symposium}, pages = {9:1--9:12}, publisher = {Society for Computer Simulation International}, series = {HPC '18}, title = {Parallel in Time Algorithms for Multiscale Dynamical Systems Using Interpolation and Neural Networks}, url = {http://dl.acm.org/citation.cfm?id=3213069.3213078}, year = {2018} }
Abstract for BibTeX entry YallaEnquist2018

The parareal algorithm allows for efficient parallel in time computation of dynamical systems. We present a novel coarse scale solver to be used in the parareal framework. The coarse scale solver can be defined through interpolation or as the output of a neural network, and accounts for slow scale motion in the system. Through a parareal scheme, we pair this coarse solver with a fine scale solver that corrects for fast scale motion. By doing so we are able to achieve the accuracy of the fine solver at the efficiency of the coarse solver. Successful tests for smaller but challenging problems are presented, which cover both highly oscillatory solutions and problems with strong forces localized in time. The results suggest significant speed up can be gained for multiscale problems when using a parareal scheme with this new coarse solver as opposed to the traditional parareal setup.
YueEtAl2018

X. Q. Yue, S. Shu, X. W. Xu, W. P. Bu, and K. J. Pan, “Parallel-in-Time with Fully Finite Element Multigrid for 2-D Space-fractional Diffusion Equations,” arXiv:1805.06688 [math.NA], 2018 [Online]. Available at: https://arxiv.org/abs/1805.06688v1
BibTeX entry YueEtAl2018

@unpublished{YueEtAl2018, author = {Yue, X.~Q. and Shu, S. and Xu, X.~W. and Bu, W.~P. and Pan, K.~J.}, howpublished = {arXiv:1805.06688 [math.NA]}, title = {Parallel-in-Time with Fully Finite Element Multigrid for 2-D Space-fractional Diffusion Equations}, url = {https://arxiv.org/abs/1805.06688v1}, year = {2018} }
Abstract for BibTeX entry YueEtAl2018

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions. We firstly obtain a fully discrete scheme via using the linear finite element method to discretize spatial and temporal derivatives to propagate solutions. Next, we present a non-intrusive time-parallelization and its two-level convergence analysis, where we algorithmically and theoretically generalize the MGRIT to time-dependent fine time-grid propagators. Finally, numerical illustrations show that the obtained numerical scheme possesses the saturation error order, theoretical results of the two-level variant deliver good predictions, and significant speedups can be achieved when compared to parareal and the sequential time-stepping approach.
ZhuWeng2018

S. Zhu and S. Weng, “A parallel spectral deferred correction method for first-order evolution problems,” BIT Numerical Mathematics, pp. 1–28, 2018, doi: 10.1007/s10543-018-0702-4. [Online]. Available at: https://doi.org/10.1007/s10543-018-0702-4
BibTeX entry ZhuWeng2018

@article{ZhuWeng2018, author = {Zhu, Shuai and Weng, Shilie}, doi = {10.1007/s10543-018-0702-4}, journal = {BIT Numerical Mathematics}, pages = {1--28}, title = {A parallel spectral deferred correction method for first-order evolution problems}, url = {https://doi.org/10.1007/s10543-018-0702-4}, year = {2018} }
Abstract for BibTeX entry ZhuWeng2018

This paper investigates a novel parallel technique based on the spectral deferred correction (SDC) method and a compensation step for solving first-order evolution problems, and we call it para-SDC method for convenience. The standard SDC method is used in parallel with a rough initial guess and a Picard integral equation with high precision initial condition is acted as a compensator. The goal of this paper is to show how these processes can be parallelized and how to improve the efficiency. During the SDC step an implicit or semi-implicit method can be used for stiff problems which is always time-consuming, therefore that’s why we do this procedure in parallel. Due to a better initial condition of parallel intervals after the SDC step, the goal of compensation step is to get a better approximation and also avoid of solving an implicit problem again. During the compensation step an explicit Picard scheme is taken based on the numerical integration with polynomial interpolation on Gauss Radau II nodes, which is almost no time consumption, obviously, that’s why we do this procedure in serial. The convergency analysis and the parallel efficiency of our method are also discussed. Several numerical experiments and an application for simulation Allen–Cahn equation are presented to show the accuracy, stability, convergence order and efficiency features of para-SDC method.

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2017

ArielEtAl2017

G. Ariel, H. Nguyen, and R. Tsai, “θ-parareal schemes,” arXiv:1704.06882 [math.NA], 2017 [Online]. Available at: https://arxiv.org/abs/1704.06882
BibTeX entry ArielEtAl2017

@unpublished{ArielEtAl2017, author = {Ariel, Gil and Nguyen, Hieu and Tsai, Richard}, howpublished = {arXiv:1704.06882 [math.NA]}, title = {$\theta$-parareal schemes}, url = {https://arxiv.org/abs/1704.06882}, year = {2017} }
Abstract for BibTeX entry ArielEtAl2017

A weighted version of the parareal method for parallel-in-time computation of time dependent problems is presented. Linear stability analysis for a scalar weighing strategy shows that the new scheme may enjoy favorable stability properties with marginal reduction in accuracy at worse. More complicated matrix-valued weights are applied in numerical examples. The weights are optimized using information from past iterations, providing a systematic framework for using the parareal iterations as an approach to multiscale coupling. The advantage of the method is demonstrated using numerical examples, including some well-studied nonlinear Hamiltonian systems.
BadiaEtAl2017

S. Badia and M. Olm, “Space-Time Balancing Domain Decomposition,” SIAM Journal on Scientific Computing, vol. 39, no. 2, pp. C194–C213, 2017, doi: 10.1137/16M1074266. [Online]. Available at: https://doi.org/10.1137/16M1074266
BibTeX entry BadiaEtAl2017

@article{BadiaEtAl2017, author = {Badia, Santiago and Olm, Marc}, doi = {10.1137/16M1074266}, journal = {SIAM Journal on Scientific Computing}, number = {2}, pages = {C194--C213}, title = {Space-Time Balancing Domain Decomposition}, url = {https://doi.org/10.1137/16M1074266}, volume = {39}, year = {2017} }
Abstract for BibTeX entry BadiaEtAl2017

In this work, we propose two-level space-time domain decomposition preconditioners for parabolic problems discretized using finite elements. They are motivated as an extension to space-time of balancing domain decomposition by constraints preconditioners. The key ingredients to be defined are the subassembled space and operator, the coarse degrees of freedom (DOFs) in which we want to enforce continuity among subdomains at the preconditioner level, and the transfer operator from the subassembled to the original finite element space. With regard to the subassembled operator, a perturbation of the time derivative is needed to end up with a well-posed preconditioner. The set of coarse DOFs includes the time average (at the space-time subdomain) of classical space constraints plus new constraints between consecutive subdomains in time. Numerical experiments show that the proposed schemes are weakly scalable in time, i.e., we can efficiently exploit increasing computational resources to solve more time steps in the same total elapsed time. Further, the scheme is also weakly space-time scalable, since it leads to asymptotically constant iterations when solving larger problems both in space and time. Excellent wall clock time weak scalability is achieved for space-time parallel solvers on some thousands of cores.
BelliveauHaber2017

P. Belliveau and E. Haber, “Coupled simulation of electromagnetic induction and IP effects using stretched exponential relaxation,” Geophysics, pp. 1–61, 2017, doi: 10.1190/geo2017-0494.1. [Online]. Available at: https://doi.org/10.1190/geo2017-0494.1
BibTeX entry BelliveauHaber2017

@article{BelliveauHaber2017, author = {Belliveau, Patrick and Haber, Eldad}, doi = {10.1190/geo2017-0494.1}, journal = {Geophysics}, pages = {1-–61}, title = {Coupled simulation of electromagnetic induction and {IP} effects using stretched exponential relaxation}, url = {https://doi.org/10.1190/geo2017-0494.1}, year = {2017} }
Abstract for BibTeX entry BelliveauHaber2017

We have developed a new algorithm for three-dimensional time-domain electromagnetic (EM) modelling, taking full account of induced polarization (IP) and the coupling between EM and IP effects. The algorithm can be used to model grounded source IP surveys that show EM induction effects as well as airborne time-domain electromagnetic surveys that exhibit IP effects. IP effects are most often approximated as static or modelled in the frequency domain, using frequency dependent electrical conductivity. It is difficult to translate the frequency dependent conductivity approach directly to the time domain in a computationally efficient manner. We take an alternative approach in which we model IP relaxations in time using the stretched exponential function. We incorporate this IP model into a direct time-stepping discretization of the quasi-static time-domain Maxwell equations. We show that modelling of IP effects with this stretched exponential approach is asymptotically equivalent to the commonly used Cole-Cole model of IP transformed to the time-domain. We have implemented our algorithm using efficient numerical methods that allow it to tackle large scale problems and are amenable to use in inversion. In particular we have developed a parallel time-stepping technique that allows us to compute transient electric fields at multiple time-steps simultaneously. We demonstrate the behaviour of the stretched exponential model of IP decay as well as the efficiency of our algorithm by applying it to synthetic numerical examples that simulate a grounded source IP survey with significant EM effects and a concentric-loop airborne EM sounding over a chargeable body.

E. Blayo, A. Rousseau, and M. Tayachi, “Boundary conditions and Schwarz waveform relaxation method for linear viscous Shallow Water equations in hydrodynamics,” The SMAI journal of computational mathematics, vol. 3, pp. 117–137, 2017, doi: 10.5802/smai-jcm.22. [Online]. Available at: https://smai-jcm.centre-mersenne.org/item/SMAI-JCM_2017__3__117_0

BoltenEtAl2017

M. Bolten, D. Moser, and R. Speck, “A multigrid perspective on the parallel full approximation scheme in space and time,” Numerical Linear Algebra with Applications, vol. 24, no. 6, p. e2110, 2017, doi: 10.1002/nla.2110. [Online]. Available at: https://dx.doi.org/10.1002/nla.2110
BibTeX entry BoltenEtAl2017

@article{BoltenEtAl2017, author = {Bolten, Matthias and Moser, Dieter and Speck, Robert}, doi = {10.1002/nla.2110}, journal = {Numerical Linear Algebra with Applications}, number = {6}, pages = {e2110}, title = {A multigrid perspective on the parallel full approximation scheme in space and time}, url = {https://dx.doi.org/10.1002/nla.2110}, volume = {24}, year = {2017} }
Abstract for BibTeX entry BoltenEtAl2017

For the numerical solution of time-dependent partial differential equations, time-parallel methods have recently been shown to provide a promising way to extend prevailing strong-scaling limits of numerical codes. One of the most complex methods in this field is the “Parallel Full Approximation Scheme in Space and Time” (PFASST). PFASST already shows promising results for many use cases and benchmarks. However, a solid and reliable mathematical foundation is still missing. We show that, under certain assumptions, the PFASST algorithm can be conveniently and rigorously described as a multigrid-in-time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using blockwise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type.
DobrevEtAl2017

V. A. Dobrev, T. Kolev, N. A. Petersson, and J. B. Schroder, “Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT),” SIAM Journal on Scientific Computing, vol. 39, no. 5, pp. S501–S527, 2017, doi: 10.1137/16M1074096. [Online]. Available at: https://doi.org/10.1137/16M1074096
BibTeX entry DobrevEtAl2017

@article{DobrevEtAl2017, author = {Dobrev, V.~A. and Kolev, Tz. and Petersson, N.~A. and Schroder, J.~B.}, doi = {10.1137/16M1074096}, journal = {SIAM Journal on Scientific Computing}, number = {5}, pages = {S501--S527}, title = {Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT)}, url = {https://doi.org/10.1137/16M1074096}, volume = {39}, year = {2017} }
Abstract for BibTeX entry DobrevEtAl2017

In this paper we develop a two-grid convergence theory for the parallel-in-time scheme known as multigrid reduction in time (MGRIT), as it is implemented in the open-source package [XBraid: Parallel Multigrid in Time, http://llnl.gov/casc/xbraid]. MGRIT is a scalable and multilevel approach to parallel-in-time simulations that nonintrusively uses existing time-stepping schemes, and in a specific two-level setting it is equivalent to the widely known parareal algorithm. The goal of this paper is twofold. First, we present a two-level MGRIT convergence analysis for linear problems where the spatial discretization matrix can be diagonalized, and then apply this analysis to our two basic model problems, the heat equation and the advection equation. One important assumption is that the coarse and fine time-grid propagators can be diagaonalized by the same set of eigenvectors, which is often the case when the same spatial discretization operator is used on the coarse and fine time grids. In many cases, the MGRIT algorithm is guaranteed to converge, and we demonstrate numerically that the theoretically predicted convergence rates are sharp in practice for our model problems. Second, we explore how the convergence of MGRIT compares to the stability of the chosen time-stepping scheme. In particular, we demonstrate that a stable time-stepping scheme does not necessarily imply convergence of MGRIT, although MGRIT with FCF-relaxation always converges for the diffusion dominated problems considered here.
FalgoutEtAl2017

R. D. Falgout, T. A. Manteuffel, B. O’Neill, and J. B. Schroder, “Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study,” SIAM Journal on Scientific Computing, vol. 39, no. 5, pp. S298–S322, 2017, doi: 10.1137/16M1082330. [Online]. Available at: https://doi.org/10.1137/16M1082330
BibTeX entry FalgoutEtAl2017

@article{FalgoutEtAl2017, author = {Falgout, R.~D. and Manteuffel, T.~A. and O'Neill, B. and Schroder, J.~B.}, doi = {10.1137/16M1082330}, journal = {SIAM Journal on Scientific Computing}, number = {5}, pages = {S298--S322}, title = {Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study}, url = {https://doi.org/10.1137/16M1082330}, volume = {39}, year = {2017} }
Abstract for BibTeX entry FalgoutEtAl2017

The need for parallelism in the time dimension is being driven by changes in computer architectures, where performance increases are now provided through greater concurrency, not faster clock speeds. This creates a bottleneck for sequential time marching schemes because they lack parallelism in the time dimension. Multigrid reduction in time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficiency is defined as achieving similar performance when compared to an equivalent linear problem. The benchmark nonlinear problem is the p-Laplacian, where p=4 corresponds to a well-known nonlinear diffusion equation and p=2 corresponds to the standard linear diffusion operator, our benchmark linear problem. The key difficulty encountered is that the nonlinear time-step solver becomes progressively more expensive on coarser time levels as the time-step size increases. To overcome such difficulties, multigrid research has historically targeted an accumulated body of experience regarding how to choose an appropriate solver for a specific problem type. To that end, this paper develops a library of MGRIT optimizations and modifications, most important an alternate initial guess for the nonlinear time-step solver and delayed spatial coarsening, that will allow many nonlinear parabolic problems to be solved with parallel scaling behavior comparable to the corresponding linear problem.
FalgoutEtAl2017b

R. D. Falgout, S. Friedhoff, T. V. Kolev, S. P. MacLachlan, J. B. Schroder, and S. Vandewalle, “Multigrid methods with space–time concurrency,” Computing and Visualization in Science, vol. 18, no. 4, pp. 123–143, 2017, doi: 10.1007/s00791-017-0283-9. [Online]. Available at: https://doi.org/10.1007/s00791-017-0283-9
BibTeX entry FalgoutEtAl2017b

@article{FalgoutEtAl2017b, author = {Falgout, R.~D. and Friedhoff, S. and Kolev, Tz.~V. and MacLachlan, S. P. and Schroder, J.~B. and Vandewalle, S.}, doi = {10.1007/s00791-017-0283-9}, journal = {Computing and Visualization in Science}, number = {4}, pages = {123--143}, title = {Multigrid methods with space--time concurrency}, url = {https://doi.org/10.1007/s00791-017-0283-9}, volume = {18}, year = {2017} }
Abstract for BibTeX entry FalgoutEtAl2017b

We consider the comparison of multigrid methods for parabolic partial differential equations that allow space–time concurrency. With current trends in computer architectures leading towards systems with more, but not faster, processors, space–time concurrency is crucial for speeding up time-integration simulations. In contrast, traditional time-integration techniques impose serious limitations on parallel performance due to the sequential nature of the time-stepping approach, allowing spatial concurrency only. This paper considers the three basic options of multigrid algorithms on space–time grids that allow parallelism in space and time: coarsening in space and time, semicoarsening in the spatial dimensions, and semicoarsening in the temporal dimension. We develop parallel software and performance models to study the three methods at scales of up to 16K cores and introduce an extension of one of them for handling multistep time integration. We then discuss advantages and disadvantages of the different approaches and their benefit compared to traditional space-parallel algorithms with sequential time stepping on modern architectures.
GanderHalpern2017

M. J. Gander and L. Halpern, “Time Parallelization for Nonlinear Problems Based on Diagonalization,” in Domain Decomposition Methods in Science and Engineering XXIII, 2017, pp. 163–170, doi: 10.1007/978-3-319-52389-7_15 [Online]. Available at: https://doi.org/10.1007/978-3-319-52389-7_15
BibTeX entry GanderHalpern2017

@inproceedings{GanderHalpern2017, author = {Gander, Martin J. and Halpern, Laurence}, booktitle = {Domain Decomposition Methods in Science and Engineering {XXIII}}, doi = {10.1007/978-3-319-52389-7_15}, editor = {Lee, Chang-Ock and Cai, Xiao-Chuan and Keyes, David E. and Kim, Hyea Hyun and Klawonn, Axel and Park, Eun-Jae and Widlund, Olof B.}, pages = {163--170}, publisher = {Springer International Publishing}, title = {Time Parallelization for Nonlinear Problems Based on Diagonalization}, url = {https://doi.org/10.1007/978-3-319-52389-7_15}, year = {2017} }
Abstract for BibTeX entry GanderHalpern2017

The direct time parallelization method based on diagonalization is only applicable to linear problems. We propose here a new method based on diagonalization which permits the direct parallelization in time of a Newton iteration that works simultaneously over several time steps. We first explain the method for a scalar model problem, and then give a formulation for a nonlinear partial differential equation based on tensorization. We illustrate the methods with numerical experiments.
GasparRodrigo2017

F. J. Gaspar and C. Rodrigo, “Multigrid Waveform Relaxation for the Time-Fractional Heat Equation,” SIAM Journal on Scientific Computing, vol. 39, no. 4, pp. A1201–A1224, 2017, doi: 10.1137/16M1090193. [Online]. Available at: https://doi.org/10.1137/16M1090193
BibTeX entry GasparRodrigo2017

@article{GasparRodrigo2017, author = {Gaspar, Francisco J. and Rodrigo, Carmen}, doi = {10.1137/16M1090193}, journal = {SIAM Journal on Scientific Computing}, number = {4}, pages = {A1201--A1224}, title = {Multigrid Waveform Relaxation for the Time-Fractional Heat Equation}, url = {https://doi.org/10.1137/16M1090193}, volume = {39}, year = {2017} }
Abstract for BibTeX entry GasparRodrigo2017

In this work, we propose an efficient and robust multigrid method for solving the time-fractional heat equation. Due to the nonlocal property of fractional differential operators, numerical methods usually generate systems of equations for which the coefficient matrix is dense. Therefore, the design of efficient solvers for the numerical simulation of these problems is a difficult task. We develop a parallel-in-time multigrid algorithm based on the waveform relaxation approach, whose application to time-fractional problems seems very natural due to the fact that the fractional derivative at each spatial point depends on the values of the function at this point at all earlier times. Exploiting the Toeplitz-like structure of the coefficient matrix, the proposed multigrid waveform relaxation method has a computational cost of O(NM\log(M)) operations, where M is the number of time steps and N is the number of spatial grid points. A semialgebraic mode analysis is also developed to theoretically confirm the good results obtained. Several numerical experiments, including examples with nonsmooth solutions and a nonlinear problem with applications in porous media, are presented.
HanEtAl2017

S. Han and O. A. Bauchau, “Parallel Time-Integration of Flexible Multibody Dynamics Based on Newton-Waveform Method,” in 13th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, 2017, vol. 6, doi: 10.1115/DETC2017-68232 [Online]. Available at: https://dx.doi.org/10.1115/DETC2017-68232
BibTeX entry HanEtAl2017

@inproceedings{HanEtAl2017, author = {Han, Shilei and Bauchau, Olivier A.}, booktitle = {13th International Conference on Multibody Systems, Nonlinear Dynamics, and Control}, doi = {10.1115/DETC2017-68232}, title = {Parallel Time-Integration of Flexible Multibody Dynamics Based on Newton-Waveform Method}, url = {https://dx.doi.org/10.1115/DETC2017-68232}, volume = {6}, year = {2017} }
Abstract for BibTeX entry HanEtAl2017

Traditionally, the time integration algorithms for multibody dynamics are in sequential. The predictions of previous time steps are necessary to get the solutions at current time step. This time-marching character impedes the application of parallel processor implementation. In this paper, the idea of computing a number of time steps concurrently is applied to flexible multi-body dynamics, which makes parallel time-integration possible. In the present method, the solution at the current time step is computed before accurate values at previous time step are available. This method is suitable for small-scale parallel analysis of flexible multibody systems.

A. J. M. Howse, “Nonlinear Preconditioning Methods for Optimization and Parallel-In-Time Methods for 1D Scalar Hyperbolic Partial Differential Equations,” PhD thesis, UWSpace, 2017 [Online]. Available at: http://hdl.handle.net/10012/12559

J. Jansson and J. Hoffman, “Direct FEM parallel-in-time computation of turbulent flow,” 2017 [Online]. Available at: http://www.csc.kth.se/ jjan/publications/pit_preprint_2017-08-09.pdf

KooijEtAl2017

G. L. Kooij, M. A. Botchev, and B. J. Geurts, “A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations,” Journal of Computational and Applied Mathematics, vol. 316, pp. 229–246, 2017, doi: 10.1016/j.cam.2016.09.036. [Online]. Available at: http://dx.doi.org/10.1016/j.cam.2016.09.036
BibTeX entry KooijEtAl2017

@article{KooijEtAl2017, author = {Kooij, G.L. and Botchev, M.A. and Geurts, B.J.}, doi = {10.1016/j.cam.2016.09.036}, journal = {Journal of Computational and Applied Mathematics}, note = {Selected Papers from NUMDIFF-14}, pages = {229--246}, title = {A block Krylov subspace implementation of the time-parallel Paraexp method and its extension for nonlinear partial differential equations}, url = {http://dx.doi.org/10.1016/j.cam.2016.09.036}, volume = {316}, year = {2017} }
Abstract for BibTeX entry KooijEtAl2017

A parallel time integration method for nonlinear partial differential equations is proposed. It is based on a new implementation of the Paraexp method for linear partial differential equations (PDEs) employing a block Krylov subspace method. For nonlinear {PDEs} the algorithm is based on our Paraexp implementation within a waveform relaxation. The initial value problem is solved iteratively on a complete time interval. Nonlinear terms are treated as source terms, provided by the solution from the previous iteration. At each iteration, the problem is decoupled into independent subproblems by the principle of superposition. The decoupled subproblems are solved fast by exponential integration, based on a block Krylov method. The new time integration is demonstrated for the one-dimensional advection–diffusion equation and the viscous Burgers equation. Numerical experiments confirm excellent parallel scaling for the linear advection–diffusion problem, and good scaling in case the nonlinear Burgers equation is simulated.
KreienbuehlEtAl2017

A. Kreienbuehl, P. Benedusi, D. Ruprecht, and R. Krause, “Time-parallel gravitational collapse simulation,” Communications in Applied Mathematics and Computational Science, vol. 12, no. 1, pp. 109–128, 2017, doi: 10.2140/camcos.2017.12.109. [Online]. Available at: http://dx.doi.org/10.2140/camcos.2017.12.109
BibTeX entry KreienbuehlEtAl2017

@article{KreienbuehlEtAl2017, author = {Kreienbuehl, Andreas and Benedusi, Pietro and Ruprecht, Daniel and Krause, Rolf}, doi = {10.2140/camcos.2017.12.109}, issue = {1}, journal = {Communications in Applied Mathematics and Computational Science}, pages = {109--128}, title = {Time-parallel gravitational collapse simulation}, url = {http://dx.doi.org/10.2140/camcos.2017.12.109}, volume = {12}, year = {2017} }
Abstract for BibTeX entry KreienbuehlEtAl2017

This article demonstrates the applicability of the parallel-in-time method Parareal to the numerical solution of the Einstein gravity equations for the spherical collapse of a massless scalar field. To account for the shrinking of the spatial domain in time, a tailored load balancing scheme is proposed and compared to load balancing based on number of time steps alone. The performance of Parareal is studied for both the sub-critical and black hole case; our experiments show that Parareal generates substantial speedup and, in the super-critical regime, can reproduce Choptuik’s black hole mass scaling law.
MasthayEtAl2017

T. M. Masthay and S. Perugini, “Parareal Algorithm Implementation and Simulation in Julia,” arXiv:1706.08569v1 [cs.MS], 2017 [Online]. Available at: https://arxiv.org/pdf/1706.08569.pdf
BibTeX entry MasthayEtAl2017

@unpublished{MasthayEtAl2017, author = {Masthay, Tyler M. and Perugini, Saverio}, howpublished = {arXiv:1706.08569v1 [cs.MS]}, title = {Parareal Algorithm Implementation and Simulation in Julia}, url = {https://arxiv.org/pdf/1706.08569.pdf}, year = {2017} }
Abstract for BibTeX entry MasthayEtAl2017

We present a full implementation of the parareal algorithm—an integration technique to solve di fferential equations in parallel— in the Julia programming language for a fully general, first-order, initial-value problem. Our implementation accepts both coarse and fine integrators as functional arguments. We use Euler’s method and another Runge-Kutta integration technique as the integrators in our experiments. We also present a simulation of the algorithm for purposes of pedagogy.
MerkelEtAl2017

M. Merkel, I. Niyonzima, and S. Schöps, “ParaExp Using Leapfrog as Integrator for High-Frequency Electromagnetic Simulations,” Radio Science, vol. 52, no. 12, pp. 1558–1569, 2017, doi: 10.1002/2017RS006357. [Online]. Available at: https://dx.doi.org/10.1002/2017RS006357
BibTeX entry MerkelEtAl2017

@article{MerkelEtAl2017, author = {Merkel, Melina and Niyonzima, Innocent and Schöps, Sebastian}, doi = {10.1002/2017RS006357}, journal = {Radio Science}, number = {12}, pages = {1558--1569}, title = {ParaExp Using Leapfrog as Integrator for High-Frequency Electromagnetic Simulations}, url = {https://dx.doi.org/10.1002/2017RS006357}, volume = {52}, year = {2017} }
Abstract for BibTeX entry MerkelEtAl2017

Abstract Recently, ParaExp was proposed for the time integration of linear hyperbolic problems. It splits the time interval of interest into subintervals and computes the solution on each subinterval in parallel. The overall solution is decomposed into a particular solution defined on each subinterval with zero initial conditions and a homogeneous solution propagated by the matrix exponential applied to the initial conditions. The efficiency of the method depends on fast approximations of this matrix exponential based on recent results from numerical linear algebra. This paper deals with the application of ParaExp in combination with Leapfrog to electromagnetic wave problems in time domain. Numerical tests are carried out for a simple toy problem and a realistic spiral inductor model discretized by the Finite Integration Technique.
Pazner2017700

W. Pazner and P.-O. Persson, “Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations,” Journal of Computational Physics, vol. 335, pp. 700–717, 2017, doi: 10.1016/j.jcp.2017.01.050. [Online]. Available at: https://doi.org/10.1016/j.jcp.2017.01.050
BibTeX entry Pazner2017700

@article{Pazner2017700, author = {Pazner, Will and Persson, Per-Olof}, doi = {10.1016/j.jcp.2017.01.050}, journal = {Journal of Computational Physics}, pages = {700--717}, title = {{Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations}}, url = {https://doi.org/10.1016/j.jcp.2017.01.050}, volume = {335}, year = {2017} }
Abstract for BibTeX entry Pazner2017700

Abstract In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge–Kutta methods. This method makes use of the iterative preconditioned {GMRES} algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge–Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge–Kutta stages simultaneously. Numerical experiments are performed on the Navier–Stokes equations using Euler vortex and 2D and 3D {NACA} airfoil test cases in serial and in parallel settings. The fully implicit Radau {IIA} Runge–Kutta methods compare favorably with equal-order {DIRK} methods in terms of accuracy, number of {GMRES} iterations, number of matrix–vector multiplications, and wall-clock time, for a wide range of time steps.
PerezEtAl2017

D. Perez, R. Huang, and A. F. Voter, “Long-time molecular dynamics simulations on massively parallel platforms: A comparison of parallel replica dynamics and parallel trajectory splicing,” Journal of Materials Research, pp. 1–10, 2017, doi: 10.1557/jmr.2017.456. [Online]. Available at: https://dx.doi.org/10.1557/jmr.2017.456
BibTeX entry PerezEtAl2017

@article{PerezEtAl2017, author = {Perez, Danny and Huang, Rao and Voter, Arthur F.}, doi = {10.1557/jmr.2017.456}, journal = {Journal of Materials Research}, pages = {1-–10}, title = {Long-time molecular dynamics simulations on massively parallel platforms: A comparison of parallel replica dynamics and parallel trajectory splicing}, url = {https://dx.doi.org/10.1557/jmr.2017.456}, year = {2017} }
Abstract for BibTeX entry PerezEtAl2017

Molecular dynamics (MD) is one of the most widely used techniques in computational materials science. By providing fully resolved trajectories, it allows for a natural description of static, thermodynamic, and kinetic properties. A major hurdle that has hampered the use of MD is the fact that the timescales that can be directly simulated are very limited, even when using massively parallel computers. In this study, we compare two time-parallelization approaches, parallel replica dynamics (ParRep) and parallel trajectory splicing (ParSplice), that were specifically designed to address this issue for rare event systems by leveraging parallel computing resources. Using simulations of the relaxation of small disordered platinum nanoparticles, a comparative performance analysis of the two methods is presented. The results show that ParSplice can significantly outperform ParRep in the common case where the trajectory remains trapped for a long time within a region of configuration space but makes rapid structural transitions within this region.
Ruprecht2017_lncs

D. Ruprecht, “Shared Memory Pipelined Parareal,” in Euro-Par 2017: Parallel Processing: 23rd International Conference on Parallel and Distributed Computing, Santiago de Compostela, Spain, August 28 – September 1, 2017, Proceedings, F. F. Rivera, T. F. Pena, and J. C. Cabaleiro, Eds. Springer International Publishing, 2017, pp. 669–681 [Online]. Available at: https://doi.org/10.1007/978-3-319-64203-1_48
BibTeX entry Ruprecht2017_lncs

@inbook{Ruprecht2017_lncs, author = {Ruprecht, Daniel}, booktitle = {Euro-Par 2017: Parallel Processing: 23rd International Conference on Parallel and Distributed Computing, Santiago de Compostela, Spain, August 28 -- September 1, 2017, Proceedings}, doi = {10.1007/978-3-319-64203-1_48}, editor = {Rivera, Francisco F. and Pena, Tom{\'a}s F. and Cabaleiro, Jos{\'e} C.}, pages = {669--681}, publisher = {Springer International Publishing}, title = {Shared Memory Pipelined Parareal}, url = {https://doi.org/10.1007/978-3-319-64203-1_48}, year = {2017} }
Abstract for BibTeX entry Ruprecht2017_lncs

For the parallel-in-time integration method Parareal, pipelining can be used to hide some of the cost of the serial correction step and improve its efficiency. The paper introduces a basic OpenMP implementation of pipelined Parareal and compares it to a standard MPI-based variant. Both versions yield almost identical runtimes, but, depending on the compiler, the OpenMP variant consumes about 7% less energy and has a significantly smaller memory footprint. However, its higher implementation complexity might make it difficult to use in legacy codes and in combination with spatial parallelisation.
SpeckRuprecht2017

R. Speck and D. Ruprecht, “Toward fault-tolerant parallel-in-time integration with PFASST ,” Parallel Computing, vol. 62, pp. 20–37, 2017, doi: 10.1016/j.parco.2016.12.001. [Online]. Available at: http://dx.doi.org/10.1016/j.parco.2016.12.001
BibTeX entry SpeckRuprecht2017

@article{SpeckRuprecht2017, author = {Speck, Robert and Ruprecht, Daniel}, doi = {10.1016/j.parco.2016.12.001}, journal = {Parallel Computing}, pages = {20--37}, title = {Toward fault-tolerant parallel-in-time integration with {PFASST} }, url = {http://dx.doi.org/10.1016/j.parco.2016.12.001}, volume = {62}, year = {2017} }
Abstract for BibTeX entry SpeckRuprecht2017

Abstract We introduce and analyze different strategies for the parallel-in-time integration method PFASST to recover from hard faults and subsequent data loss. Since PFASST stores solutions at multiple time steps on different processors, information from adjacent steps can be used to recover after a processor has failed. PFASST’s multi-level hierarchy allows to use the coarse level for correcting the reconstructed solution, which can help to minimize overhead. A theoretical model is devised linking overhead to the number of additional PFASST iterations required for convergence after a fault. The potential efficiency of different strategies is assessed in terms of required additional iterations for examples of diffusive and advective type.
WangPeng2017

S. Wang and Z. Peng, “Space-time parallel computation for time-domain Maxwell’s equations,” in 2017 International Conference on Electromagnetics in Advanced Applications (ICEAA), 2017, pp. 1680–1683, doi: 10.1109/ICEAA.2017.8065615 [Online]. Available at: http://ieeexplore.ieee.org/document/8065615/
BibTeX entry WangPeng2017

@inproceedings{WangPeng2017, author = {Wang, S. and Peng, Z.}, booktitle = {2017 International Conference on Electromagnetics in Advanced Applications (ICEAA)}, doi = {10.1109/ICEAA.2017.8065615}, month = sep, number = {}, pages = {1680--1683}, title = {Space-time parallel computation for time-domain Maxwell's equations}, url = {http://ieeexplore.ieee.org/document/8065615/}, volume = {}, year = {2017} }
Abstract for BibTeX entry WangPeng2017

This work addresses a growing need for the parallel-in-time simulation capability in electromagnetics (EM) applications. Currently time-dependent EM solvers are typically parallel only in space. The sequential-in-time nature of these solvers can achieve good parallel scaling when the number of spatial mesh points per core is large. But the parallel efficiency quickly deteriorates and even saturates if spatial parallelism has been fully exploited. We proposed a new time domain EM solver to harvest parallelism in both spatial and temporal dimension. The spatial parallelism is achieved by discontinuous Galerkin formulation, and the temporal parallelism is enabled by Krylov subspace method based exponential integrator. This work results in a highly scalable parallel time domain solver which can amend the scalability issue for traditional ones. The convergence and parallel performance are validated through numerical experiments.

S.-L. Wu, “Three rapidly convergent parareal solvers with application to time-dependent PDEs with fractional Laplacian,” Mathematical Methods in the Applied Sciences, 2017, doi: 10.1002/mma.4273. [Online]. Available at: http://dx.doi.org/10.1002/mma.4273

S.-L. Wu, “An efficient parareal algorithm for a class of time-dependent problems with fractional Laplacian,” Applied Mathematics and Computation, vol. 307, pp. 329–341, 2017, doi: 10.1016/j.amc.2017.02.012. [Online]. Available at: http://dx.doi.org/10.1016/j.amc.2017.02.012

S.-L. Wu and T.-Z. Huang, “A fast second-order parareal solver for fractional optimal control problems,” Journal of Vibration and Control, vol. 0, no. 0, p. 1077546317705557, 2017, doi: 10.1177/1077546317705557. [Online]. Available at: http://dx.doi.org/10.1177/1077546317705557

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2016

AlhubailEtAl2016

M. Alhubail and Q. Wang, “The swept rule for breaking the latency barrier in time advancing PDEs,” Journal of Computational Physics, vol. 307, pp. 110–121, 2016, doi: 10.1016/j.jcp.2015.11.026. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2015.11.026
BibTeX entry AlhubailEtAl2016

@article{AlhubailEtAl2016, author = {Alhubail, Maitham and Wang, Qiqi}, doi = {10.1016/j.jcp.2015.11.026}, journal = {Journal of Computational Physics}, pages = {110--121}, title = {The swept rule for breaking the latency barrier in time advancing {PDEs}}, url = {http://dx.doi.org/10.1016/j.jcp.2015.11.026}, volume = {307}, year = {2016} }
Abstract for BibTeX entry AlhubailEtAl2016

This article investigates the swept rule of space-time domain decomposition, an idea to break the latency barrier via communicating less often when explicitly solving time-dependent PDEs. The swept rule decomposes space and time among computing nodes in ways that exploit the domains of influence and the domain of dependency, making it possible to communicate once per many timesteps without redundant computation. The article presents simple theoretical analysis to the performance of the swept rule which then was shown to be accurate by conducting numerical experiments.

G. Ariel, S. J. Kim, and R. Tsai, “Parareal Multiscale Methods for Highly Oscillatory Dynamical Systems,” SIAM Journal on Scientific Computing, vol. 38, no. 6, pp. A3540–A3564, Jan. 2016, doi: 10.1137/15m1011044. [Online]. Available at: https://doi.org/10.1137/15m1011044

Astorino2016

M. Astorino, F. Chouly, and A. Quarteroni, “A Time-Parallel Framework for Coupling Finite Element and Lattice Boltzmann Methods,” Applied Mathematics Research eXpress, vol. 2016, no. 1, pp. 24–67, 2016, doi: 10.1093/amrx/abv009. [Online]. Available at: http://dx.doi.org/10.1093/amrx/abv009
BibTeX entry Astorino2016

@article{Astorino2016, author = {Astorino, Matteo and Chouly, Franz and Quarteroni, Alfio}, doi = {10.1093/amrx/abv009}, journal = {Applied Mathematics Research eXpress}, number = {1}, pages = {24--67}, title = {A Time-Parallel Framework for Coupling Finite Element and Lattice Boltzmann Methods}, url = {http://dx.doi.org/10.1093/amrx/abv009}, volume = {2016}, year = {2016} }
Abstract for BibTeX entry Astorino2016

In this work, we propose a new numerical procedure for the simulation of time-dependent problems based on the coupling between the finite element method (FEM) and the lattice Boltzmann method. The procedure exploits the Parareal paradigm to efficiently couple the two numerical methods, allowing independent grid size and time-step size. The motivations behind this approach are wide-ranging. In particular, one technique may be more efficient or physically more appropriate or less memory consuming than the other depending on the target of the simulation and/or on the sub-region of the computational domain. Furthermore, the coupling with FEM may circumvent some difficulties inherent to lattice Boltzmann discretization, for some domains with complex boundaries, or for some kind of boundary conditions. The theoretical and numerical framework is presented for the time-dependent heat equation in order to describe and validate numerically the methodology in a simple situation.
Beck2016

T. Beck, “In-Time Parallelization Of Atmospheric Chemical Kinetics,” PhD thesis, Ruprecht-Karls-Universität Heidelberg, 2016 [Online]. Available at: http://archiv.ub.uni-heidelberg.de/volltextserver/20092/1/TBeck_Phd_a.pdf
BibTeX entry Beck2016

@phdthesis{Beck2016, author = {Beck, Teresa}, school = {Ruprecht-Karls-Universit\"{a}t Heidelberg}, title = {In-Time Parallelization Of Atmospheric Chemical Kinetics}, url = {http://archiv.ub.uni-heidelberg.de/volltextserver/20092/1/TBeck_Phd_a.pdf}, year = {2016} }
Abstract for BibTeX entry Beck2016

This work investigates the potential of an in-time parallelization of atmospheric chemical kinetics. Its numerical calculation is one time-consuming step within the numerical prediction of the air quality. The widely used parallelization strategies only allow a limited potential level of parallelism. A higher level of parallelism within the codes will be necessary to enable benefits from future exa-scale computing architectures. In air quality prediction codes, chemical kinetics is typically considered to react in isolated boxes over short splitting intervals. This allows their trivial parallelization in space, which however is limited by the number of grid entities. This work pursues a parallelization beyond this trivial potential and investigates a parallelization across time using the so called “parareal algorithm”. The latter is an iterative prediction-correction scheme, whose efficiency strongly depends on the choice of the predictor. For that purpose, different options are being investigate and compared: Time-stepping schemes with fixed step size, adaptive time-stepping schemes and repro-models, functional representations, that map a given state to a later state in time. Only the choice of repromodels leads to a speed-up through parallelism, compared to the sequential reference for the scenarios considered here.
BenedusiEtAl2016

P. Benedusi, D. Hupp, P. Arbenz, and R. Krause, “A Parallel Multigrid Solver for Time–periodic Incompressible Navier–Stokes Equations in 3D,” in Numerical Mathematics and Advanced Applications ENUMATH 2015, 2016, pp. 265–273, doi: 10.1007/978-3-319-39929-4_26 [Online]. Available at: https://doi.org/10.1007/978-3-319-39929-4_26
BibTeX entry BenedusiEtAl2016

@inproceedings{BenedusiEtAl2016, author = {Benedusi, P. and Hupp, D. and Arbenz, P. and Krause, R.}, booktitle = {Numerical Mathematics and Advanced Applications ENUMATH 2015}, doi = {10.1007/978-3-319-39929-4_26}, organization = {Springer}, pages = {265--273}, title = {{A Parallel Multigrid Solver for Time--periodic Incompressible Navier--Stokes Equations in 3D}}, url = {https://doi.org/10.1007/978-3-319-39929-4_26}, year = {2016} }
Abstract for BibTeX entry BenedusiEtAl2016

We present a parallel and efficient multilevel solution method for the nonlinear systems arising from the discretization of Navier–Stokes (N-S) equations with finite differences. In particular we study the incompressible, unsteady N-S equations with periodic boundary condition in time. A sequential time integration limits the parallelism of the solver to the spatial variables and can therefore be an obstacle to parallel scalability. Time periodicity allows for a space-time discretization, which adds time as an additional direction for parallelism and thus can improve parallel scalability. To achieve fast convergence, we used a space-time multigrid algorithm with a SCGS smoothing procedure (symmetrical coupled Gauss–Seidel, a.k.a. box smoothing). This technique, proposed by Vanka (J Comput Phys 65:138–156, 1986), for the steady viscous incompressible Navier–Stokes equations is extended to the unsteady case and its properties are studied using local Fourier analysis. We used numerical experiments to analyze the scalability and the convergence of the solver, focusing on the case of a pulsatile flow.
ChaudryEtAl2016

J. H. Chaudhry, D. Estep, S. Tavener, V. Carey, and J. Sandelin, “A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm,” SIAM Journal on Numerical Analysis, vol. 54, no. 5, pp. 2974–3002, 2016, doi: 10.1137/16M1079014. [Online]. Available at: http://dx.doi.org/10.1137/16M1079014
BibTeX entry ChaudryEtAl2016

@article{ChaudryEtAl2016, author = {Chaudhry, Jehanzeb Hameed and Estep, Don and Tavener, Simon and Carey, Varis and Sandelin, Jeff}, doi = {10.1137/16M1079014}, journal = {SIAM Journal on Numerical Analysis}, number = {5}, pages = {2974--3002}, title = {A Posteriori Error Analysis of Two-Stage Computation Methods with Application to Efficient Discretization and the Parareal Algorithm}, url = {http://dx.doi.org/10.1137/16M1079014}, volume = {54}, year = {2016} }
Abstract for BibTeX entry ChaudryEtAl2016

We consider numerical methods for initial value problems that employ a two-stage approach consisting of solution on a relatively coarse discretization followed by solution on a relatively fine discretization. Examples include adaptive error control, parallel-in-time solution schemes, and efficient solution of adjoint problems for computing a posteriori error estimates. We describe a general formulation of two-stage computations and then perform a general a posteriori error analysis based on computable residuals and solution of an adjoint problem. The analysis accommodates various variations in the two-stage computation and in the formulation of the adjoint problems. We apply the analysis to computing “dual-weighted” a posteriori error estimates, developing novel algorithms for efficient solution that take into account cancellation of error, and to the Parareal algorithm. We test the various results using several numerical examples.

F. De Vuyst, “Efficient solvers for time-dependent problems: a review of IMEX, LATIN, PARAEXP and PARAREAL algorithms for heat-type problems with potential use of approximate exponential integrators and reduced-order models,” Advanced Modeling and Simulation in Engineering Sciences, pp. 3–8, 2016, doi: 10.1186/s40323-016-0063-y. [Online]. Available at: http://dx.doi.org/10.1186/s40323-016-0063-y

EghbalEtAl2016

A. Eghbal, A. G. Gerber, and E. Aubanel, “Acceleration of Unsteady Hydrodynamic Simulations Using the Parareal Algorithm,” Journal of Computational Science , vol. 19, pp. 57–76, 2016, doi: 10.1016/j.jocs.2016.12.006. [Online]. Available at: http://dx.doi.org/10.1016/j.jocs.2016.12.006
BibTeX entry EghbalEtAl2016

@article{EghbalEtAl2016, author = {Eghbal, Araz and Gerber, Andrew G. and Aubanel, Eric}, doi = {10.1016/j.jocs.2016.12.006}, journal = {Journal of Computational Science }, pages = {57--76}, title = {Acceleration of Unsteady Hydrodynamic Simulations Using the Parareal Algorithm}, url = {http://dx.doi.org/10.1016/j.jocs.2016.12.006}, volume = {19}, year = {2016} }
Abstract for BibTeX entry EghbalEtAl2016

The parareal algorithm is used to obtain temporal parallelization added to the parallelism obtained from the conventional spatial domain decomposition techniques for hydrodynamic problems. Parareal solution becomes unstable at high Reynolds numbers where the non-linear convection term in the Navier-Stokes equations becomes much larger than the diffusion term. A new framework that allows using parareal for unsteady high Reynolds number hydrodynamic problems is proposed where parareal coarse and fine time integration operators are incorporated with coarse and fine spatial grids respectively and {RANS} or {DES} turbulence models are employed with a blended filter that can be tuned for the stability of the method. This framework is composed of three solution stages where parareal serves as a transitional stage that bridges a coarse grid solution to a fine grid one. While in lower Reynolds number problems parareal solution can serve as a final solution, in higher Reynolds number problems with high degree of non-linearity parareal provides a shorter path to the final solution. Anticipating a parareal stage in a transitional sense allows a looser convergence requirement which leads to high speedup gains in that stage. On the other hand improved initial values at the beginning of the last stage yields a shorter final fine stage solution. A windowing technique is employed in this methodology to further control the parareal instabilities by keeping the parareal corrections smaller while still being able to cover an arbitrary simulation time with given computational resources. Application of this methodology has been illustrated with a fully turbulent vortex shedding from a cylinder and a flow from the Grand Passage tidal zone in the Bay of Fundy. It is concluded that a tuned turbulence model may sufficiently stabilize the parareal methodology for many practical problems such that it becomes applicable in the initialization process if not accurate enough as a final solution. {MPI} and OpenMP programming paradigms are used for temporal parallelism introduced by parareal and data parallelism obtained via spatial domain decomposition methods respectively. Also all computational tasks are accelerated using {CUDA} compatible GPGPUs.
FalgoutEtAl2016

R. D. Falgout, T. A. Manteuffel, B. Southworth, and J. B. Schroder, “Parallel-In-Time For Moving Meshes,” Lawrence Livermore National Laboratory, 2016 [Online]. Available at: http://www.osti.gov/scitech/servlets/purl/1239230
BibTeX entry FalgoutEtAl2016

@techreport{FalgoutEtAl2016, author = {Falgout, R.~D. and Manteuffel, T.~A. and Southworth, B. and Schroder, J. B.}, doi = {10.2172/1239230}, institution = {Lawrence Livermore National Laboratory}, title = {Parallel-In-Time For Moving Meshes}, url = {http://www.osti.gov/scitech/servlets/purl/1239230}, year = {2016} }
Abstract for BibTeX entry FalgoutEtAl2016

With steadily growing computational resources available, scientists must develop effective ways to utilize the increased resources. High performance, highly parallel software has become a standard. However until recent years parallelism has focused primarily on the spatial domain. When solving a space-time partial differential equation (PDE), this leads to a sequential bottleneck in the temporal dimension, particularly when taking a large number of time steps. The XBraid parallel-in-time library was developed as a practical way to add temporal parallelism to existing sequential codes with only minor modi cations. In this work, a rezoning-type moving mesh is applied to a diffusion problem and formulated in a parallel-in-time framework. Tests and scaling studies are run using XBraid and demonstrate excellent results for the simple model problem considered herein.
GahvariEtAl2016

H. Gahvari et al., “A Performance Model for Allocating the Parallelism in a Multigrid-in-Time Solver,” in 7th International Workshop on Performance Modeling, Benchmarking and Simulation of High Performance Computer Systems, 2016, doi: 10.1109/PMBS.2016.8 [Online]. Available at: http://dx.doi.org/10.1109/PMBS.2016.8
BibTeX entry GahvariEtAl2016

@inproceedings{GahvariEtAl2016, author = {Gahvari, Hormozd and Dobrev, Veselin A. and Falgout, Rob D. and Kolev, Tzanio V. and Schroder, Jacob B. and Schulz, Martin and {Meier Yang}, Ulrike}, booktitle = {7th International Workshop on Performance Modeling, Benchmarking and Simulation of High Performance Computer Systems}, doi = {10.1109/PMBS.2016.8}, title = {A Performance Model for Allocating the Parallelism in a Multigrid-in-Time Solver}, url = {http://dx.doi.org/10.1109/PMBS.2016.8}, year = {2016} }
Abstract for BibTeX entry GahvariEtAl2016

The traditional way to numerically solve time-dependent problems is to sequentially march through time, solving for one time step and then the next. The parallelism in this approach is limited to the spatial dimension, which is quickly exhausted, causing the gain from using more processors to solve a problem to diminish. One approach to overcome this barrier is to use methods that are parallel in time. These methods have the potential to achieve dramatically better performance compared to time-stepping approaches, but achieving this performance requires carefully choosing the amount of parallelism devoted to space versus the amount devoted to time. Here, we present a performance model that, for a multigrid-in-time solver, makes the decision on when to switch to parallel-in-time and on how much parallelism to devote to space vs. time. In our experiments, the model selects the best parallel configuration in most of our test cases and a configuration close to the best one in all other cases

M. J. Gander, L. Halpern, J. Ryan, and T. T. B. Tran, “A Direct Solver for Time Parallelization,” in Domain Decomposition Methods in Science and Engineering XXII, 2016, pp. 491–499, doi: 10.1007/978-3-319-18827-0_50 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-18827-0_50

R. GUETAT, “Coupling Parareal with Non-Overlapping Domain Decomposition Method,” Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, vol. Volume 23 - 2016 - Special..., Dec. 2016, doi: 10.46298/arima.1474. [Online]. Available at: https://doi.org/10.46298/arima.1474

GurralaEtAl2016

G. Gurrala, A. Dimitrovski, S. Pannala, S. Simunovic, and M. Starke, “Parareal in Time for Fast Power System Dynamic Simulations,” IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 1820–1830, 2016, doi: 10.1109/TPWRS.2015.2434833. [Online]. Available at: http://dx.doi.org/10.1109/TPWRS.2015.2434833
BibTeX entry GurralaEtAl2016

@article{GurralaEtAl2016, author = {{Gurrala}, G. and {Dimitrovski}, A. and {Pannala}, S. and {Simunovic}, S. and {Starke}, M.}, doi = {10.1109/TPWRS.2015.2434833}, journal = {IEEE Transactions on Power Systems}, number = {3}, pages = {1820--1830}, title = {{Parareal in Time for Fast Power System Dynamic Simulations}}, url = {http://dx.doi.org/10.1109/TPWRS.2015.2434833}, volume = {31}, year = {2016} }
Abstract for BibTeX entry GurralaEtAl2016

Recent advancements in high-performance parallel computing platforms and parallel algorithms have significantly enhanced the opportunities for real-time power system protection and control. This paper investigates application of Parareal in time algorithm for fast dynamic simulations. Parareal algorithm belongs to the class of temporal decomposition methods which divide the time interval into sub-intervals and solve them concurrently. Time-parallel algorithms face the difficulty of providing correct initial conditions for all the sub-intervals which impact the convergence rates. Parareal overcomes this difficulty by using an approximate trajectory. It has become popular in recent years for long transient simulations (e.g., molecular dynamics, fusion, reacting flows). This paper presents an approach for reliable implementation of Parareal with detailed models of power systems including saturation. Windowing approach is proposed for improving the convergence. Parareal is compared with the Newton-based time-parallel method. Effectiveness of the algorithm is analyzed by parallel emulation using extensive case studies on 10-generator 39-bus system and 327-generator 2383-bus system for various disturbances. Parareal with simulation windows of 1 s have shown convergence in 1 to 3 iterations for majority of the simulated cases, irrespective of the size of the system and nature of the disturbance. All the cases tested have converged with the proposed implementation.
Haut2016

T. S. Haut, T. Babb, P. G. Martinsson, and B. A. Wingate, “A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator,” IMA Journal of Numerical Analysis, vol. 36, no. 2, pp. 688–716, 2016, doi: 10.1093/imanum/drv021. [Online]. Available at: http://dx.doi.org/10.1093/imanum/drv021
BibTeX entry Haut2016

@article{Haut2016, author = {Haut, T.~S. and Babb, T. and Martinsson, P.~G. and Wingate, B.~A.}, doi = {10.1093/imanum/drv021}, journal = {IMA Journal of Numerical Analysis}, number = {2}, pages = {688--716}, title = {A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator}, url = {http://dx.doi.org/10.1093/imanum/drv021}, volume = {36}, year = {2016} }
Abstract for BibTeX entry Haut2016

The manuscript presents a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u /∂t = \mathcal Lu, where \mathcal L is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator \exp (τ\mathcal L) for a relatively large time-step τ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.
LapinEtAl2016

A. Lapin and A. Romanenko, “Udzawa-type iterative method with parareal preconditioner for a parabolic optimal control problem,” IOP Conference Series: Materials Science and Engineering, vol. 158, no. 1, p. 012059, 2016, doi: 10.1088/1757-899X/158/1/012059. [Online]. Available at: http://dx.doi.org/10.1088/1757-899X/158/1/012059
BibTeX entry LapinEtAl2016

@article{LapinEtAl2016, author = {Lapin, A and Romanenko, A}, doi = {10.1088/1757-899X/158/1/012059}, journal = {IOP Conference Series: Materials Science and Engineering}, number = {1}, pages = {012059}, title = {Udzawa-type iterative method with parareal preconditioner for a parabolic optimal control problem}, url = {http://dx.doi.org/10.1088/1757-899X/158/1/012059}, volume = {158}, year = {2016} }
Abstract for BibTeX entry LapinEtAl2016

The article deals with the optimal control problem with the parabolic equation as state problem. There are point-wise constraints on the state and control functions. The objective functional involves the observation given in the domain at each moment. The conditions for convergence Udzawa’s type iterative method are given. The parareal method to inverse preconditioner is given. The results of calculations are presented.
Lecouvez2016

M. Lecouvez, R. D. Falgout, C. S. Woodward, and P. Top, “A parallel multigrid reduction in time method for power systems,” in 2016 IEEE Power and Energy Society General Meeting (PESGM), 2016, pp. 1–5, doi: 10.1109/PESGM.2016.7741520 [Online]. Available at: https://dx.doi.org/10.1109/PESGM.2016.7741520
BibTeX entry Lecouvez2016

@inproceedings{Lecouvez2016, author = {{Lecouvez}, M. and {Falgout}, R.~D. and {Woodward}, C.~S. and {Top}, P.}, booktitle = {2016 IEEE Power and Energy Society General Meeting (PESGM)}, doi = {10.1109/PESGM.2016.7741520}, pages = {1--5}, title = {A parallel multigrid reduction in time method for power systems}, url = {https://dx.doi.org/10.1109/PESGM.2016.7741520}, year = {2016} }
Abstract for BibTeX entry Lecouvez2016

This paper presents a fully multilevel approach to parallel in time solution of transient power system simulations. The method employs a multigrid reduction algorithm in time parallelized using the MPI distributed memory programming model. The method is demonstrated on a simple Single Machine Infinite Bus differential-algebraic equation model problem, for which speedup is obtained for as few as 8 processing cores on a problem with 10,000 time steps. Speedup of a factor of 13 is observed for a 100,000 step version of this simple problem. Based on these results, we expect significantly better speedup on larger problems where more work is available to each processor allowing greater amortization of the parallel communication.
Lederman2016

C. Lederman, R. Martin, and J.-L. Cambier, “Time-parallel solutions to differential equations via functional optimization,” Computational and Applied Mathematics, pp. 1–25, 2016, doi: 10.1007/s40314-016-0319-7. [Online]. Available at: http://dx.doi.org/10.1007/s40314-016-0319-7
BibTeX entry Lederman2016

@article{Lederman2016, author = {Lederman, C. and Martin, R. and Cambier, J.-L.}, doi = {10.1007/s40314-016-0319-7}, journal = {Computational and Applied Mathematics}, pages = {1--25}, title = {Time-parallel solutions to differential equations via functional optimization}, url = {http://dx.doi.org/10.1007/s40314-016-0319-7}, year = {2016} }
Abstract for BibTeX entry Lederman2016

We describe an approach to solving a generic time-dependent differential equation (DE) that recasts the problem as a functional optimization one. The techniques employed to solve for the functional minimum, which we relate to the Sobolev Gradient method, allow for large-scale parallelization in time, and therefore potential faster “wall-clock” time computing on machines with significant parallel computing capacity. We are able to come up with numerous different discretizations and approximations for our optimization-derived equations, each of which either puts an existing approach, the Parareal method, in an optimization context, or provides a new time-parallel (TP) scheme with potentially faster convergence to the DE solution. We describe how the approach is particularly effective for solving multiscale DEs and present TP schemes that incorporate two different solution scales. Sample results are provided for three differential equations, solved with TP schemes, and we discuss how the choice of TP scheme can have an orders of magnitude effect on the accuracy or convergence rate.

J. I. Leffell, J. Sitaraman, V. K. Lakshminarayan, and A. M. Wissink, “Towards Efficient Parallel-in-Time Simulation of Periodic Flows,” in 54th AIAA Aerospace Sciences Meeting, 2016, doi: 10.2514/6.2016-0066 [Online]. Available at: http://dx.doi.org/10.2514/6.2016-0066

MatsuokaEtAl2016

S. Matsuoka et al., “From FLOPS to BYTES: Disruptive Change in High-performance Computing Towards the Post-moore Era,” in Proceedings of the ACM International Conference on Computing Frontiers, New York, NY, USA, 2016, pp. 274–281, doi: 10.1145/2903150.2906830 [Online]. Available at: http://dx.doi.org/10.1145/2903150.2906830
BibTeX entry MatsuokaEtAl2016

@inproceedings{MatsuokaEtAl2016, address = {New York, NY, USA}, author = {Matsuoka, Satoshi and Amano, Hideharu and Nakajima, Kengo and Inoue, Koji and Kudoh, Tomohiro and Maruyama, Naoya and Taura, Kenjiro and Iwashita, Takeshi and Katagiri, Takahiro and Hanawa, Toshihiro and Endo, Toshio}, booktitle = {Proceedings of the ACM International Conference on Computing Frontiers}, doi = {10.1145/2903150.2906830}, location = {Como, Italy}, numpages = {8}, pages = {274--281}, publisher = {ACM}, series = {CF '16}, title = {From FLOPS to BYTES: Disruptive Change in High-performance Computing Towards the Post-moore Era}, url = {http://dx.doi.org/10.1145/2903150.2906830}, year = {2016} }
Abstract for BibTeX entry MatsuokaEtAl2016

Slowdown and inevitable end in exponential scaling of processor performance, the end of the so-called "Moore’s Law" is predicted to occur around 2025–2030 timeframe. Because CMOS semiconductor voltage is also approaching its limits, this means that logic transistor power will become constant, and as a result, the system FLOPS will cease to improve, resulting in serious consequences for IT in general, especially supercomputing. Existing attempts to overcome the end of Moore’s law are rather limited in their future outlook or applicability. We claim that data-oriented parameters, such as bandwidth and capacity, or BYTES, are the new parameters that will allow continued performance gains for periods even after computing performance or FLOPS ceases to improve, due to continued advances in storage device technologies and optics, and manufacturing technologies including 3-D packaging. Such transition from FLOPS to BYTES will lead to disruptive changes in the overall systems from applications, algorithms, software to architecture, as to what parameter to optimize for, in order to achieve continued performance growth over time. We are launching a new set of research efforts to investigate and devise new technologies to enable such disruptive changes from FLOPS to BYTES in the Post-Moore era, focusing on HPC, where there is extreme sensitivity to performance, and expect the results to disseminate to the rest of IT.
MerkelEtAl2016

M. Merkel, I. Niyonzima, and S. Schöps, “An Application of ParaExp to Electromagnetic Wave Problems,” in Proceedings of 2016 URSI International Symposium on Electromagnetic Theory (EMTS), 2016, doi: 10.1109/URSI-EMTS.2016.7571330 [Online]. Available at: https://doi.org/10.1109/URSI-EMTS.2016.7571330
BibTeX entry MerkelEtAl2016

@inproceedings{MerkelEtAl2016, author = {Merkel, Melina and Niyonzima, Innocent and Schöps, Sebastian}, booktitle = {Proceedings of 2016 URSI International Symposium on Electromagnetic Theory (EMTS)}, doi = {10.1109/URSI-EMTS.2016.7571330}, editor = {Sihvola, Ari}, note = {arXiv:1607.00368 [math.NA]}, publisher = {IEEE}, title = {An Application of ParaExp to Electromagnetic Wave Problems}, url = {https://doi.org/10.1109/URSI-EMTS.2016.7571330}, year = {2016} }
Abstract for BibTeX entry MerkelEtAl2016

Recently, ParaExp was proposed for the time integration of hyperbolic problems. It splits the time interval of interest into sub-intervals and computes the solution on each sub-interval in parallel. The overall solution is decomposed into a particular solution defined on each sub-interval with zero initial conditions and a homogeneous solution propagated by the matrix exponential applied to the initial conditions. The efficiency of the method results from fast approximations of this matrix exponential using tools from linear algebra. This paper deals with the application of ParaExp to electromagnetic wave problems in time-domain. Numerical tests are carried out for an electric circuit and an electromagnetic wave problem discretized by the Finite Integration Technique.
NeumuellerGander2016

M. J. Gander and M. Neumüller, “Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems,” SIAM Journal on Scientific Computing, vol. 38, no. 4, pp. A2173–A2208, 2016, doi: 10.1137/15M1046605. [Online]. Available at: http://dx.doi.org/10.1137/15M1046605
BibTeX entry NeumuellerGander2016

@article{NeumuellerGander2016, author = {Gander, Martin J. and Neum\"uller, Martin}, doi = {10.1137/15M1046605}, journal = {SIAM Journal on Scientific Computing}, number = {4}, pages = {A2173--A2208}, title = {Analysis of a New Space-Time Parallel Multigrid Algorithm for Parabolic Problems}, url = {http://dx.doi.org/10.1137/15M1046605}, volume = {38}, year = {2016} }
Abstract for BibTeX entry NeumuellerGander2016

We present and analyze a new space-time parallel multigrid method for parabolic equations. The method is based on arbitrarily high order discontinuous Galerkin discretizations in time, and a finite element discretization in space. The key ingredient of the new algorithm is a block Jacobi smoother. We present a detailed convergence analysis when the algorithm is applied to the heat equation, and determine asymptotically optimal smoothing parameters, a precise criterion for semi-coarsening in time or full coarsening, and give an asymptotic two grid contraction factor estimate. We then explain how to implement the new multigrid algorithm in parallel, and show with numerical experiments its excellent strong and weak scalability properties.
NielsenHesthaven2016

A. S. Nielsen and J. S. Hesthaven, “Fault Tolerance in the Parareal Method,” in Proceedings of the ACM Workshop on Fault-Tolerance for HPC at Extreme Scale, New York, NY, USA, 2016, pp. 1–8, doi: 10.1145/2909428.2909431 [Online]. Available at: http://dx.doi.org/10.1145/2909428.2909431
BibTeX entry NielsenHesthaven2016

@inproceedings{NielsenHesthaven2016, acmid = {2909431}, address = {New York, NY, USA}, author = {Nielsen, Allan S. and Hesthaven, Jan S.}, booktitle = {Proceedings of the ACM Workshop on Fault-Tolerance for HPC at Extreme Scale}, doi = {10.1145/2909428.2909431}, isbn = {978-1-4503-4349-7}, location = {Kyoto, Japan}, numpages = {8}, pages = {1--8}, publisher = {ACM}, series = {FTXS '16}, title = {Fault Tolerance in the Parareal Method}, url = {http://dx.doi.org/10.1145/2909428.2909431}, year = {2016} }
Abstract for BibTeX entry NielsenHesthaven2016

Parallel-in-time integration is an often advocated approach for extracting parallelism in the solution of PDEs beyond what is possible using spacial domain decomposition tech- niques. Due to the comparatively low parallel efficiency of parallel-in-time integration techniques, they are primar- ily of interest as an extension for classical approaches at parallelism. As such, potential applications are expected to scale across several hundreds, or possibly thousands of nodes, making algorithmic resilience towards hardware in- duced errors highly relevant. In this work we develop a scheduling scheme for the parareal algorithm that is resilient to node-loss. The fault-tolerant scheme is based on a popu- lar approach introduced by E. Aubanel in [1], modified with a set of MPI interface extensions for implementing recov- ery strategies available in the ULFM framework. In ad- dition, we demonstrate how the parareal algorithm may be made resilient towards Silent-Data-Corruption (SDC) errors by viewing it as a point-iterative method, locally monitor- ing the residual between consecutive iterations so to discard potentially corrupt iterations.
OngEtAl2016

B. W. Ong, R. D. Haynes, and K. Ladd, “Algorithm 965: RIDC Methods: A Family of Parallel Time Integrators,” ACM Trans. Math. Softw., vol. 43, no. 1, pp. 8:1–8:13, 2016, doi: 10.1145/2964377. [Online]. Available at: http://dx.doi.org/10.1145/2964377
BibTeX entry OngEtAl2016

@article{OngEtAl2016, articleno = {8}, author = {Ong, Benjamin W. and Haynes, Ronald D. and Ladd, Kyle}, doi = {10.1145/2964377}, journal = {ACM Trans. Math. Softw.}, number = {1}, numpages = {13}, pages = {8:1--8:13}, title = {Algorithm 965: {RIDC} Methods: A Family of Parallel Time Integrators}, url = {http://dx.doi.org/10.1145/2964377}, volume = {43}, year = {2016} }
Abstract for BibTeX entry OngEtAl2016

Revisionist integral deferred correction methods are a family of parallel-in-time methods to solve systems of initial values problems. The approach is able to bootstrap lower-order time integrators to provide high-order approximations in approximately the same wall-clock time, hence providing a multiplicative increase in the number of compute cores utilized. Here we provide a library that automatically produces a parallel-in-time solution of a system of initial value problems given user-supplied code for the right-hand side of the system and a sequential code for a first-order timestep. The user-supplied timestep routine may be explicit or implicit and may make use of any auxiliary libraries that take care of the solution of any nonlinear algebraic systems that may arise or the numerical linear algebra required.
RuprechtEtAl2016

D. Ruprecht, R. Speck, and R. Krause, “Parareal for Diffusion Problems with Space- and Time-Dependent Coefficients,” in Domain Decomposition Methods in Science and Engineering XXII, 2016, pp. 371–378, doi: 10.1007/978-3-319-18827-0_37 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-18827-0_37
BibTeX entry RuprechtEtAl2016

@inproceedings{RuprechtEtAl2016, author = {Ruprecht, Daniel and Speck, Robert and Krause, Rolf}, booktitle = {Domain Decomposition Methods in Science and Engineering {XXII}}, doi = {10.1007/978-3-319-18827-0_37}, editor = {Dickopf, Thomas and Gander, J. Martin and Halpern, Laurence and Krause, Rolf and Pavarino, F. Luca}, pages = {371--378}, publisher = {Springer International Publishing}, title = {Parareal for Diffusion Problems with Space- and Time-Dependent Coefficients}, url = {http://dx.doi.org/10.1007/978-3-319-18827-0_37}, year = {2016} }
Abstract for BibTeX entry RuprechtEtAl2016

For the time-parallel Parareal method, there exists both numerical and analytical proof that it converges very well for diffusive problems like the heat equation. Many applications, however, do not lead to simple homogeneous diffusive problems but feature strongly inhomogeneous and possibly anisotropic coefficients. In the talk, we will present results from a numerical study of how non-constant coefficients in a diffusion problem influence the convergence behaviour of Parareal. Further, the effect of different parameters like e.g. temporal and spatial resolution and geometry is explored.
SekineEtAl2016

T. Sekine, T. Tsuji, T. Oyama, F. Magoulès, and K. Uchida, “Speedup of parallel computing by parareal method in transient stability analysis of Japanese power system,” in 2016 IEEE Innovative Smart Grid Technologies - Asia (ISGT-Asia), 2016, pp. 1177–1182, doi: 10.1109/ISGT-Asia.2016.7796552 [Online]. Available at: http://dx.doi.org/10.1109/ISGT-Asia.2016.7796552
BibTeX entry SekineEtAl2016

@inproceedings{SekineEtAl2016, author = {Sekine, T. and Tsuji, T. and Oyama, T. and Magoulès, F. and Uchida, K.}, booktitle = {2016 IEEE Innovative Smart Grid Technologies - Asia (ISGT-Asia)}, doi = {10.1109/ISGT-Asia.2016.7796552}, pages = {1177--1182}, title = {Speedup of parallel computing by parareal method in transient stability analysis of Japanese power system}, url = {http://dx.doi.org/10.1109/ISGT-Asia.2016.7796552}, year = {2016} }
Abstract for BibTeX entry SekineEtAl2016

Long computation time is required for transient stability analysis of power systems because it can be expressed as combination of differential and algebraic equations. In order to finish the calculation within a practical amount of time, it is an effective approach that applying Parareal method as one of parallel computing technique to transient stability analysis. In the Parareal method, the accuracy of the result is gradually updated through iterative procedures in combination with rough estimation of entire waveform and detailed calculation for each decomposed time window. Based on the Parareal method, the authors have developed a new stability assessment method by using chaos theory. If an operating point is in unstable region, the trajectory has the chaotic property, “sensitive dependence on initial conditions”. Therefore, it is possible to assess the stability by detecting the convergence characteristics of the estimated waveform at each iterative procedure by the Parareal method. The effectiveness of the proposed method was tested by using IEEJ WEST10 system model.

S.-L. Wu, “A second-order parareal algorithm for fractional PDEs,” Journal of Computational Physics, vol. 307, pp. 280–290, 2016, doi: 10.1016/j.jcp.2015.12.007. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2015.12.007

Wu2016_JCAM

S.-L. Wu, “Towards essential improvement for the Parareal-TR and Parareal-Gauss4 algorithms,” Journal of Computational and Applied Mathematics, vol. 308, pp. 391–407, 2016, doi: 10.1016/j.cam.2016.05.036. [Online]. Available at: http://dx.doi.org/10.1016/j.cam.2016.05.036
BibTeX entry Wu2016_JCAM

@article{Wu2016_JCAM, author = {Wu, Shu-Lin}, doi = {10.1016/j.cam.2016.05.036}, journal = {Journal of Computational and Applied Mathematics}, pages = {391--407}, title = {Towards essential improvement for the Parareal-TR and Parareal-Gauss4 algorithms}, url = {http://dx.doi.org/10.1016/j.cam.2016.05.036}, volume = {308}, year = {2016} }
Abstract for BibTeX entry Wu2016_JCAM

Abstract Parareal is an iterative algorithm and is characterized by two propagators G and F , which are respectively associated with large step size Δ T and small step size Δ t , where Δ T = J Δ t and J ≥ 2 is an integer. For symmetric positive definite (SPD) system u ′ ( t ) + A u ( t ) = g ( t ) arising from semi-discretizing time-dependent PDEs, if we fix the G -propagator to the Backward-Euler method and choose for F some L -stable time-integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy ρ ≈ 1 3 , ∀ J ≥ 2 and ∀ σ ( A ) ⊂ [ 0 , + ∞ ) , where σ ( A ) is the spectrum of the matrix A . However, this result does not hold when time-integrators that lack L -stability, such as the Trapezoidal rule and the 4th-order Gauss {RK} method, are chosen as the F -propagator. The parareal algorithms using these two methods for the F -propagator are denoted by Parareal-TR and Parareal-Gauss4. In this paper, we propose a strategy to let these two parareal algorithms possess such a uniform convergence property. The idea is to choose an L -stable propagator F ˜ and on each coarse time-interval [ T n , T n + 1 ] we perform first two steps of F ˜ , then followed by J − 2 steps of F . Precisely, for the Trapezoidal rule we select the 2nd-order {SDIRK} method as the F ˜ -propagator, and for the 4th-order Gauss {RK} method we select the 4th-order Lobatto III-C method as the F ˜ -propagator. Numerical results are given to support our theoretical conclusions.

S.-L. Wu, “Convergence Analysis of the Parareal-Euler Algorithm for Systems of ODEs with Complex Eigenvalues,” Journal of Scientific Computing, vol. 67, no. 2, pp. 644–668, 2016, doi: 10.1007/s10915-015-0100-x. [Online]. Available at: http://dx.doi.org/10.1007/s10915-015-0100-x

S.-L. Wu and T. Zhou, “Fast parareal iterations for fractional diffusion equations,” Journal of Computational Physics, vol. 329, pp. 210–226, 2016, doi: 10.1016/j.jcp.2016.10.046. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2016.10.046

top

2015

Ariel2015

G. Ariel, S. J. Kim, and R. Tsai, “Parareal methods for highly oscillatory ordinary differential equations.” arXiv:1503.02094 [math.NA], 2015 [Online]. Available at: http://arxiv.org/abs/1503.02094v1
BibTeX entry Ariel2015

@misc{Ariel2015, author = {Ariel, G. and Kim, Seong Jun and Tsai, Richard}, howpublished = {arXiv:1503.02094 [math.NA]}, title = {{Parareal methods for highly oscillatory ordinary differential equations}}, url = {http://arxiv.org/abs/1503.02094v1}, year = {2015} }
Abstract for BibTeX entry Ariel2015

We introduce a new parallel in time (parareal) algorithm which couples multiscale integrators with fully resolved fine scale integration and computes highly oscillatory solutions for a class of ordinary differential equations in parallel. The algorithm computes a low-cost approximation of all slow variables in the system. Then, fast phase-like variables are obtained using the parareal iterative methodology and an alignment algorithm. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. Convergence of the parareal iterations is proved and demonstrated in numerical examples.
ArteagaEtAl2015

A. Arteaga, D. Ruprecht, and R. Krause, “A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA,” Applied Mathematics and Computation, vol. 267, pp. 727–741, 2015, doi: 10.1016/j.amc.2014.12.055. [Online]. Available at: http://dx.doi.org/10.1016/j.amc.2014.12.055
BibTeX entry ArteagaEtAl2015

@article{ArteagaEtAl2015, author = {Arteaga, A. and Ruprecht, Daniel and Krause, Rolf}, doi = {10.1016/j.amc.2014.12.055}, journal = {Applied Mathematics and Computation}, pages = {727--741}, title = {{A stencil-based implementation of Parareal in the {C++} domain specific embedded language {STELLA}}}, url = {http://dx.doi.org/10.1016/j.amc.2014.12.055}, volume = {267}, year = {2015} }
Abstract for BibTeX entry ArteagaEtAl2015

In view of the rapid rise of the number of cores in modern supercomputers, time-parallel methods that introduce concurrency along the temporal axis are becoming increasingly popular. For the solution of time-dependent partial differential equations, these methods can add another direction for concurrency on top of spatial parallelization. The paper presents an implementation of the time-parallel Parareal method in a C++ domain specific language for stencil computations (STELLA). STELLA provides both an OpenMP and a CUDA backend for a shared memory parallelization, using the CPU or GPU inside a node for the spatial stencils. Here, we intertwine this node-wise spatial parallelism with the time-parallel Parareal. This is done by adding an MPI-based implementation of Parareal, which allows us to parallelize in time across nodes. The performance of Parareal with both backends is analyzed in terms of speedup, parallel efficiency and energy-to-solution for an advection-diffusion problem with a time-dependent diffusion coefficient.
Bedez2015

M. Bedez et al., “A fully parallel in time and space algorithm for simulating the electrical activity of a neural tissue,” Journal of Neuroscience Methods, vol. 257, pp. 17–25, 2015, doi: 10.1016/j.jneumeth.2015.09.017. [Online]. Available at: http://dx.doi.org/10.1016/j.jneumeth.2015.09.017
BibTeX entry Bedez2015

@article{Bedez2015, author = {Bedez, Mathieu and Belhachmi, Zakaria and Haeberl\'e, Olivier and Greget, Renaud and Moussaoui, Saliha and Bouteiller, Jean-Marie and Bischoff, Serge}, doi = {10.1016/j.jneumeth.2015.09.017}, journal = {{Journal of Neuroscience Methods}}, note = {in press}, pages = {17--25}, title = {A fully parallel in time and space algorithm for simulating the electrical activity of a neural tissue}, url = {http://dx.doi.org/10.1016/j.jneumeth.2015.09.017}, volume = {257}, year = {2015} }
Abstract for BibTeX entry Bedez2015

The resolution of a model describing the electrical activity of neural tissue and its propagation within this tissue is highly consuming in term of computing time and requires strong computing power to achieve good results. New method in this study, we present a method to solve a model describing the electrical propagation in neuronal tissue, using parareal algorithm, coupling with parallelization space using CUDA in graphical processing unit (GPU). Results we applied the method of resolution to different dimensions of the geometry of our model (1-D, 2-D and 3-D). The GPU results are compared with simulations from a multi-core processor cluster, using message-passing interface (MPI), where the spatial scale was parallelized in order to reach a comparable calculation time than that of the presented method using GPU. A gain of a factor 100 in term of computational time between sequential results and those obtained using the GPU has been obtained, in the case of 3-D geometry. Given the structure of the GPU, this factor increases according to the fineness of the geometry used in the computation. Comparison with existing method(s) To the best of our knowledge, it is the first time such a method is used, even in the case of neuroscience. Conclusion Parallelization time coupled with GPU parallelization space allows for drastically reducing computational time with a fine resolution of the model describing the propagation of the electrical signal in a neuronal tissue.
CarracciuoloEtAl2015

L. Carracciuolo, L. D’Amore, and V. Mele, “Toward a fully parallel multigrid in time algorithm in PETSc environment: A case study in ocean models,” in High Performance Computing Simulation (HPCS), 2015 International Conference on, 2015, pp. 595–598, doi: 10.1109/HPCSim.2015.7237098 [Online]. Available at: http://dx.doi.org/10.1109/HPCSim.2015.7237098
BibTeX entry CarracciuoloEtAl2015

@inproceedings{CarracciuoloEtAl2015, author = {Carracciuolo, L. and D'Amore, L. and Mele, V.}, booktitle = {High Performance Computing Simulation (HPCS), 2015 International Conference on}, doi = {10.1109/HPCSim.2015.7237098}, pages = {595--598}, title = {Toward a fully parallel multigrid in time algorithm in PETSc environment: A case study in ocean models}, url = {http://dx.doi.org/10.1109/HPCSim.2015.7237098}, year = {2015} }
Abstract for BibTeX entry CarracciuoloEtAl2015

We consider linear systems that arise from the discretization of evolutionary models. Typically, solution algorithms are based on a time-stepping approach, solving for one time step after the other. Parallelism is limited to the spatial dimension only. Because time is sequential in nature, the idea of simultaneously solving along time steps is not intuitive. One approach to achieve parallelism in time direction is MGRIT algorithm [7], based on multigrid reduction (MGR) techniques. Here we refer to this approach as MGR-1D. Other kind of approach is the space-time multigrid, where time is simply another dimension in the grid. Analougsly, we refer to this approach as MGR-4D. In this work, motivated by the need of maximizing the availability of new algorithms to climate science, we propose a new parallel approach that mixes both the MGR-1D idea and classical space multigrid methods. We refer to it as the MGR3D+1 approach. Moreover, we discuss their implementation in the high performance scientific library PETSc, as starting point to develope more efficient and scalable algorithms in ocean models.
ChristliebEtAl2015

A. J. Christlieb, C. B. MacDonald, B. W. Ong, and R. J. Spiteri, “Revisionist integral deferred correction with adaptive step-size control,” Communications in Applied Mathematics and Computational Science, vol. 10, no. 1, pp. 1–25, 2015, doi: 10.2140/camcos.2015.10.1. [Online]. Available at: http://dx.doi.org/10.2140/camcos.2015.10.1
BibTeX entry ChristliebEtAl2015

@article{ChristliebEtAl2015, author = {Christlieb, Andrew J. and MacDonald, Colin B. and Ong, Benjamin W. and Spiteri, Raymond J.}, doi = {10.2140/camcos.2015.10.1}, issue = {1}, journal = {Communications in Applied Mathematics and Computational Science}, pages = {1--25}, title = {Revisionist integral deferred correction with adaptive step-size control}, url = {http://dx.doi.org/10.2140/camcos.2015.10.1}, volume = {10}, year = {2015} }
Abstract for BibTeX entry ChristliebEtAl2015

Adaptive step-size control is a critical feature for the robust and efficient numerical solution of initial-value problems in ordinary differential equations. In this paper, we show that adaptive step-size control can be incorporated within a family of parallel time integrators known as revisionist integral deferred correction (RIDC) methods. The RIDC framework allows for various strategies to implement step-size control, and we report results from exploring a few of them.
FengEtAl2015

F. Chen, J. S. Hesthaven, Y. Maday, and A. S. Nielsen, “An Adjoint Approach for Stabilizing the Parareal Method,” EPFL-ARTICLE-211097, 2015 [Online]. Available at: http://infoscience.epfl.ch/record/211097
BibTeX entry FengEtAl2015

@unpublished{FengEtAl2015, author = {Chen, Feng and Hesthaven, Jan S. and Maday, Yvon and Nielsen, Allan S.}, howpublished = {EPFL-ARTICLE-211097}, title = {An Adjoint Approach for Stabilizing the Parareal Method}, url = {http://infoscience.epfl.ch/record/211097}, year = {2015} }
Abstract for BibTeX entry FengEtAl2015

The parareal algorithm seeks to extract parallelism in the time-integration direction of time-dependent differential equations. While it has been applied with success to a wide range of problems, it suffers from some stability issues when applied to non-dissipative problems. We express the method through an iteration matrix and show that the problematic behavior is related to the non-normal structure of the iteration matrix. To enforce monotone convergence we propose an adjoint parareal algorithm, accelerated by the Conjugate Gradient Method. Numerical experiments confirm the stability and suggest directions for further improving the performance.
Gander2015_Review

M. J. Gander, “50 years of Time Parallel Time Integration,” in Multiple Shooting and Time Domain Decomposition, Springer, 2015 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-23321-5_3
BibTeX entry Gander2015_Review

@incollection{Gander2015_Review, author = {Gander, Martin J.}, booktitle = {Multiple Shooting and Time Domain Decomposition}, doi = {10.1007/978-3-319-23321-5_3}, editors = {Carraro, T. and Geiger, M. and K\"orkel, S. and Rannacher, R.}, publisher = {Springer}, title = {{50 years of Time Parallel Time Integration}}, url = {http://dx.doi.org/10.1007/978-3-319-23321-5_3}, year = {2015} }
Abstract for BibTeX entry Gander2015_Review

Time parallel time integration methods have received renewed interest over the last decade because of the advent of massively parallel computers, which is mainly due to the clock speed limit reached on today’s processors. When solving time dependent partial differential equations, the time direction is usually not used for parallelization. But when parallelization in space saturates, the time direction offers itself as a further direction for parallelization. The time direction is however special, and for evolution problems there is a causality principle: the solution later in time is affected (it is even determined) by the solution earlier in time, but not the other way round. Algorithms trying to use the time direction for parallelization must therefore be special, and take this very different property of the time dimension into account. We show in this chapter how time domain decomposition methods were invented, and give an overview of the existing techniques. Time parallel methods can be classified into four different groups: methods based on multiple shooting, methods based on domain decomposition and waveform relaxation, space-time multigrid methods and direct time parallel methods. We show for each of these techniques the main inventions over time by choosing specific publications and explaining the core ideas of the authors. This chapter is for people who want to quickly gain an overview of the exciting and rapidly developing area of research of time parallel methods.
GurralaEtAl2015

G. Gurrala, A. Dimitrovski, P. Sreekanth, S. Simunovic, and M. Starke, “Parareal in Time for Dynamic Simulations of Power Systems,” in Proceedings of the International Conference on Power Systems Transients (IPST2015) in Cavtat, Croatia June 15-18, 2015, 2015 [Online]. Available at: http://www.ipstconf.org/papers/Proc_IPST2015/15IPST073.pdf
BibTeX entry GurralaEtAl2015

@inproceedings{GurralaEtAl2015, author = {Gurrala, Gurunath and Dimitrovski, Aleksandar and Sreekanth, Pannala and Simunovic, Srdjan and Starke, Michael}, booktitle = {{Proceedings of the International Conference on Power Systems Transients (IPST2015) in Cavtat, Croatia June 15-18, 2015}}, title = {{Parareal in Time for Dynamic Simulations of Power Systems}}, url = {http://www.ipstconf.org/papers/Proc_IPST2015/15IPST073.pdf}, year = {2015} }
Abstract for BibTeX entry GurralaEtAl2015

In recent years, there have been significant developments in parallel algorithms and high performance parallelm computing platforms. Parareal in time algorithm has become popular for long transient simulations (e.g., molecular dynamics, fusion, reacting flows). Parareal is a arallel algorithm which divides the time interval into sub-intervals and solves them concurrently. This paper investigates the applicability of the parareal algorithm to power system dynamic simulations. Preliminary results on the application of parareal for multi-machine power systems are reported in this paper. Two widely used test systems, WECC 3-generator 9-bus system, New England 10-generator 39-bus system, is used to explore the effectiveness of the parareal. Severe 3 phase bus faults are simulated using both the classical and detailed models of multi-machine power systems. Actual Speedup of 5-7 times is observed assuming ideal parallelization. It has been observed that the speedup factors of the order of 20 can be achieved by using fast coarse approximations of power system models. Dependency of parareal convergence on fault duration and location has been observed.
HautEtAl2015

T. S. Haut, T. Babb, P. G. Martinsson, and B. A. Wingate, “A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator,” IMA Journal of Numerical Analysis, 2015, doi: 10.1093/imanum/drv021. [Online]. Available at: http://dx.doi.org/10.1093/imanum/drv021
BibTeX entry HautEtAl2015

@article{HautEtAl2015, author = {Haut, T.~S. and Babb, T. and Martinsson, P.~G. and Wingate, B.~A.}, doi = {10.1093/imanum/drv021}, journal = {IMA Journal of Numerical Analysis}, title = {A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator}, url = {http://dx.doi.org/10.1093/imanum/drv021}, year = {2015} }
Abstract for BibTeX entry HautEtAl2015

The manuscript presents a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u /∂t = \mathcal Lu, where \mathcal L is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator \exp (τ\mathcal L) for a relatively large time-step τ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge?Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.
KreienbuehlEtAl2015

A. Kreienbuehl, A. Naegel, D. Ruprecht, R. Speck, G. Wittum, and R. Krause, “Numerical simulation of skin transport using Parareal,” Computing and Visualization in Science, vol. 17, no. 2, pp. 99–108, 2015, doi: 10.1007/s00791-015-0246-y. [Online]. Available at: http://dx.doi.org/10.1007/s00791-015-0246-y
BibTeX entry KreienbuehlEtAl2015

@article{KreienbuehlEtAl2015, author = {Kreienbuehl, Andreas and Naegel, Arne and Ruprecht, Daniel and Speck, Robert and Wittum, Gabriel and Krause, Rolf}, doi = {10.1007/s00791-015-0246-y}, issue = {2}, journal = {Computing and Visualization in Science}, pages = {99--108}, title = {{Numerical simulation of skin transport using Parareal}}, url = {http://dx.doi.org/10.1007/s00791-015-0246-y}, volume = {17}, year = {2015} }
Abstract for BibTeX entry KreienbuehlEtAl2015

In-silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination’s performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal’s convergence. The influence of load imbalances in time due to differences in number of iterations required by the spatial solver as well as spatio-temporal weak scaling is discussed.
MinionEtAl2015

M. L. Minion, R. Speck, M. Bolten, M. Emmett, and D. Ruprecht, “Interweaving PFASST and parallel multigrid,” SIAM Journal on Scientific Computing, vol. 37, no. 5, pp. S244–S263, 2015, doi: 10.1137/14097536X. [Online]. Available at: http://dx.doi.org/10.1137/14097536X
BibTeX entry MinionEtAl2015

@article{MinionEtAl2015, author = {Minion, Michael L. and Speck, Robert and Bolten, Matthias and Emmett, Matthew and Ruprecht, Daniel}, doi = {10.1137/14097536X}, issue = {5}, journal = {{SIAM} Journal on Scientific Computing}, pages = {S244--S263}, title = {{Interweaving {PFASST} and parallel multigrid}}, url = {http://dx.doi.org/10.1137/14097536X}, volume = {37}, year = {2015} }
Abstract for BibTeX entry MinionEtAl2015

The parallel full approximation scheme in space and time (PFASST) introduced by Emmett and Minion in 2012 is an iterative strategy for the temporal parallelization of ODEs and discretized PDEs. As the name suggests, PFASST is similar in spirit to a space-time FAS multigrid method performed over multiple time-steps in parallel. However, since the original focus of PFASST has been on the performance of the method in terms of time parallelism, the solution of any spatial system arising from the use of implicit or semi-implicit temporal methods within PFASST have simply been assumed to be solved to some desired accuracy completely at each sub-step and each iteration by some unspecified procedure. It hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the PFASST iterations and whether this strategy leads to a more efficient overall approach. This paper presents an initial investigation on the relative performance of different strategies for coupling PFASST iterations with multigrid methods for the implicit treatment of diffusion terms in PDEs. In particular, we compare full accuracy multigrid solves at each sub-step with a small fixed number of multigrid V-cycles. This reduces the cost of each PFASST iteration at the possible expense of a corresponding increase in the number of PFASST iterations needed for convergence. Parallel efficiency of the resulting methods is explored through numerical examples.

B. Ong, F. Kwok, and S. High, “Pipeline Schwarz Waveform Relaxation,” in Methods in Science and Engineering XXII, 2015.

PerezEtAl2015

D. Perez, E. D. Cubuk, A. Waterland, E. Kaxiras, and A. F. Voter, “Long-time dynamics through parallel trajectory splicing,” Journal of Chemical Theory and Computation, 2015, doi: 10.1021/acs.jctc.5b00916. [Online]. Available at: http://dx.doi.org/10.1021/acs.jctc.5b00916
BibTeX entry PerezEtAl2015

@article{PerezEtAl2015, author = {Perez, Danny and Cubuk, Ekin Dogus and Waterland, Amos and Kaxiras, Efthimios and Voter, Arthur F.}, doi = {10.1021/acs.jctc.5b00916}, journal = {Journal of Chemical Theory and Computation}, title = {Long-time dynamics through parallel trajectory splicing}, url = {http://dx.doi.org/10.1021/acs.jctc.5b00916}, year = {2015} }
Abstract for BibTeX entry PerezEtAl2015

Simulating the atomistic evolution of materials over long timescales is a longstanding challenge, especially for complex systems where the distribution of barrier heights is very heterogeneous. Such systems are difficult to investigate using conventional long-timescale techniques and the fact that they tend to remain trapped in small regions of configuration space for extended periods of time strongly limits the physical insights gained from short simulations. We introduce a novel simulation technique, Parallel Trajectory Splicing (ParSplice), that aims at addressing this problem through the timewise parallelization of long trajectories. The computational efficiency of ParSplice stems from a speculation strategy whereby predictions of the future evolution of the system are leveraged to increase the amount of work that can be concurrently performed at any one time, hence improving the scalability of the method. ParSplice is also able to accurately account for, and potentially reuse, a substantial fraction of the computational work invested in the simulation. We validate the method on a simple Ag surface system and demonstrate substantial increases in efficiency compared to previous methods. We then demonstrate the power of ParSplice through the study of topology changes in Ag42Cu13 core-shell nanoparticles.

T. D. Scheibe et al., “An Analysis Platform for Multiscale Hydrogeologic Modeling with Emphasis on Hybrid Multiscale Methods,” Groundwater, vol. 53, no. 1, pp. 38–56, 2015, doi: 10.1111/gwat.12179. [Online]. Available at: http://dx.doi.org/10.1111/gwat.12179

SchreiberEtAl2015

M. Schreiber, A. Peddle, T. Haut, and B. Wingate, “A Decentralized Parallelization-in-Time Approach with Parareal,” arXiv:1506.05157 [cs.DC], 2015 [Online]. Available at: http://arxiv.org/abs/1506.05157
BibTeX entry SchreiberEtAl2015

@unpublished{SchreiberEtAl2015, author = {Schreiber, Martin and Peddle, Adam and Haut, Terry and Wingate, Beth}, howpublished = {arXiv:1506.05157 [cs.DC]}, title = {A Decentralized Parallelization-in-Time Approach with Parareal}, url = {http://arxiv.org/abs/1506.05157}, year = {2015} }
Abstract for BibTeX entry SchreiberEtAl2015

With steadily increasing parallelism for high-performance architectures, simulations requiring a good strong scalability are prone to be limited in scalability with standard spatial-decomposition strategies at a certain amount of parallel processors. This can be a show-stopper if the simulation results have to be computed with wallclock time restrictions or as fast as possible. Here, the time-dimension is the only one left for parallelisation and we focus on Parareal as one particular parallelisationin-time method. We present a software approach for making Parareal parallelisation transparent for application developers, hence allowing fast prototyping for Parareal. Further, we introduce a decentralized Parareal which results in autonomous simulation instances which only require communicating with the previous and next simulation instances. This concept is evaluated by solving the rotational shallow water equations parallel-in-time: We provide speedup benchmarks and an in-depth analysis of our results based on state-plots and a performance model. This allows us to show the applicability of the Parareal approach with the rotational shallow water equations and also to evaluate the limitations of Parareal.

B. Song and Y.-L. Jiang, “A new parareal waveform relaxation algorithm for time-periodic problems,” International Journal of Computer Mathematics, vol. 92, no. 2, pp. 377–393, 2015, doi: 10.1080/00207160.2014.891734. [Online]. Available at: http://dx.doi.org/10.1080/00207160.2014.891734

SpeckEtAl2015_BIT

R. Speck, D. Ruprecht, M. Emmett, M. L. Minion, M. Bolten, and R. Krause, “A multi-level spectral deferred correction method,” BIT Numerical Mathematics, vol. 55, no. 3, pp. 843–867, 2015, doi: 10.1007/s10543-014-0517-x. [Online]. Available at: http://dx.doi.org/10.1007/s10543-014-0517-x
BibTeX entry SpeckEtAl2015_BIT

@article{SpeckEtAl2015_BIT, author = {Speck, Robert and Ruprecht, Daniel and Emmett, Matthew and Minion, Michael L. and Bolten, Matthias and Krause, Rolf}, doi = {10.1007/s10543-014-0517-x}, issue = {3}, journal = {{BIT} Numerical Mathematics}, pages = {843--867}, title = {{A multi-level spectral deferred correction method}}, url = {http://dx.doi.org/10.1007/s10543-014-0517-x}, volume = {55}, year = {2015} }
Abstract for BibTeX entry SpeckEtAl2015_BIT

The spectral deferred correction (SDC) method is an iterative scheme for computing a higher-order collocation solution to an ODE by performing a series of correction sweeps using a low-order timestepping method. This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. Three different strategies to reduce the computational cost of correction sweeps on the coarser levels are examined: reducing the degrees of freedom, reducing the order of the spatial discretization, and reducing the accuracy when solving linear systems arising in implicit temporal integration. Several numerical examples demonstrate the effect of multi-level coarsening on the convergence and cost of SDC integration. In particular, MLSDC can provide significant savings in compute time compared to SDC for a three-dimensional problem.
SteinerEtAl2015

J. Steiner, D. Ruprecht, R. Speck, and R. Krause, “Convergence of Parareal for the Navier-Stokes equations depending on the Reynolds number,” in Numerical Mathematics and Advanced Applications - ENUMATH 2013, vol. 103, A. Abdulle, S. Deparis, D. Kressner, F. Nobile, and M. Picasso, Eds. Springer International Publishing, 2015, pp. 195–202 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-10705-9_19
BibTeX entry SteinerEtAl2015

@incollection{SteinerEtAl2015, author = {Steiner, J. and Ruprecht, Daniel and Speck, Robert and Krause, Rolf}, booktitle = {Numerical Mathematics and Advanced Applications - {ENUMATH} 2013}, doi = {10.1007/978-3-319-10705-9_19}, editor = {Abdulle, Assyr and Deparis, Simone and Kressner, Daniel and Nobile, Fabio and Picasso, Marco}, pages = {195--202}, publisher = {Springer International Publishing}, series = {{Lecture Notes in Computational Science and Engineering}}, title = {{Convergence of {P}arareal for the {N}avier-{S}tokes equations depending on the {R}eynolds number}}, url = {http://dx.doi.org/10.1007/978-3-319-10705-9_19}, volume = {103}, year = {2015} }
Abstract for BibTeX entry SteinerEtAl2015

The paper presents first a linear stability analysis for the time-parallel Parareal method, using an IMEX Euler as coarse and a Runge-Kutta-3 method as fine propagator, confirming that dominant imaginary eigenvalues negatively affect Parareal’s convergence. This suggests that when Parareal is applied to the nonlinear Navier-Stokes equations, problems for small viscosities could arise. Numerical results for a driven cavity benchmark are presented, confirming that Parareal’s convergence can indeed deteriorate as viscosity decreases and the flow becomes increasingly dominated by convection. The effect is found to strongly depend on the spatial resolution.

S. Ulbrich, “Preconditioners Based on ‘Parareal’ Time-Domain Decomposition for Time-Dependent PDE-Constrained Optimization,” in Multiple Shooting and Time Domain Decomposition Methods: MuS-TDD, Heidelberg, May 6-8, 2013, T. Carraro, M. Geiger, S. Körkel, and R. Rannacher, Eds. Cham: Springer International Publishing, 2015, pp. 203–232 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-23321-5_8

Wang2015

Z. Wang and S.-L. Wu, “Parareal Algorithms Implemented with IMEX Runge-Kutta Methods,” Mathematical Problems in Engineering, vol. 2015, 2015, doi: 10.1155/2015/395340. [Online]. Available at: http://dx.doi.org/10.1155/2015/395340
BibTeX entry Wang2015

@article{Wang2015, author = {Wang, Zhiyong and Wu, Shu-Lin}, doi = {10.1155/2015/395340}, journal = {Mathematical Problems in Engineering}, title = {Parareal Algorithms Implemented with {IMEX} Runge-Kutta Methods}, url = {http://dx.doi.org/10.1155/2015/395340}, volume = {2015}, year = {2015} }
Abstract for BibTeX entry Wang2015

Parareal algorithm is a very powerful parallel computation method and has received much interest from many researchers over the past few years. The aim of this paper is to investigate the performance of parareal algorithm implemented with IMEX Runge-Kutta (RK) methods. A stability criterion of the parareal algorithm coupled with IMEX RK methods is established and the advantage (in the sense of stability) of implementing with this kind of RK methods is numerically investigated. Finally, numerical examples are given to illustrate the efficiency and performance of different parareal methods.

S.-L. Wu and T. Zhou, “Convergence Analysis for Three Parareal Solvers,” SIAM Journal on Scientific Computing, vol. 37, no. 2, pp. A970–A992, 2015, doi: 10.1137/140970756. [Online]. Available at: http://dx.doi.org/10.1137/140970756

S.-L. Wu, “Convergence Analysis of the Parareal-Euler Algorithm for Systems of ODEs with Complex Eigenvalues,” Journal of Scientific Computing, pp. 1–25, 2015, doi: 10.1007/s10915-015-0100-x. [Online]. Available at: http://dx.doi.org/10.1007/s10915-015-0100-x

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2014

P. Arbenz, D. Hupp, and D. Obrist, “A Parallel Solver for the Time-Periodic Navier-Stokes Equations,” in Parallel Processing and Applied Mathematics, R. Wyrzykowski, J. Dongarra, K. Karczewski, and J. Waśniewski, Eds. Springer Berlin Heidelberg, 2014, pp. 291–300 [Online]. Available at: http://dx.doi.org/10.1007/978-3-642-55195-6_27

A. T. Barker, “A minimal communication approach to parallel time integration,” International Journal of Computer Mathematics, vol. 91, no. 3, pp. 601–615, 2014, doi: 10.1080/00207160.2013.800193. [Online]. Available at: http://dx.doi.org/10.1080/00207160.2013.800193

A.-M. Baudron, J.-J. Lautard, Y. Maday, M. K. Riahi, and J. Salomon, “Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model,” Journal of Computational Physics, vol. 279, no. 0, pp. 67–79, 2014, doi: 10.1016/j.jcp.2014.08.037. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2014.08.037

A.-M. Baudron, J.-J. Lautard, Y. Maday, and O. Mula, “The parareal in time algorithm applied to the kinetic neutron diffusion equation,” in Domain Decomposition Methods in Science and Engineering XXI, 2014, pp. 437–445, doi: 10.1007/978-3-319-05789-7_41 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-05789-7_41

S. Bu and J.-Y. Lee, “An enhanced parareal algorithm based on the deferred correction methods for a stiff system,” Journal of Computational and Applied Mathematics, vol. 255, no. 0, pp. 297–305, 2014, doi: 10.1016/j.cam.2013.05.001. [Online]. Available at: http://dx.doi.org/10.1016/j.cam.2013.05.001

J. J. Caceres Silva, B. Baran, and C. E. Schaerer, “Implementation of a distributed parallel in time scheme using PETSc for a parabolic optimal control problem,” in Computer Science and Information Systems (FedCSIS), 2014 Federated Conference on, 2014, pp. 577–586, doi: 10.15439/2014F340 [Online]. Available at: http://dx.doi.org/10.15439/2014F340

F. Chen, J. S. Hesthaven, and X. Zhu, “On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method,” in Reduced Order Methods for Modeling and Computational Reduction, vol. 9, A. Quarteroni and G. Rozza, Eds. Springer International Publishing, 2014, pp. 187–214 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-02090-7_7

F. Chouly and A. Lozinski, “Parareal multi-model numerical zoom for parabolic multiscale problems,” Comptes Rendus Mathematique, vol. 352, no. 6, pp. 535–540, 2014, doi: 10.1016/j.crma.2014.03.018. [Online]. Available at: http://dx.doi.org/10.1016/j.crma.2014.03.018

CroceEtAl2014

R. Croce, D. Ruprecht, and R. Krause, “Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady Navier-Stokes Equations for Incompressible Flow,” in Modeling, Simulation and Optimization of Complex Processes – HPSC 2012, H. G. Bock, X. P. Hoang, R. Rannacher, and J. P. Schlöder, Eds. Springer International Publishing, 2014, pp. 13–23 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-09063-4_2
BibTeX entry CroceEtAl2014

@incollection{CroceEtAl2014, author = {Croce, Roberto and Ruprecht, Daniel and Krause, Rolf}, booktitle = {Modeling, Simulation and Optimization of Complex Processes -- {HPSC} 2012}, doi = {10.1007/978-3-319-09063-4_2}, editor = {Bock, Hans Georg and Hoang, Xuan Phu and Rannacher, Rolf and Schlöder, Johannes P.}, pages = {13--23}, publisher = {Springer International Publishing}, title = {{Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady {N}avier-{S}tokes Equations for Incompressible Flow}}, url = {http://dx.doi.org/10.1007/978-3-319-09063-4_2}, year = {2014} }
Abstract for BibTeX entry CroceEtAl2014

In this paper we combine the Parareal parallel-in-time method together with spatial parallelization and investigate this space-time parallel scheme by means of solving the three-dimensional incompressible Navier-Stokes equations. Parallelization of time stepping provides a new direction of parallelization and allows to employ additional cores to further speed up simulations after spatial parallelization has saturated. We report on numerical experiments performed on a Cray XE6, simulating a driven cavity flow with and without obstacles. Distributed memory parallelization is used in both space and time, featuring up to 2,048 cores in total. It is confirmed that the space-time-parallel method can provide speedup beyond the saturation of the spatial parallelization.

J. Dongarra and al., “Applied Mathematics Research for Exascale Computing,” Lawrence Livermore National Laboratory, LLNL-TR-651000, 2014 [Online]. Available at: http://science.energy.gov/ /media/ascr/pdf/research/am/docs/EMWGreport.pdf

M. Emmett and M. L. Minion, “Efficient implementation of a multi-level parallel in time algorithm,” in Domain Decomposition Methods in Science and Engineering XXI, 2014, vol. 98, pp. 359–366, doi: 10.1007/978-3-319-05789-7_33 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-05789-7_33

R. D. Falgout, A. Katz, T. V. Kolev, J. B. Schroder, A. M. Wissink, and U. M. Yang, “Parallel Time Integration with Multigrid Reduction for a Compressible Fluid Dynamics Application,” Lawrence Livermore National Laboratory, 2014 [Online]. Available at: https://computation.llnl.gov/project/parallel-time-integration/pubs/strand2d-pit.pdf

R. D. Falgout, S. Friedhoff, T. V. Kolev, S. P. MacLachlan, and J. B. Schroder, “Parallel time integration with multigrid,” SIAM Journal on Scientific Computing, vol. 36, no. 6, pp. C635–C661, 2014, doi: 10.1137/130944230. [Online]. Available at: http://dx.doi.org/10.1137/130944230

M. J. Gander and E. Hairer, “Analysis for parareal algorithms applied to Hamiltonian differential equations,” Journal of Computational and Applied Mathematics, vol. 259, Part A, no. 0, pp. 2–13, 2014, doi: 10.1016/j.cam.2013.01.011. [Online]. Available at: http://dx.doi.org/10.1016/j.cam.2013.01.011

T. Haut and B. Wingate, “An asymptotic parallel-in-time method for highly oscillatory PDEs,” SIAM Journal on Scientific Computing, vol. 36, no. 2, pp. A693–A713, 2014, doi: 10.1137/130914577. [Online]. Available at: http://dx.doi.org/10.1137/130914577

R. D. Haynes and B. W. Ong, “MPI-OpenMP algorithms for the parallel space-time solution of time dependent PDEs,” in Domain Decomposition Methods in Science and Engineering XXI, 2014, vol. 98, pp. 179–187, doi: 10.1007/978-3-319-05789-7_14 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-05789-7_14

T. Loderer, V. Heuveline, and R. Lohner, “The parareal algorithm as a new approach for numerical integration of ODEs in real-time simulations in automotive industry,” PAMM, vol. 14, no. 1, pp. 1027–1030, 2014, doi: 10.1002/pamm.201410489. [Online]. Available at: http://dx.doi.org/10.1002/pamm.201410489

N. Makhoul-Karam, N. R. Nassif, and J. Erhel, “An Adaptive Parallel-in-Time Method with application to a membrane problem,” in Domain Decomposition Methods in Science and Engineering XXI, 2014, vol. 98, pp. 707–717, doi: 10.1007/978-3-319-05789-7_68 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-05789-7_68

O. Mula, “Some contributions towards the parallel simulation of time dependent neutron transport and the integration of observed data in real time,” PhD thesis, Université Pierre et Marie Curie - Paris VI, 2014 [Online]. Available at: https://tel.archives-ouvertes.fr/tel-01081601

Neumueller2014

M. J. Gander and M. Neumueller, “Analysis of a Time Multigrid Algorithm for DG-Discretizations in Time,” 2014 [Online]. Available at: http://arxiv.org/abs/1409.5254
BibTeX entry Neumueller2014

@unpublished{Neumueller2014, author = {Gander, Martin J. and Neumueller, M.}, title = {{Analysis of a Time Multigrid Algorithm for {DG}-Discretizations in Time}}, url = {http://arxiv.org/abs/1409.5254}, year = {2014} }
Abstract for BibTeX entry Neumueller2014

We present and analyze for a scalar linear evolution model problem a time multigrid algorithm for DG-discretizations in time. We derive asymptotically optimized parameters for the smoother, and also an asymptotically sharp convergence estimate for the two grid cycle. Our results hold for any A-stable time stepping scheme and represent the core component for space-time multigrid methods for parabolic partial differential equations. Our time multigrid method has excellent strong and weak scaling properties for parallelization in time, which we show with numerical experiments.
Randles2014

A. Randles and E. Kaxiras, “Parallel in time approximation of the lattice Boltzmann method for laminar flows,” Journal of Computational Physics, vol. 270, pp. 577–586, 2014, doi: 10.1016/j.jcp.2014.04.006. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2014.04.006
BibTeX entry Randles2014

@article{Randles2014, author = {Randles, Amanda and Kaxiras, Efthimios}, doi = {10.1016/j.jcp.2014.04.006}, journal = {Journal of Computational Physics}, pages = {577--586}, title = {{Parallel in time approximation of the lattice {B}oltzmann method for laminar flows}}, url = {http://dx.doi.org/10.1016/j.jcp.2014.04.006}, volume = {270}, year = {2014} }
Abstract for BibTeX entry Randles2014

Fluid dynamics simulations using grid-based methods, such as the lattice Boltzmann equation, can benefit from parallel-in-space computation. However, for a fixed-size simulation of this type, the efficiency of larger processor counts will saturate when the number of grid points per core becomes too small. To overcome this fundamental strong scaling limit in space-parallel approaches, we present a novel time-parallel version of the lattice Boltzmann method using the parareal algorithm. This method is based on a predictor–corrector scheme combined with mesh refinement to enable the simulation of larger number of time steps. We present results of up to a 32× increase in speed for a model system consisting of a cylinder with conditions for laminar flow. The parallel gain obtainable is predicted with strong accuracy, providing a quantitative understanding of the potential impact of this method.

A. Randles and E. Kaxiras, “A Spatio-temporal Coupling Method to Reduce the Time-to-Solution of Cardiovascular Simulations,” in Parallel and Distributed Processing Symposium, 2014 IEEE 28th International, 2014, pp. 593–602, doi: 10.1109/IPDPS.2014.68 [Online]. Available at: http://dx.doi.org/10.1109/IPDPS.2014.68

V. Rao and A. Sandu, “An adjoint-based scalable algorithm for time-parallel integration,” Journal of Computational Science, vol. 5, no. 2, pp. 76–84, 2014, doi: 10.1016/j.jocs.2013.03.004. [Online]. Available at: http://dx.doi.org/10.1016/j.jocs.2013.03.004

Ruprecht2014_GAMM

D. Ruprecht, “Convergence of Parareal with spatial coarsening,” PAMM, vol. 14, no. 1, pp. 1031–1034, 2014, doi: 10.1002/pamm.201410490. [Online]. Available at: http://dx.doi.org/10.1002/pamm.201410490
BibTeX entry Ruprecht2014_GAMM

@article{Ruprecht2014_GAMM, author = {Ruprecht, Daniel}, doi = {10.1002/pamm.201410490}, journal = {PAMM}, number = {1}, pages = {1031--1034}, title = {{Convergence of Parareal with spatial coarsening}}, url = {http://dx.doi.org/10.1002/pamm.201410490}, volume = {14}, year = {2014} }
Abstract for BibTeX entry Ruprecht2014_GAMM

The effect is investigated of using a reduced spatial resolution in the coarse propagator of the time-parallel Parareal method for a finite difference discretization of the linear advection-diffusion equation. It is found that convergence can critically depend on the order of the interpolation used to transfer the coarse propagator solution to the fine mesh in the correction step. The effect also strongly depends on the employed spatial and temporal resolution.
RuprechtKrause2014_DDM

R. Krause and D. Ruprecht, “Hybrid Space-Time Parallel Solution of Burgers’ Equation,” in Domain Decomposition Methods in Science and Engineering XXI, 2014, vol. 98, pp. 647–655, doi: 10.1007/978-3-319-05789-7_62 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-05789-7_62
BibTeX entry RuprechtKrause2014_DDM

@inproceedings{RuprechtKrause2014_DDM, author = {Krause, Rolf and Ruprecht, Daniel}, booktitle = {Domain Decomposition Methods in Science and Engineering {XXI}}, doi = {10.1007/978-3-319-05789-7_62}, editors = {Erhel, J. and Gander, M.~J. and Halpern, L. and Pichot, G. and Sassi, T. and Widlund, O.}, pages = {647--655}, publisher = {Springer International Publishing}, series = {{Lecture Notes in Computational Science and Engineering}}, title = {{Hybrid Space-Time Parallel Solution of {B}urgers' Equation}}, url = {http://dx.doi.org/10.1007/978-3-319-05789-7_62}, volume = {98}, year = {2014} }
Abstract for BibTeX entry RuprechtKrause2014_DDM

An OpenMP-based shared memory implementation of the Parareal parallel-in-time integration scheme using explicit integrators is combined with a standard MPI-based spatial parallelization of a finite difference method into a hybrid space-time parallel scheme. The capability of this approach to achieve speedups beyond the saturation of the pure space-parallel scheme is demonstrated for the two-dimensional Burgers equation.

D. Samaddar et al., “Poster: Greater than 10x Acceleration of fusion plasma edge simulations using the Parareal algorithm,” in Proceedings of the 2014 Conference on High Performance Computing Networking, Storage and Analysis Companion, 2014 [Online]. Available at: http://sc14.supercomputing.org/sites/all/themes/sc14/files/archive/tech_poster/poster_files/post163s2-file3.pdf

B. Song and Y.-L. Jiang, “Analysis of a new parareal algorithm based on waveform relaxation method for time-periodic problems,” Numerical Algorithms, vol. 67, no. 3, pp. 599–622, 2014, doi: 10.1007/s11075-013-9810-z. [Online]. Available at: http://dx.doi.org/10.1007/s11075-013-9810-z

SpeckEtAl2014_DDM2012

R. Speck et al., “Integrating an N-body problem with SDC and PFASST,” in Domain Decomposition Methods in Science and Engineering XXI, 2014, vol. 98, pp. 637–645, doi: 10.1007/978-3-319-05789-7_61 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-05789-7_61
BibTeX entry SpeckEtAl2014_DDM2012

@inproceedings{SpeckEtAl2014_DDM2012, author = {Speck, Robert and Ruprecht, Daniel and Krause, Rolf and Emmett, Matthew and Minion, Michael L. and Winkel, Mathias and Gibbon, Paul}, booktitle = {Domain Decomposition Methods in Science and Engineering {XXI}}, doi = {10.1007/978-3-319-05789-7_61}, editors = {Erhel, J. and Gander, M.~J. and Halpern, L. and Pichot, G. and Sassi, T. and Widlund, O.}, pages = {637--645}, publisher = {Springer International Publishing}, series = {{Lecture Notes in Computational Science and Engineering}}, title = {{Integrating an {N}-body problem with {SDC} and {PFASST}}}, url = {http://dx.doi.org/10.1007/978-3-319-05789-7_61}, volume = {98}, year = {2014} }
Abstract for BibTeX entry SpeckEtAl2014_DDM2012

Vortex methods for the Navier-Stokes equations are based on a Lagrangian particle discretization, which reduces the governing equations to a first-order initial value system of ordinary differential equations for the position and vorticity of N particles. In this paper, the accuracy of solving this system by time-serial spectral deferred corrections (SDC) as well as by the time-parallel Parallel Full Approximation Scheme in Space And Time (PFASST) is investigated. PFASST is based on intertwining SDC iterations with differing resolution in a manner similar to the Parareal algorithm and uses a Full Approximation Scheme (FAS) correction to improve the accuracy of coarser SDC iterations. It is demonstrated that SDC and PFASST can generate highly accurate solutions, and the performance in terms of function evaluations required for a certain accuracy is analyzed and compared to a standard Runge-Kutta method.
SpeckEtAl2014_Parco

R. Speck, D. Ruprecht, M. Emmett, M. Bolten, and R. Krause, “A space-time parallel solver for the three-dimensional heat equation,” in Parallel Computing: Accelerating Computational Science and Engineering (CSE), 2014, vol. 25, pp. 263–272, doi: 10.3233/978-1-61499-381-0-263 [Online]. Available at: http://dx.doi.org/10.3233/978-1-61499-381-0-263
BibTeX entry SpeckEtAl2014_Parco

@inproceedings{SpeckEtAl2014_Parco, author = {Speck, Robert and Ruprecht, Daniel and Emmett, Matthew and Bolten, Matthias and Krause, Rolf}, booktitle = {Parallel Computing: Accelerating Computational Science and Engineering ({CSE})}, doi = {10.3233/978-1-61499-381-0-263}, editors = {Bader, M. and Bode, A. and Bungartz, H.-J. and Gerndt, M. and Joubert, G.R. and Peters, F.}, pages = {263--272}, publisher = {IOS Press}, series = {{Advances in Parallel Computing}}, title = {{A space-time parallel solver for the three-dimensional heat equation}}, url = {http://dx.doi.org/10.3233/978-1-61499-381-0-263}, volume = {25}, year = {2014} }
Abstract for BibTeX entry SpeckEtAl2014_Parco

The paper presents a combination of the time-parallel “parallel full approximation scheme in space and time” (PFASST) with a parallel multigrid method (PMG) in space, resulting in a mesh-based solver for the three-dimensional heat equation with a uniquely high degree of efficient concurrency. Parallel scaling tests are reported on the Cray XE6 machine “Monte Rosa” on up to 16,384 cores and on the IBM Blue Gene/Q system “JUQUEEN” on up to 65,536 cores. The efficacy of the combined spatial- and temporal parallelization is shown by demonstrating that using PFASST in addition to PMG significantly extends the strong-scaling limit. Implications of using spatial coarsening strategies in PFASST’s multi-level hierarchy in large-scale parallel simulations are discussed.

T. Takami and D. Fukudome, “An Identity Parareal Method for Temporal Parallel Computations,” in Parallel Processing and Applied Mathematics, R. Wyrzykowski, J. Dongarra, K. Karczewski, and J. Waśniewski, Eds. Springer Berlin Heidelberg, 2014, pp. 67–75 [Online]. Available at: http://dx.doi.org/10.1007/978-3-642-55224-3_7

T. Takami and D. Fukudome, “An Efficient Pipelined Implementation of Space-Time Parallel Applications,” in Parallel Computing: Accelerating Computational Science and Engineering (CSE), vol. 25, M. Bader, A. Bode, H.-J. Bungartz, M. Gerndt, G. R. Joubert, and F. Peters, Eds. 2014, pp. 273–281 [Online]. Available at: http://dx.doi.org/10.3233/978-1-61499-381-0-273

P. L. C. van der Valk and D. J. Rixen, “Towards a Parallel Time Integration Method for Nonlinear Systems,” in Dynamics of Coupled Structures, Volume 1: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, M. Allen, R. Mayes, and D. Rixen, Eds. Cham: Springer International Publishing, 2014, pp. 135–145 [Online]. Available at: http://dx.doi.org/10.1007/978-3-319-04501-6_12

S.-L. Wu, “Convergence analysis of some second-order parareal algorithms,” IMA Journal of Numerical Analysis, 2014, doi: 10.1093/imanum/dru031. [Online]. Available at: http://dx.doi.org/10.1093/imanum/dru031

Q. Xu, J. S. Hesthaven, and F. Chen, “A parareal method for time-fractional differential equations,” Journal of Computational Physics, 2014, doi: 10.1016/j.jcp.2014.11.034. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2014.11.034

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2013

E. J. Bylaska, J. Q. Weare, and J. H. Weare, “Extending molecular simulation time scales: Parallel in time integrations for high-level quantum chemistry and complex force representations,” The Journal of Chemical Physics, vol. 139, no. 7, p. 074114, 2013, doi: 10.1063/1.4818328. [Online]. Available at: http://dx.doi.org/10.1063/1.4818328

X. Dai and Y. Maday, “Stable Parareal in Time Method for First- and Second-Order Hyperbolic Systems,” SIAM Journal on Scientific Computing, vol. 35, no. 1, pp. A52–A78, 2013, doi: 10.1137/110861002. [Online]. Available at: http://dx.doi.org/10.1137/110861002

X. Dai, C. Le Bris, F. Legoll, and Y. Maday, “Symmetric parareal algorithms for Hamiltonian systems,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 47, no. 03, pp. 717–742, Apr. 2013, doi: 10.1051/m2an/2012046. [Online]. Available at: http://dx.doi.org/10.1051/m2an/2012046

X. Du, M. Sarkis, C. E. Schaerer, and D. B. Szyld, “Inexact and truncated parareal-in-time Krylov subspace methods for parabolic optimal control problems,” Electrontic Transactions on Numerical Analysis, vol. 40, pp. 36–57, 2013 [Online]. Available at: http://etna.mcs.kent.edu/vol.40.2013/pp36-57.dir/pp36-57.pdf

S. Friedhoff, R. D. Falgout, T. V. Kolev, S. P. MacLachlan, and J. B. Schroder, “A Multigrid-in-Time Algorithm for Solving Evolution Equations in Parallel,” in Presented at: Sixteenth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, United States, Mar 17 - Mar 22, 2013, 2013 [Online]. Available at: http://www.osti.gov/scitech/servlets/purl/1073108

D. Fukudome and T. Takami, “Parallel bucket-brigade communication interface for scientific applications,” in Proceedings of the 20th European MPI Users’ Group Meeting, New York, NY, USA, 2013, pp. 135–136, doi: 10.1145/2488551.2488595 [Online]. Available at: http://dx.doi.org/10.1145/2488551.2488595

M. J. Gander, Y.-L. Jiang, and R.-J. Li, “Parareal Schwarz Waveform Relaxation Methods,” in Domain Decomposition Methods in Science and Engineering XX, vol. 91, R. Bank, M. Holst, O. Widlund, and J. Xu, Eds. Springer Berlin Heidelberg, 2013, pp. 451–458 [Online]. Available at: http://dx.doi.org/10.1007/978-3-642-35275-1_53

M. J. Gander and S. Güttel, “PARAEXP: A Parallel Integrator for Linear Initial-Value Problems,” SIAM Journal on Scientific Computing, vol. 35, no. 2, pp. C123–C142, 2013, doi: 10.1137/110856137. [Online]. Available at: http://dx.doi.org/10.1137/110856137

F. Legoll, T. Lelièvre, and G. Samaey, “A Micro-Macro Parareal Algorithm: Application to Singularly Perturbed Ordinary Differential Equations,” SIAM Journal on Scientific Computing, vol. 35, no. 4, pp. A1951–A1986, 2013, doi: 10.1137/120872681. [Online]. Available at: http://dx.doi.org/10.1137/120872681

J. R. McClean, J. A. Parkhill, and A. Aspuru-Guzik, “Feynman’s clock, a new variational principle, and parallel-in-time quantum dynamics,” Proceedings of the National Academy of Sciences, vol. 110, no. 41, pp. E3901–E3909, 2013, doi: 10.1073/pnas.1308069110. [Online]. Available at: http://dx.doi.org/10.1073/pnas.1308069110

D. Ruprecht, R. Speck, M. Emmett, M. Bolten, and R. Krause, “Poster: Extreme-scale space-time parallelism,” in Proceedings of the 2013 Conference on High Performance Computing Networking, Storage and Analysis Companion, 2013 [Online]. Available at: http://sc13.supercomputing.org/sites/default/files/PostersArchive/tech_posters/post148s2-file3.pdf

D. Samaddar et al., “Time parallelization of advanced operation scenario simulations of ITER plasma,” Journal of Physics: Conference Series, vol. 410, no. 1, p. 012032, 2013, doi: 10.1088/1742-6596/410/1/012032. [Online]. Available at: http://dx.doi.org/10.1088/1742-6596/410/1/012032

WangEtAl2013

Q. Wang, S. A. Gomez, P. J. Blonigan, A. L. Gregory, and E. Y. Qian, “Towards scalable parallel-in-time turbulent flow simulations,” Physics of Fluids (1994-present), vol. 25, no. 11, p. 110818, 2013, doi: 10.1063/1.4819390. [Online]. Available at: https://doi.org/10.1063/1.4819390
BibTeX entry WangEtAl2013

@article{WangEtAl2013, author = {Wang, Qiqi and Gomez, Steven A and Blonigan, Patrick J and Gregory, Alastair L and Qian, Elizabeth Y}, doi = {10.1063/1.4819390}, journal = {Physics of Fluids (1994-present)}, number = {11}, pages = {110818}, title = {Towards scalable parallel-in-time turbulent flow simulations}, url = {https://doi.org/10.1063/1.4819390}, volume = {25}, year = {2013} }
Abstract for BibTeX entry WangEtAl2013

We present a reformulation of unsteady turbulent flow simulations. The initial condition is relaxed and information is allowed to propagate both forward and backward in time. Simulations of chaotic dynamical systems with this reformulation can be proven to be well-conditioned time domain boundary value problems. The reformulation can enable scalable parallel-in-time simulation of turbulent flows.

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2012

P. Arbenz, A. Hiltebrand, and D. Obrist, “A Parallel Space-Time Finite Difference Solver for Periodic Solutions of the Shallow-Water Equation,” in Parallel Processing and Applied Mathematics, vol. 7204, R. Wyrzykowski, J. Dongarra, K. Karczewski, and J. Waśniewski, Eds. Springer Berlin Heidelberg, 2012, pp. 302–312 [Online]. Available at: http://dx.doi.org/10.1007/978-3-642-31500-8_31

L. A. Berry, W. R. Elwasif, J. M. Reynolds-Barredo, D. Samaddar, R. S. Sánchez, and D. E. Newman, “Event-based parareal: A data-flow based implementation of parareal,” Journal of Computational Physics, vol. 231, no. 17, pp. 5945–5954, 2012, doi: 10.1016/j.jcp.2012.05.016. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2012.05.016

A. J. Christlieb, R. D. Haynes, and B. W. Ong, “A Parallel Space-Time Algorithm,” SIAM Journal on Scientific Computing, vol. 34, no. 5, pp. C233–C248, 2012, doi: 10.1137/110843484. [Online]. Available at: http://dx.doi.org/10.1137/110843484

M. Emmett and M. L. Minion, “Toward an Efficient Parallel in Time Method for Partial Differential Equations,” Communications in Applied Mathematics and Computational Science, vol. 7, pp. 105–132, 2012, doi: 10.2140/camcos.2012.7.105. [Online]. Available at: http://dx.doi.org/10.2140/camcos.2012.7.105

S. S. Foley, W. R. Elwasif, and D. E. Bernholdt, “The integrated plasma simulator: A flexible python framework for coupled multiphysics simulation,” Oak Ridge National Laboratory, ORNL/TM-2012/57, 2012 [Online]. Available at: http://info.ornl.gov/sites/publications/files/Pub34832.pdf

J. Geiser and S. Güttel, “Coupling methods for heat transfer and heat flow: Operator splitting and the parareal algorithm,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 873–887, 2012, doi: 10.1016/j.jmaa.2011.10.030. [Online]. Available at: http://dx.doi.org/10.1016/j.jmaa.2011.10.030

He2012

L.-P. He and M. He, “Parareal in Time Simulation Of Morphological Transformation in Cubic Alloys with Spatially Dependent Composition,” Communications in Computational Physics, vol. 11, no. 5, pp. 1697–1717, 2012, doi: 10.4208/cicp.110310.090911a. [Online]. Available at: http://dx.doi.org/10.4208/cicp.110310.090911a
BibTeX entry He2012

@article{He2012, author = {He, Li-Ping and He, Minxin}, doi = {10.4208/cicp.110310.090911a}, issue = {5}, journal = {Communications in Computational Physics}, pages = {1697--1717}, title = {Parareal in Time Simulation Of Morphological Transformation in Cubic Alloys with Spatially Dependent Composition}, url = {http://dx.doi.org/10.4208/cicp.110310.090911a}, volume = {11}, year = {2012} }
Abstract for BibTeX entry He2012

In this paper, a reduced morphological transformation model with spatially dependent composition and elastic modulus is considered. The parareal in time algorithm introduced by Lions et al. is developed for longer-time simulation. The fine solver is based on a second-order scheme in reciprocal space, and the coarse solver is based on a multi-model backward Euler scheme, which is fast and less expensive. Numerical simulations concerning the composition with a random noise and a discontinuous curve are performed. Some microstructure characteristics at very low temperature are obtained by a variable temperature technique.

J. Liu and Y.-L. Jiang, “A parareal algorithm based on waveform relaxation,” Mathematics and Computers in Simulation, vol. 82, no. 11, pp. 2167–2181, 2012, doi: 10.1016/j.matcom.2012.05.017. [Online]. Available at: http://dx.doi.org/10.1016/j.matcom.2012.05.017

J. Liu and Y.-L. Jiang, “A parareal waveform relaxation algorithm for semi-linear parabolic partial differential equations,” Journal of Computational and Applied Mathematics, vol. 236, no. 17, pp. 4245–4263, 2012, doi: 10.1016/j.cam.2012.05.014. [Online]. Available at: http://dx.doi.org/10.1016/j.cam.2012.05.014

Loic2012

M. Loïc, “Semi-explicit Parareal method based on convergence acceleration technique,” arXiv:1212.4703 [cs.SY], 2012 [Online]. Available at: https://arxiv.org/abs/1212.4703
BibTeX entry Loic2012

@unpublished{Loic2012, author = {Lo{\"i}c, Michel}, howpublished = {arXiv:1212.4703 [cs.SY]}, title = {Semi-explicit Parareal method based on convergence acceleration technique}, url = {https://arxiv.org/abs/1212.4703}, year = {2012} }
Abstract for BibTeX entry Loic2012

The Parareal algorithm is used to solve time-dependent problems considering multiple solvers that may work in parallel. The key feature is a initial rough approximation of the solution that is iteratively refined by the parallel solvers. We report a derivation of the Parareal method that uses a convergence acceleration technique to improve the accuracy of the solution. Our approach uses firstly an explicit ODE solver to perform the parallel computations with different time-steps and then, a decomposition of the solution into specific convergent series, based on an extrapolation method, allows to refine the precision of the solution. Our proposed method exploits basic explicit integration methods, such as for example the explicit Euler scheme, in order to preserve the simplicity of the global parallel algorithm. The first part of the paper outlines the proposed method applied to the simple explicit Euler scheme and then the derivation of the classical Parareal algorithm is discussed and illustrated with numerical examples.

B. W. Ong, A. Melfi, and A. J. Christlieb, “Parallel Semi-Implicit Time Integrators,” 2012 [Online]. Available at: http://arxiv.org/abs/1209.4297

V. Rao, A. Cioaca, and A. Sandu, “A Highly Scalable Approach for Time Parallelization of Long Range Forecasts,” in High Performance Computing, Networking, Storage and Analysis (SCC), 2012 SC Companion: 2012, pp. 609–616, doi: 10.1109/SC.Companion.2012.85 [Online]. Available at: http://dx.doi.org/10.1109/SC.Companion.2012.85

J. M. Reynolds-Barredo, D. E. Newman, R. S. Sánchez, D. Samaddar, L. A. Berry, and W. R. Elwasif, “Mechanisms for the convergence of time-parallelized, parareal turbulent plasma simulations,” Journal of Computational Physics, vol. 231, no. 23, pp. 7851–7867, 2012, doi: 10.1016/j.jcp.2012.07.028. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2012.07.028

J. M. Reynolds-Barredo, D. E. Newman, R. S. Sánchez, and L. A. Berry, “Modelling parareal convergence in 2D drift wave plasma turbulence,” in High Performance Computing and Simulation (HPCS), 2012 International Conference on, 2012, pp. 726–727, doi: 10.1109/HPCSim.2012.6267004 [Online]. Available at: http://dx.doi.org/10.1109/HPCSim.2012.6267004

RuprechtKrause2012

D. Ruprecht and R. Krause, “Explicit parallel-in-time integration of a linear acoustic-advection system,” Computers & Fluids, vol. 59, no. 0, pp. 72–83, 2012, doi: 10.1016/j.compfluid.2012.02.015. [Online]. Available at: http://dx.doi.org/10.1016/j.compfluid.2012.02.015
BibTeX entry RuprechtKrause2012

@article{RuprechtKrause2012, author = {Ruprecht, Daniel and Krause, Rolf}, doi = {10.1016/j.compfluid.2012.02.015}, journal = {Computers \& Fluids}, number = {0}, pages = {72--83}, title = {{Explicit parallel-in-time integration of a linear acoustic-advection system}}, url = {http://dx.doi.org/10.1016/j.compfluid.2012.02.015}, volume = {59}, year = {2012} }
Abstract for BibTeX entry RuprechtKrause2012

The applicability of the Parareal parallel-in-time integration scheme for the solution of a linear, two-dimensional hyperbolic acoustic-advection system, which is often used as a test case for integration schemes for numerical weather prediction (NWP), is addressed. Parallel-in-time schemes are a possible way to increase, on the algorithmic level, the amount of parallelism, a requirement arising from the rapidly growing number of CPUs in high performance computer systems. A recently introduced modification of the "parallel implicit time-integration algorithm" could successfully solve hyperbolic problems arising in structural dynamics. It has later been cast into the framework of Parareal. The present paper adapts this modified Parareal and employs it for the solution of a hyperbolic flow problem, where the initial value problem solved in parallel arises from the spatial discretization of a partial differential equation by a finite difference method. It is demonstrated that the modified Parareal is stable and can produce reasonably accurate solutions while allowing for a noticeable reduction of the time-to-solution. The implementation relies on integration schemes already widely used in NWP (RK-3, partially split forward Euler, forward-backward). It is demonstrated that using an explicit partially split scheme for the coarse integrator allows to avoid the use of an implicit scheme while still achieving speedup.

H. Samuel, “Time domain parallelization for computational geodynamics,” Geochemistry, Geophysics, Geosystems, vol. 13, no. 1, 2012, doi: 10.1029/2011GC003905. [Online]. Available at: http://dx.doi.org/10.1029/2011GC003905

SpeckEtAl2012

R. Speck et al., “A massively space-time parallel N-body solver,” in Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, Los Alamitos, CA, USA, 2012, pp. 92:1–92:11, doi: 10.1109/SC.2012.6 [Online]. Available at: http://dx.doi.org/10.1109/SC.2012.6
BibTeX entry SpeckEtAl2012

@inproceedings{SpeckEtAl2012, address = {Los Alamitos, CA, USA}, articleno = {92}, author = {Speck, Robert and Ruprecht, Daniel and Krause, Rolf and Emmett, Matthew and Minion, Michael L. and Winkel, Mathias and Gibbon, Paul}, booktitle = {Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis}, doi = {10.1109/SC.2012.6}, location = {Salt Lake City, Utah}, numpages = {11}, pages = {92:1--92:11}, publisher = {IEEE Computer Society Press}, series = {{SC '12}}, title = {{A massively space-time parallel {N}-body solver}}, url = {http://dx.doi.org/10.1109/SC.2012.6}, year = {2012} }
Abstract for BibTeX entry SpeckEtAl2012

We present a novel space-time parallel version of the Barnes-Hut tree code PEPC using PFASST, the Parallel Full Approximation Scheme in Space and Time. The naive use of increasingly more processors for a fixed-size N-body problem is prone to saturate as soon as the number of unknowns per core becomes too small. To overcome this intrinsic strong-scaling limit, we introduce temporal parallelism on top of pepc’s existing hybrid MPI/PThreads spatial decomposition. Here, we use PFASST which is based on a combination of the iterations of the parallel-in-time algorithm parareal with the sweeps of spectral deferred correction (SDC) schemes. By combining these sweeps with multiple space-time discretization levels, PFASST relaxes the theoretical bound on parallel efficiency in parareal. We present results from runs on up to 262,144 cores on the IBM Blue Gene/P installation JUGENE, demonstrating that the space-time parallel code provides speedup beyond the saturation of the purely space-parallel approach.

T. Takami and A. Nishida, “Parareal Acceleration of Matrix Multiplication,” in Applications, Tools and Techniques on the Road to Exascale Computing, 2012, vol. 22, pp. 437–444, doi: 10.3233/978-1-61499-041-3-437 [Online]. Available at: http://dx.doi.org/10.3233/978-1-61499-041-3-437

XiaoAubanel2012

H. Xiao and E. Aubanel, “Scheduling of Tasks in the Parareal Algorithm for Heterogeneous Cloud Platforms,” in Parallel and Distributed Processing Symposium Workshops PhD Forum (IPDPSW), 2012 IEEE 26th International, 2012, pp. 1440–1448, doi: 10.1109/IPDPSW.2012.181 [Online]. Available at: http://dx.doi.org/10.1109/IPDPSW.2012.181
BibTeX entry XiaoAubanel2012

@inproceedings{XiaoAubanel2012, author = {Xiao, Hongtao and Aubanel, E.}, booktitle = {{Parallel and Distributed Processing Symposium Workshops PhD Forum (IPDPSW), 2012 IEEE 26th International}}, doi = {10.1109/IPDPSW.2012.181}, pages = {1440--1448}, title = {{Scheduling of Tasks in the Parareal Algorithm for Heterogeneous Cloud Platforms}}, url = {http://dx.doi.org/10.1109/IPDPSW.2012.181}, year = {2012} }
Abstract for BibTeX entry XiaoAubanel2012

Parallelization of time-dependent partial differential equations (PDEs) can be accomplished by time decomposition using the parareal algorithm. While the parareal algorithm was designed to enable real-time simulations, it holds particular promise for long time simulations on computational grids and clouds, due its low communication overhead and potential for adaptation to heterogeneous processors. This contribution extends previous work on the scheduling of tasks of the parareal algorithm to resources with heterogeneous CPU performance. Experiments on Amazon’s EC2 show the suitability of this algorithm for execution on a heterogeneous cloud platform and its insensitivity to network latency.

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2011

E. Aubanel, “Scheduling of Tasks in the Parareal Algorithm,” Parallel Computing, vol. 37, pp. 172–182, 2011, doi: 10.1016/j.parco.2010.10.004. [Online]. Available at: http://dx.doi.org/10.1016/j.parco.2010.10.004

X. Bai and J. L. Junkins, “Modified Chebyshev-Picard Iteration Methods for Orbit Propagation,” The Journal of the Astronautical Sciences, vol. 58, no. 4, pp. 583–613, Oct. 2011, doi: 10.1007/bf03321533. [Online]. Available at: https://doi.org/10.1007/bf03321533

T. Cadeau and F. Magoules, “Coupling the Parareal Algorithm with the Waveform Relaxation Method for the Solution of Differential Algebraic Equations,” in Distributed Computing and Applications to Business, Engineering and Science (DCABES), 2011 Tenth International Symposium on, 2011, pp. 15–19, doi: 10.1109/DCABES.2011.34 [Online]. Available at: http://dx.doi.org/10.1109/DCABES.2011.34

A. J. Christlieb and B. W. Ong, “Implicit parallel time integrators,” Journal of Scientific Computing, vol. 49, no. 2, pp. 167–179, 2011, doi: 10.1007/s10915-010-9452-4. [Online]. Available at: http://dx.doi.org/10.1007/s10915-010-9452-4

C. Douglas, I. Kim, H. Lee, and D. Sheen, “Higher-order schemes for the Laplace transformation method for parabolic problems,” Computing and Visualization in Science, vol. 14, no. 1, pp. 39–47, 2011, doi: 10.1007/s00791-011-0156-6. [Online]. Available at: https://doi.org/10.1007/s00791-011-0156-6

M. Duarte, M. Massot, and S. Descombes, “Parareal operator splitting techniques for multi-scale reaction waves: Numerical analysis and strategies,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 45, no. 05, pp. 825–852, Aug. 2011, doi: 10.1051/m2an/2010104. [Online]. Available at: http://dx.doi.org/10.1051/m2an/2010104

W. R. Elwasif et al., “A dependency-driven formulation of parareal: parallel-in-time solution of PDEs as a many-task application,” in Proceedings of the 2011 ACM international workshop on many task computing on grids and supercomputers, 2011, pp. 15–24, doi: 10.1145/2132876.2132883 [Online]. Available at: http://dx.doi.org/10.1145/2132876.2132883

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2010

A. Blouza, B. Laurent, and S. M. Kaber, “Parallel in time algorithms with reduction methods for solving chemical kinetics,” Communications in Applied Mathematics and Computational Science, vol. 5, no. 2, pp. 241–263, 2010, doi: 10.2140/camcos.2010.5.241. [Online]. Available at: http://dx.doi.org/10.2140/camcos.2010.5.241

A. J. Christlieb, C. B. Macdonald, and B. W. Ong, “Parallel high-order integrators,” SIAM Journal on Scientific Computing, vol. 32, no. 2, pp. 818–835, 2010, doi: 10.1137/09075740X. [Online]. Available at: http://dx.doi.org/10.1137/09075740X

FuEtAl2010

N. K. Fu and N. H. M. Ali, “Improving Pipelined Time Stepping Algorithm for Distributed Memory Multicomputers,” Sains Malaysiana, vol. 39, no. 6, pp. 1041–1048, 2010 [Online]. Available at: http://www.ukm.my/jsm/pdf_files/SM-PDF-39-6-2010/25 Ng Kok Fu.pdf
BibTeX entry FuEtAl2010

@article{FuEtAl2010, author = {Fu, Ng Kok and Ali, Norhashidah Hj. Mohd}, journal = {Sains Malaysiana}, number = {6}, pages = {1041--1048}, title = {Improving Pipelined Time Stepping Algorithm for Distributed Memory Multicomputers}, url = {http://www.ukm.my/jsm/pdf_files/SM-PDF-39-6-2010/25 Ng Kok Fu.pdf}, volume = {39}, year = {2010} }
Abstract for BibTeX entry FuEtAl2010

Time stepping algorithm with spatial parallelisation is commonly used to solve time dependent partial differential equations. Computation in each time step is carried out using all processors available before sequentially advancing to the next time step. In cases where few spatial components are involved and there are relatively many processors available for use, this will result in fine granularity and decreased scalability. Naturally one alternative is to parallelise the temporal domain. Several time parallelisation algorithms have been suggested for the past two decades. One of them is the pipelined iterations across time steps. In this pipelined time stepping method, communication however is extensive between time steps during the pipelining process. This causes a decrease in performance on distributed memory environment which often has high message latency. We present a modified pipelined time stepping algorithm based on delayed pipelining and reduced communication strategies to improve overall execution time on a distributed memory environment using MPI. Our goal is to reduce the inter-time step communications while providing adequate information for the next time step to converge. Numerical result confirms that the improved algorithm is faster than the original pipelined algorithm and sequential time stepping algorithm with spatial parallelisation alone. The improved algorithm is most beneficial for fine granularity time dependent problems with limited spatial parallelisation.

C. H. Lai, “On Transformation Methods and the Induced Parallel Properties for the Temporal Domain,” in Substructing Techniques and Domain Decomposition Methods, 2010, pp. 45–70, doi: 10.4203/csets.24.3 [Online]. Available at: http://dx.doi.org/10.4203/csets.24.3

LepsaSandu2010

B. Lepsa and A. Sandu, “An efficient error control mechanism for the adaptive ’parareal’ time discretization algorithm,” in Proceedings of the 2010 Spring Simulation Multiconference, San Diego, CA, USA, 2010, pp. 87:1–87:7, doi: 10.1145/1878537.1878628 [Online]. Available at: http://dx.doi.org/10.1145/1878537.1878628
BibTeX entry LepsaSandu2010

@inproceedings{LepsaSandu2010, acmid = {1878628}, address = {San Diego, CA, USA}, articleno = {87}, author = {Lepsa, Bianca and Sandu, Adrian}, booktitle = {{Proceedings of the 2010 Spring Simulation Multiconference}}, doi = {10.1145/1878537.1878628}, location = {Orlando, Florida}, numpages = {7}, pages = {87:1--87:7}, publisher = {Society for Computer Simulation International}, series = {{SpringSim '10}}, title = {{An efficient error control mechanism for the adaptive 'parareal' time discretization algorithm}}, url = {http://dx.doi.org/10.1145/1878537.1878628}, year = {2010} }
Abstract for BibTeX entry LepsaSandu2010

’Parareal’ attempts to speed up the solution of ordinary differential equations (ODEs) by parallelizing the time dimension. Different solutions are obtained in parallel on different subintervals. An iterative procedure is employed to match the solution values at the end of each subinterval with those at the beginning of the next one. It has been shown that insufficient ’parareal’ iterations lead to an inaccurate solution, while too many iterations waste CPU cycles without bringing any improvement to the solution. In this paper we discuss an error control mechanism for both the classical ’parareal’ time discretization method and its adaptive variant. This mechanism can be effectively used as a mean of controlling the number of ’parareal’ iterations. We show that bounding the difference between the solution of the fine integrator and the ’parareal’ solution by the local truncation error of the fine grid leads to a sufficient convergence criterion that guaranties a solution that is accurate enough in a minimum number of iterations. Tests on a nonlinear problem illustrate the effectiveness of the proposed error control mechanism on the classical automatic adaptive time stepping formulation of an embedded method on the two-level ’parareal’ algorithm.
MathewEtAl2010

T. Mathew, M. Sarkis, and C. E. Schaerer, “Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems,” SIAM Journal on Scientific Computing, vol. 32, no. 3, pp. 1180–1200, 2010, doi: 10.1137/080717481. [Online]. Available at: http://dx.doi.org/10.1137/080717481
BibTeX entry MathewEtAl2010

@article{MathewEtAl2010, author = {Mathew, Tarek and Sarkis, Marcus and Schaerer, Christian E.}, doi = {10.1137/080717481}, journal = {SIAM Journal on Scientific Computing}, number = {3}, pages = {1180--1200}, title = {{Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems}}, url = {http://dx.doi.org/10.1137/080717481}, volume = {32}, year = {2010} }
Abstract for BibTeX entry MathewEtAl2010

In this paper, we describe block matrix algorithms for the iterative solution of a large-scale linear-quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon. We consider an "all at once" discretization of the problem and formulate three iterative algorithms. The first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables; however inner-outer iterations are required. The second algorithm modifies the first algorithm to avoid inner-outer iterations by introducing an auxiliary variable. It yields a symmetric indefinite system with a positive definite block preconditioner. The third algorithm is the central focus of this paper. It modifies the preconditioner in the second algorithm by a parallel-in-time preconditioner based on the parareal algorithm. Theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability. Numerical experiments confirm the theoretical results.

M. L. Minion, “A Hybrid Parareal Spectral Deferred Corrections Method,” Communications in Applied Mathematics and Computational Science, vol. 5, no. 2, pp. 265–301, 2010, doi: 10.2140/camcos.2010.5.265. [Online]. Available at: http://dx.doi.org/10.2140/camcos.2010.5.265

S. Mitran, “Time parallel kinetic-molecular interaction algorithm for CPU/GPU computers,” Procedia Computer Science, vol. 1, no. 1, pp. 745–752, 2010, doi: 10.1016/j.procs.2010.04.080. [Online]. Available at: http://dx.doi.org/10.1016/j.procs.2010.04.080

D. Samaddar, D. E. Newman, and R. S. Sánchez, “Parallelization in time of numerical simulations of fully-developed plasma turbulence using the parareal algorithm,” Journal of Computational Physics, vol. 229, no. 18, pp. 6558–6573, 2010, doi: 10.1016/j.jcp.2010.05.012. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2010.05.012

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2009

P. Amodio and L. Brugnano, “Parallel solution in time of ODEs: some achievements and perspectives,” Applied Numerical Mathematics, vol. 59, no. 3–4, pp. 424–435, 2009, doi: 10.1016/j.apnum.2008.03.024. [Online]. Available at: http://dx.doi.org/10.1016/j.apnum.2008.03.024

A. Borzì and G. von Winckel, “Multigrid Methods and Sparse-Grid Collocation Techniques for Parabolic Optimal Control Problems with Random Coefficients,” SIAM Journal on Scientific Computing, vol. 31, no. 3, pp. 2172–2192, 2009, doi: 10.1137/070711311. [Online]. Available at: http://dx.doi.org/10.1137/070711311

E. Celledoni and T. Kvamsdal, “Parallelization in time for thermo-viscoplastic problems in extrusion of aluminium,” International Journal for Numerical Methods in Engineering, vol. 79, no. 5, pp. 576–598, 2009, doi: 10.1002/nme.2585. [Online]. Available at: http://dx.doi.org/10.1002/nme.2585

J. Cortial and C. Farhat, “A time-parallel implicit method for accelerating the solution of non-linear structural dynamics problems,” International Journal for Numerical Methods in Engineering, vol. 77, no. 4, pp. 451–470, 2009, doi: 10.1002/nme.2418. [Online]. Available at: http://dx.doi.org/10.1002/nme.2418

S. Engblom, “Parallel in Time Simulation of Multiscale Stochastic Chemical Kinetics,” Multiscale Modeling & Simulation, vol. 8, no. 1, pp. 46–68, 2009, doi: 10.1137/080733723. [Online]. Available at: http://dx.doi.org/10.1137/080733723

G. Frantziskonis, K. Muralidharan, P. Deymier, S. Simunovic, P. Nukala, and S. Pannala, “Time-parallel multiscale/multiphysics framework,” Journal of Computational Physics, vol. 228, no. 21, pp. 8085–8092, 2009, doi: 10.1016/j.jcp.2009.07.035. [Online]. Available at: http://dx.doi.org/10.1016/j.jcp.2009.07.035

Y. Maday, “Symposium: Recent Advances on the Parareal in Time Algorithms,” AIP Conference Proceedings, vol. 1168, no. 1, pp. 1515–1516, 2009, doi: 10.1063/1.3241386. [Online]. Available at: http://dx.doi.org/10.1063/1.3241386

D. Mercerat, L. Guillot, and J.-P. Vilotte, “Application of the parareal algorithm for acoustic wave propagation,” in AIP Conference Proceedings, 2009, vol. 1168, pp. 1521–1524, doi: 10.1063/1.3241388 [Online]. Available at: http://dx.doi.org/10.1063/1.3241388

N. R. Nassif, N. Makhoul-Karam, and Y. Soukiassian, “Computation of blowing-up solutions for second-order differential equations using re-scaling techniques,” Journal of Computational and Applied Mathematics, vol. 227, no. 1, pp. 185–195, 2009, doi: 10.1016/j.cam.2008.07.020. [Online]. Available at: http://dx.doi.org/10.1016/j.cam.2008.07.020

Wu2009

S. Wu, B. Shi, and C. Huang, “Parareal-Richardson Algorithm for Solving Nonlinear ODEs and PDEs,” Communications in Computational Physics, vol. 6, no. 4, pp. 883–902, 2009, doi: 10.4208/cicp.2009.v6.p883. [Online]. Available at: http://dx.doi.org/10.4208/cicp.2009.v6.p883
BibTeX entry Wu2009

@article{Wu2009, author = {Wu, Shulin and Shi, Baochang and Huang, Chengming}, doi = {10.4208/cicp.2009.v6.p883}, issue = {4}, journal = {Communications in Computational Physics}, pages = {883--902}, title = {Parareal-Richardson Algorithm for Solving Nonlinear ODEs and PDEs}, url = {http://dx.doi.org/10.4208/cicp.2009.v6.p883}, volume = {6}, year = {2009} }
Abstract for BibTeX entry Wu2009

The parareal algorithm, proposed firstly by Lions et al., is an effective algorithm to solve the time-dependent problems parallel in time. This algorithm has received much interest from many researchers in the past years. We present in this paper a new variant of the parareal algorithm, which is derived by combining the original parareal algorithm and the Richardson extrapolation, for the numerical solution of the nonlinear ODEs and PDEs. Several nonlinear problems are tested to show the advantage of the new algorithm. The accuracy of the obtained numerical solution is compared with that of its original version (i.e., the parareal algorithm based on the same numerical method).

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2008

P. Amodio and L. Brugnano, “Recent Advances in the Parallel Solution in Time of ODEs,” AIP Conference Proceedings, vol. 1048, no. 1, pp. 867–870, 2008, doi: 10.1063/1.2991069. [Online]. Available at: http://dx.doi.org/10.1063/1.2991069

G. Bal and Q. Wu, “Symplectic Parareal,” in Domain Decomposition Methods in Science and Engineering XVII, vol. 60, U. Langer, M. Discacciati, D. E. Keyes, O. B. Widlund, and W. Zulehner, Eds. Springer Berlin Heidelberg, 2008, pp. 401–408 [Online]. Available at: http://dx.doi.org/10.1007/978-3-540-75199-1_51

M. J. Gander, “Analysis of the Parareal Algorithm Applied to Hyperbolic Problems using Characteristics,” Bol. Soc. Esp. Mat. Apl., vol. 42, pp. 21–35, 2008.

M. J. Gander and E. Hairer, “Nonlinear Convergence Analysis for the Parareal Algorithm,” in Domain Decomposition Methods in Science and Engineering, 2008, vol. 60, pp. 45–56, doi: 10.1007/978-3-540-75199-1_4 [Online]. Available at: http://dx.doi.org/10.1007/978-3-540-75199-1_4

M. J. Gander and M. Petcu, “Analysis of a Krylov Subspace Enhanced Parareal Algorithm for Linear Problem,” ESAIM: Proc., vol. 25, pp. 114–129, 2008, doi: 10.1051/proc:082508. [Online]. Available at: http://dx.doi.org/10.1051/proc:082508

Y. Liu and J. Hu, “Modified propagators of parareal in time algorithm and application to Princeton Ocean model,” Int. J. for Numerical Methods in Fluids, vol. 57, no. 12, pp. 1793–1804, 2008, doi: 10.1002/fld.1703. [Online]. Available at: http://dx.doi.org/10.1002/fld.1703

Y. Maday and E. M. Rønquist, “Parallelization in time through tensor-product space-time solvers,” Comptes Rendus Mathematique, vol. 346, no. 1–2, pp. 113–118, 2008, doi: 10.1016/j.crma.2007.09.012. [Online]. Available at: http://dx.doi.org/10.1016/j.crma.2007.09.012

M. L. Minion and S. A. Williams, “Parareal and spectral deferred corrections,” in AIP Conference Proceedings, 2008, vol. 1048, p. 388, doi: 10.1063/1.2990941 [Online]. Available at: http://dx.doi.org/10.1063/1.2990941

M. Sarkis, C. E. Schaerer, and T. Mathew, “Block Diagonal Parareal Preconditioner for Parabolic Optimal Control Problems,” in Domain Decomposition Methods in Science and Engineering XVII, vol. 60, U. Langer and al., Eds. Springer Berlin Heidelberg, 2008, pp. 409–416 [Online]. Available at: http://dx.doi.org/10.1007/978-3-540-75199-1_52

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2007

D. S. Daoud, “Stability of the Parareal Time Discretization for Parabolic Inverse Problems,” in Domain Decomposition Methods in Science and Engineering XVI, vol. 55, O. B. Widlund and D. E. Keyes, Eds. Springer Berlin Heidelberg, 2007, pp. 275–282 [Online]. Available at: http://dx.doi.org/10.1007/978-3-540-34469-8_32

M. J. Gander and M. Petcu, “Analysis of a Modified Parareal Algorithm for Second-Order Ordinary Differential Equations,” in AIP Conference Proceedings, 2007, vol. 936, p. 233, doi: 10.1063/1.2790116 [Online]. Available at: http://dx.doi.org/10.1063/1.2790116

M. J. Gander and S. Vandewalle, “On the Superlinear and Linear Convergence of the Parareal Algorithm,” in Domain Decomposition Methods in Science and Engineering, vol. 55, O. B. Widlund and D. E. Keyes, Eds. Springer Berlin Heidelberg, 2007, pp. 291–298 [Online]. Available at: http://dx.doi.org/10.1007/978-3-540-34469-8_34

M. J. Gander and S. Vandewalle, “Analysis of the Parareal Time-Parallel Time-Integration Method,” SIAM Journal on Scientific Computing, vol. 29, no. 2, pp. 556–578, 2007, doi: 10.1137/05064607X. [Online]. Available at: http://dx.doi.org/10.1137/05064607X

D. Guibert and D. Tromeur-Dervout, “Adaptive Parareal for Systems of ODEs,” in Domain Decomposition Methods in Science and Engineering XVI, vol. 55, O. B. Widlund and D. E. Keyes, Eds. Springer Berlin Heidelberg, 2007, pp. 587–594 [Online]. Available at: http://dx.doi.org/10.1007/978-3-540-34469-8_73

D. Guibert and D. Tromeur-Dervout, “Parallel adaptive time domain decomposition for stiff systems of ODEs/DAEs,” Computers & Structures, vol. 85, no. 9, pp. 553–562, 2007, doi: 10.1016/j.compstruc.2006.08.040. [Online]. Available at: http://dx.doi.org/10.1016/j.compstruc.2006.08.040

D. Guibert and D. Tromeur-Dervout, “Parallel deferred correction method for CFD problems,” in Parallel Computational Fluid Dynamics 2006, J. H. Kwon, A. Ecer, N. Satofuka, J. Periaux, and P. Fox, Eds. Amsterdam: Elsevier, 2007, pp. 131–138 [Online]. Available at: http://dx.doi.org/10.1016/B978-044453035-6/50019-5

S. M. Kaber and Y. Maday, “Parareal in time approximation of the Korteveg-deVries-Burgers’ equations,” PAMM, vol. 7, no. 1, pp. 1026403–1026404, 2007, doi: 10.1002/pamm.200700574. [Online]. Available at: http://dx.doi.org/10.1002/pamm.200700574

Y. Maday, J. Salomon, and G. Turinici, “Monotonic parareal control for quantum systems,” SIAM Journal on Numerical Analysis, vol. 45, no. 6, pp. 2468–2482, 2007, doi: 10.1137/050647086. [Online]. Available at: http://dx.doi.org/10.1137/050647086

S. Ulbrich, “7. Generalized SQP Methods with ‘Parareal’ Time-Domain Decomposition for Time-Dependent PDE-Constrained Optimization,” in Real-Time PDE-Constrained Optimization, SIAM, 2007, pp. 145–168 [Online]. Available at: https://dx.doi.org/10.1137/1.9780898718935.ch7

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2006

C. Farhat, J. Cortial, C. Dastillung, and H. Bavestrello, “Time-parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses,” International Journal for Numerical Methods in Engineering, vol. 67, no. 5, pp. 697–724, 2006, doi: 10.1002/nme.1653. [Online]. Available at: http://dx.doi.org/10.1002/nme.1653

I. Garrido, B. Lee, G. E. Fladmark, and M. S. Espedal, “Convergent iterative schemes for time parallelization,” Mathematics of Computation, vol. 75, no. 255, pp. 1403–1429, Feb. 2006, doi: 10.1090/s0025-5718-06-01832-1. [Online]. Available at: https://doi.org/10.1090/s0025-5718-06-01832-1

N. R. Nassif, N. M. Karam, and Y. Soukiassian, “A New Approach for Solving Evolution Problems in Time-Parallel Way,” in Computational Science – ICCS 2006, vol. 3991, V. N. Alexandrov, G. D. Albada, P. M. A. Sloot, and J. Dongarra, Eds. Springer Berlin Heidelberg, 2006, pp. 148–155 [Online]. Available at: http://dx.doi.org/10.1007/11758501_24

J. M. F. Trindade and J. C. F. Pereira, “Parallel-in-Time Simulation of Two-Dimensional, Unsteady, Incompressible Laminar Flows,” Numerical Heat Transfer, Part B: Fundamentals, vol. 50, no. 1, pp. 25–40, 2006, doi: 10.1080/10407790500459379. [Online]. Available at: http://dx.doi.org/10.1080/10407790500459379

Y. Yu, A. Srinivasan, and N. Chandra, “Scalable Time-Parallelization of Molecular Dynamics Simulations in Nano Mechanics,” in Parallel Processing, 2006. ICPP 2006. International Conference on, 2006, pp. 119–126, doi: 10.1109/ICPP.2006.64 [Online]. Available at: http://dx.doi.org/10.1109/ICPP.2006.64

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2005

G. Bal, “On the convergence and the stability of the parareal algorithm to solve partial differential equations,” in Domain Decomposition Methods in Science and Engineering, Berlin, 2005, vol. 40, pp. 426–432, doi: 10.1007/3-540-26825-1_43 [Online]. Available at: http://dx.doi.org/10.1007/3-540-26825-1_43

A. Borzì and R. Griesse, “Experiences with a space–time multigrid method for the optimal control of a chemical turbulence model,” International Journal for Numerical Methods in Fluids, vol. 47, no. 8-9, pp. 879–885, 2005, doi: 10.1002/fld.904. [Online]. Available at: http://dx.doi.org/10.1002/fld.904

P. F. Fischer, F. Hecht, and Y. Maday, “A parareal in time semi-implicit approximation of the Navier-Stokes equations,” in Domain Decomposition Methods in Science and Engineering, Berlin, 2005, vol. 40, pp. 433–440, doi: 10.1007/3-540-26825-1_44 [Online]. Available at: http://dx.doi.org/10.1007/3-540-26825-1_44

I. Garrido, M. S. Espedal, and G. E. Fladmark, “A Convergent Algorithm for Time Parallelization Applied to Reservoir Simulation,” in Domain Decomposition Methods in Science and Engineering, vol. 40, T. J. Barth and al., Eds. Springer Berlin Heidelberg, 2005, pp. 469–476 [Online]. Available at: http://dx.doi.org/10.1007/3-540-26825-1_48

Y. Maday and G. Turinici, “The parareal in time iterative solver: A further direction to parallel implementation,” in Domain Decomposition Methods in Science and Engineering, Berlin, 2005, vol. 40, pp. 441–448, doi: 10.1007/3-540-26825-1_45 [Online]. Available at: http://dx.doi.org/10.1007/3-540-26825-1_45

SchmittEtAl2005

B. A. Schmitt, R. Weiner, and H. Podhaisky, “Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration,” BIT Numerical Mathematics, vol. 45, no. 1, pp. 197–217, 2005, doi: 10.1007/s10543-005-2635-y. [Online]. Available at: http://dx.doi.org/10.1007/s10543-005-2635-y
BibTeX entry SchmittEtAl2005

@article{SchmittEtAl2005, author = {Schmitt, Bernhard A. and Weiner, Ruediger and Podhaisky, Helmut}, doi = {10.1007/s10543-005-2635-y}, journal = {BIT Numerical Mathematics}, number = {1}, pages = {197--217}, title = {Multi-Implicit Peer Two-Step W-Methods for Parallel Time Integration}, url = {http://dx.doi.org/10.1007/s10543-005-2635-y}, volume = {45}, year = {2005} }
Abstract for BibTeX entry SchmittEtAl2005

Peer two-step W-methods are designed for integration of stiff initial value problems with parallelism across the method. The essential feature is that in each time step s ’peer’ approximations are employed having similar properties. In fact, no primary solution variable is distinguished. Parallel implementation of these stages is easy since information from one previous time step is used only and the different linear systems may be solved simultaneously. This paper introduces a subclass having order s-1 where optimal damping for stiff problems is obtained by using different system parameters in different stages. Favourable properties of this subclass are uniform stability for realistic stepsize sequences and a superconvergence property which is proved using a polynomial collocation formulation. Numerical tests on a shared memory computer of a matrix-free implementation with Krylov methods are included.

A. Srinivasan and N. Chandra, “Latency tolerance through parallelization of time in scientific applications,” Parallel Computing, vol. 31, no. 7, pp. 777–796, 2005, doi: 10.1016/j.parco.2005.04.008. [Online]. Available at: http://dx.doi.org/10.1016/j.parco.2005.04.008

A. Srinivasan, Y. Yu, and N. Chandra, “Application of Reduce Order Modeling to Time Parallelization,” in High Performance Computing – HiPC 2005, vol. 3769, D. A. Bader, M. Parashar, V. Sridhar, and V. K. Prasanna, Eds. Springer Berlin Heidelberg, 2005, pp. 106–117 [Online]. Available at: http://dx.doi.org/10.1007/11602569_15

G. A. Staff and E. M. Rønquist, “Stability of the parareal algorithm,” in Domain Decomposition Methods in Science and Engineering, Berlin, 2005, vol. 40, pp. 449–456, doi: 10.1007/3-540-26825-1_46 [Online]. Available at: http://dx.doi.org/10.1007/3-540-26825-1_46

V. Thomée, “A high order parallel method for time discretization of parabolic type equations based on Laplace transformation and quadrature,” International Journal of Numerical Analysis and Modeling, vol. 2, no. 1, pp. 85–96, 2005.

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2000 - 2004

D. Sheen, I. H. Sloan, and V. Thomée, “A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature,” Mathematics of Computation, vol. 69, no. 229, pp. 177–195, 2000, doi: 10.1090/S0025-5718-99-01098-4. [Online]. Available at: https://doi.org/10.1090/S0025-5718-99-01098-4

M. A. Botchev and H. A. van der Vorst, “A parallel nearly implicit time-stepping scheme,” Journal of Computational and Applied Mathematics, vol. 137, no. 2, pp. 229–243, 2001, doi: 10.1016/S0377-0427(01)00358-2. [Online]. Available at: http://dx.doi.org/10.1016/S0377-0427(01)00358-2

LionsEtAl2001

J.-L. Lions, Y. Maday, and G. Turinici, “A ‘parareal’ in time discretization of PDE’s,” Comptes Rendus de l’Académie des Sciences - Series I - Mathematics, vol. 332, pp. 661–668, 2001, doi: 10.1016/S0764-4442(00)01793-6. [Online]. Available at: http://dx.doi.org/10.1016/S0764-4442(00)01793-6
BibTeX entry LionsEtAl2001

@article{LionsEtAl2001, author = {Lions, J.-L. and Maday, Yvon and Turinici, Gabriel}, doi = {10.1016/S0764-4442(00)01793-6}, journal = {Comptes Rendus de l'Académie des Sciences - Series I - Mathematics}, pages = {661--668}, title = {{A "parareal" in time discretization of {PDE}'s}}, url = {http://dx.doi.org/10.1016/S0764-4442(00)01793-6}, volume = {332}, year = {2001} }
Abstract for BibTeX entry LionsEtAl2001

The purpose of this Note is to propose a time discretization of a partial differential evolution equation that allows for parallel implementations. The method, based on an Euler scheme, combines coarse resolutions and independent fine resolutions in time in the same spirit as standard spacial approximations. The resulting parallel implementation is done in the non standard time direction. Its main goal concerns real time problems, hence the proposed terminology of “parareal” algorithm.

L. Baffico, S. Bernard, Y. Maday, G. Turinici, and G. Zérah, “Parallel-in-time molecular-dynamics simulations,” Phys. Rev. E, vol. 66, no. 5, p. 057701, 2002, doi: 10.1103/PhysRevE.66.057701. [Online]. Available at: http://link.aps.org/doi/10.1103/PhysRevE.66.057701

G. Bal and Y. Maday, “A ‘Parareal’ time discretization for non-linear PDE’s with application to the pricing of an American Put,” in Recent Developments in Domain Decomposition Methods, vol. 23, L. Pavarino and A. Toselli, Eds. Springer Berlin, 2002, pp. 189–202 [Online]. Available at: http://dx.doi.org/10.1007/978-3-642-56118-4_12

E. Giladi and H. B. Keller, “Space-time domain decomposition for parabolic problems,” Numerische Mathematik, vol. 93, no. 2, pp. 279–313, 2002, doi: 10.1007/s002110100345. [Online]. Available at: https://doi.org/10.1007/s002110100345

Y. Maday and G. Turinici, “A parareal in time procedure for the control of partial differential equations,” Comptes Rendus Mathématique, vol. 335, no. 4, pp. 387–392, 2002, doi: 10.1016/S1631-073X(02)02467-6. [Online]. Available at: http://dx.doi.org/10.1016/S1631-073X(02)02467-6

C. Farhat and M. Chandesris, “Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications,” International Journal for Numerical Methods in Engineering, vol. 58, no. 9, pp. 1397–1434, 2003, doi: 10.1002/nme.860. [Online]. Available at: http://dx.doi.org/10.1002/nme.860

Y. Maday and G. Turinici, “Parallel in time algorithms for quantum control: Parareal time discretization scheme,” Int. J. Quant. Chem., vol. 93, no. 3, pp. 223–228, 2003, doi: 10.1002/qua.10554. [Online]. Available at: http://dx.doi.org/10.1002/qua.10554

D. Sheen, I. H. Sloan, and V. Thomée, “A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature,” IMA Journal of Numerical Analysis, vol. 23, no. 2, pp. 269–299, 2003, doi: 10.1093/imanum/23.2.269. [Online]. Available at: https://doi.org/10.1093/imanum/23.2.269

J. M. F. Trindade and J. C. F. Pereira, “Parallel-in-time simulation of the unsteady Navier-Stokes equations for incompressible flow,” International Journal for Numerical Methods in Fluids, vol. 45, no. 10, pp. 1123–1136, 2004, doi: 10.1002/fld.732. [Online]. Available at: http://dx.doi.org/10.1002/fld.732

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1995 - 1999

K. Burrage, Parallel and sequential methods for ordinary differential equations. The Clarendon Press, Oxford University Press, New York, 1995, p. xvi+446.

A. Deshpande, S. Malhotra, M. Schultz, and C. Douglas, “A rigorous analysis of time domain parallelism,” Parallel Algorithms and Applications, vol. 6, no. 1, pp. 53–62, 1995, doi: 10.1080/10637199508915498. [Online]. Available at: http://dx.doi.org/10.1080/10637199508915498

G. Horton, S. Vandewalle, and P. Worley, “An Algorithm with Polylog Parallel Complexity for Solving Parabolic Partial Differential Equations,” SIAM Journal on Scientific Computing, vol. 16, no. 3, pp. 531–541, 1995, doi: 10.1137/0916034. [Online]. Available at: http://dx.doi.org/10.1137/0916034

G. Horton and S. Vandewalle, “A Space-Time Multigrid Method for Parabolic Partial Differential Equations,” SIAM Journal on Scientific Computing, vol. 16, no. 4, pp. 848–864, 1995, doi: 10.1137/0916050. [Online]. Available at: http://dx.doi.org/10.1137/0916050

JacksonEtAl1995

K. R. Jackson and S. P. Nørsett, “The Potential for Parallelism in Runge–Kutta Methods. Part 1: RK Formulas in Standard Form,” SIAM Journal on Numerical Analysis, vol. 32, no. 1, pp. 49–82, 1995, doi: 10.1137/0732002. [Online]. Available at: http://dx.doi.org/10.1137/0732002
BibTeX entry JacksonEtAl1995

@article{JacksonEtAl1995, author = {Jackson, K. R. and N{\o}rsett, S. P.}, doi = {10.1137/0732002}, journal = {SIAM Journal on Numerical Analysis}, number = {1}, pages = {49--82}, title = {The Potential for Parallelism in Runge–Kutta Methods. Part 1: RK Formulas in Standard Form}, url = {http://dx.doi.org/10.1137/0732002}, volume = {32}, year = {1995} }
Abstract for BibTeX entry JacksonEtAl1995

The authors examine the potential for parallelism in Runge–Kutta (RK) methods based on formulas in standard one-step form. Both negative and positive results are presented. Many of the negative results are based on a theorem that bounds the order of an RK formula in terms of the minimal polynomial associated with its coefficient matrix. The positive results are largely examples of prototypical formulas that offer a potential for effective “coarse-grain” parallelism on machines with a few processors.

S. Vandewalle and G. Horton, “Fourier mode analysis of the multigrid waveform relaxation and time-parallel multigrid methods,” Computing, vol. 54, no. 4, pp. 317–330, 1995, doi: 10.1007/BF02238230. [Online]. Available at: http://dx.doi.org/10.1007/BF02238230

K. Burrage, “Parallel methods for systems of ordinary differential equations,” in Applications on Advanced Architecture Computers, G. Astfalk, Ed. Society for Industrial and Applied Mathematics, 1996, pp. 101–120 [Online]. Available at: http://dx.doi.org/10.1137/1.9780898719659.ch10

J. Janssen and S. Vandewalle, “Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case,” SIAM Journal on Scientific Computing, vol. 17, no. 1, pp. 133–155, 1996, doi: 10.1137/0917011. [Online]. Available at: http://dx.doi.org/10.1137/0917011

T. Rauber and G. Rünger, “Parallel Implementations of Iterated Runge-Kutta Methods,” The International Journal of Supercomputer Applications and High Performance Computing, vol. 10, no. 1, pp. 62–90, Mar. 1996, doi: 10.1177/109434209601000103. [Online]. Available at: https://doi.org/10.1177/109434209601000103

S. Ta’asan and H. Zhang, “Fourier-Laplace analysis of the multigrid waveform relaxation method for hyperbolic equations,” BIT Numerical Mathematics, vol. 36, no. 4, pp. 831–841, 1996, doi: 10.1007/BF01733794. [Online]. Available at: http://dx.doi.org/10.1007/BF01733794

K. Burrage, “Parallel methods for ODEs,” Advances in Computational Mathematics, vol. 7, pp. 1–3, 1997, doi: 10.1023/A:1018997130884. [Online]. Available at: http://dx.doi.org/10.1023/A:1018997130884

M. J. Gander and A. M. Stuart, “Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation,” SIAM Journal on Scientific Computing, vol. 19, no. 6, pp. 2014–2031, 1998, doi: 10.1137/S1064827596305337. [Online]. Available at: http://dx.doi.org/10.1137/S1064827596305337

Wang1998

F. Z. Wang, “Parallel-in-time relaxed Newton method for transient stability analysis,” IEE Proceedings - Generation, Transmission and Distribution, vol. 145, no. 2, pp. 155–159, 1998, doi: 10.1049/ip-gtd:19981836. [Online]. Available at: https://dx.doi.org/10.1049/ip-gtd:19981836
BibTeX entry Wang1998

@article{Wang1998, author = {Wang, F.~Z.}, doi = {10.1049/ip-gtd:19981836}, journal = {IEE Proceedings - Generation, Transmission and Distribution}, number = {2}, pages = {155--159}, title = {Parallel-in-time relaxed Newton method for transient stability analysis}, url = {https://dx.doi.org/10.1049/ip-gtd:19981836}, volume = {145}, year = {1998} }
Abstract for BibTeX entry Wang1998

The paper presents a new parallel method for the transient stability analysis of power systems. For simplicity, the classical power system model is adopted. The implicit trapezoidal rule is used to discretise the set of differential equations which describes the transient stability problem. As is well known, these discretised nonlinear algebraic equations are almost invariably solved by a Newton procedure. A parallel-in-time relaxed Newton method is proposed to solve the overall set of discretised equations concurrently on all the time steps. The proposed method has been tested on Cray-T3D by the use of PVM system. The test results show that the proposed method presents a good compromise between parallelism-in-time and convergence. Some important aspects for parallel transient stability analysis have been clarified

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1990 - 1994

A. Bellen, R. Vermiglio, and M. Zennaro, “Parallel ODE-solvers with stepsize control,” Journal of Computational and Applied Mathematics, vol. 31, no. 2, pp. 277–293, Aug. 1990, doi: 10.1016/0377-0427(90)90170-5. [Online]. Available at: https://doi.org/10.1016/0377-0427(90)90170-5

“On the theory of parallel Runge-Kutta methods,” IMA Journal of Numerical Analysis, vol. 10, no. 4, pp. 463–488, 1990, doi: 10.1093/imanum/10.4.463. [Online]. Available at: https://doi.org/10.1093/imanum/10.4.463

ScalaEtAl1990

M. La Scala, A. Bose, D. J. Tylavsky, and J. S. Chai, “A highly parallel method for transient stability analysis,” IEEE Transactions on Power Systems, vol. 5, no. 4, pp. 1439–1446, 1990, doi: 10.1109/59.99398. [Online]. Available at: http://dx.doi.org/10.1109/59.99398
BibTeX entry ScalaEtAl1990

@article{ScalaEtAl1990, author = {{La Scala}, M. and {Bose}, A. and {Tylavsky}, D. J. and {Chai}, J. S.}, doi = {10.1109/59.99398}, journal = {IEEE Transactions on Power Systems}, number = {4}, pages = {1439--1446}, title = {A highly parallel method for transient stability analysis}, url = {http://dx.doi.org/10.1109/59.99398}, volume = {5}, year = {1990} }
Abstract for BibTeX entry ScalaEtAl1990

A method for transient stability simulation is presented that aims to exploit the maximum degree of parallelism that the problem presents. The transient stability problem is viewed as a coupled set of nonlinear algebraic and differential equations; by applying a discretization method such as the trapezoidal rule, the overall algebraic-differential set of equations is thus transformed into an unique algebraic problem at each time step. A solution that considers every time step, not in a sequential way but concurrently, is suggested. The solution of this set of equations with a relaxation-type indirect method gives rise to a highly parallel algorithm. The method can handle all the typical dynamic models of realistic power system components. Test results are presented and shown to favorably compare with those obtained with the sequential dishonest Newton algorithm for realistic power systems
VanderHouwen1990

P. J. Van Der Houwen and B. P. Sommeijer, “Parallel iteration of high-order Runge-Kutta methods with stepsize control,” Journal of Computational and Applied Mathematics, vol. 29, no. 1, pp. 111–127, 1990, doi: 10.1016/0377-0427(90)90200-J. [Online]. Available at: http://dx.doi.org/10.1016/0377-0427(90)90200-J
BibTeX entry VanderHouwen1990

@article{VanderHouwen1990, author = {Van Der Houwen, P.~J. and Sommeijer, B.~P.}, doi = {10.1016/0377-0427(90)90200-J}, journal = {Journal of Computational and Applied Mathematics}, number = {1}, pages = {111--127}, title = {{Parallel iteration of high-order Runge-Kutta methods with stepsize control}}, url = {http://dx.doi.org/10.1016/0377-0427(90)90200-J}, volume = {29}, year = {1990} }
Abstract for BibTeX entry VanderHouwen1990

This paper investigates iterated Runge-Kutta methods of high order designed in such a way that the right-hand side evaluations can be computed in parallel. Using stepsize control based on embedded formulas a highly efficient code is developed. On parallel computers, the 8th-order mode of this code is more efficient than the DOPR18 implementation of the formulas of Prince and Dormand. The 10th-order mode is about twice as cheap for comparable accuracies.

D. E. Womble, “A time-stepping algorithm for parallel computers,” SIAM Journal on Scientific and Statistical Computing, vol. 11, no. 5, pp. 824–837, 1990, doi: 10.1137/0911049. [Online]. Available at: http://dx.doi.org/10.1137/0911049

C. W. Gear, “Waveform methods for space and time parallelism,” in Proceedings of the International Symposium on Computational Mathematics (Matsuyama, 1990), 1991, vol. 38, pp. 137–147.

G. Horton, “Time-Parallel Multigrid Solution of the Navier-Stokes Equations,” in Applications of Supercomputers in Engineering II, C. A. Brebbia, A. Peters, and D. Howard, Eds. Springer Netherlands, 1991, pp. 435–445 [Online]. Available at: http://dx.doi.org/10.1007/978-94-011-3660-0_31

Jackson1991

K. R. Jackson, “A SURVEY OF PARALLEL NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS,” IEEE Transactions on Magnetics, vol. 27, no. 5, pp. 3792–3797, 1991, doi: 10.1109/20.104928. [Online]. Available at: http://dx.doi.org/10.1109/20.104928
BibTeX entry Jackson1991

@article{Jackson1991, author = {Jackson, Kenneth R.}, doi = {10.1109/20.104928}, issue = {5}, journal = {IEEE Transactions on Magnetics}, pages = {3792--3797}, title = {A SURVEY OF PARALLEL NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS}, url = {http://dx.doi.org/10.1109/20.104928}, volume = {27}, year = {1991} }
Abstract for BibTeX entry Jackson1991

The parallel solution of Initial Value Problems for Ordinary Differential Equations has become an active area of research during the past few years. We briefly survey the recent developments in this area, with particular emphasis on traditional forward-step methods that offer the potential for effective small-scale parallelism on currently existing machines.

S. Murata, N. Satofuka, and T. Kushiyama, “Parabolic multi-grid method for incompressible viscous flows using a group explicit relaxation scheme,” Computers & Fluids, vol. 19, no. 1, pp. 33–41, 1991, doi: 10.1016/0045-7930(91)90005-3. [Online]. Available at: http://dx.doi.org/10.1016/0045-7930(91)90005-3

P. J. van der Houwen and B. P. Sommeijer, “Iterated Runge–Kutta Methods on Parallel Computers,” SIAM Journal on Scientific and Statistical Computing, vol. 12, no. 5, pp. 1000–1028, 1991, doi: 10.1137/0912054. [Online]. Available at: http://dx.doi.org/10.1137/0912054

G. Horton, “The time-parallel multigrid method,” Communications in Applied Numerical Methods, vol. 8, no. 9, pp. 585–595, 1992, doi: 10.1002/cnm.1630080906. [Online]. Available at: http://dx.doi.org/10.1002/cnm.1630080906

G. Horton, R. Knirsch, and H. Vollath, “The time-parallel solution of parabolic partial differential equations using the frequency-filtering method,” in Parallel Processing: CONPAR 92 –VAPP V, vol. 634, L. Bougé, M. Cosnard, Y. Robert, and D. Trystram, Eds. Springer Berlin Heidelberg, 1992, pp. 205–216 [Online]. Available at: http://dx.doi.org/10.1007/3-540-55895-0_415

G. Horton and R. Knirsch, “A time-parallel multigrid-extrapolation method for parabolic partial differential equations,” Parallel Computing, vol. 18, no. 1, pp. 21–29, 1992, doi: 10.1016/0167-8191(92)90108-J. [Online]. Available at: http://dx.doi.org/10.1016/0167-8191(92)90108-J

S. Vandewalle and R. Piessens, “Efficient Parallel Algorithms for Solving Initial-Boundary Value and Time-Periodic Parabolic Partial Differential Equations,” SIAM Journal on Scientific and Statistical Computing, vol. 13, no. 6, pp. 1330–1346, 1992, doi: 10.1137/0913075. [Online]. Available at: http://dx.doi.org/10.1137/0913075

K. Burrage, “Parallel methods for initial value problems,” Applied Numerical Mathematics, vol. 11, no. 1–3, pp. 5–25, 1993, doi: 10.1016/0168-9274(93)90037-R. [Online]. Available at: http://dx.doi.org/10.1016/0168-9274(93)90037-R

P. Chartier and B. Philippe, “A parallel shooting technique for solving dissipative ODE’s,” Computing, vol. 51, no. 3-4, pp. 209–236, 1993, doi: 10.1007/BF02238534. [Online]. Available at: http://dx.doi.org/10.1007/BF02238534

A. Fijany, “Time Parallel Algorithms for Solution of Linear Parabolic PDEs,” in Parallel Processing, 1993. ICPP 1993. International Conference on, 1993, vol. 3, pp. 51–56, doi: 10.1109/ICPP.1993.179 [Online]. Available at: http://dx.doi.org/10.1109/ICPP.1993.179

C. W. Gear and X. Xuhai, “Parallelism across time in ODEs,” Applied Numerical Mathematics. An IMACS Journal, vol. 11, no. 1-3, pp. 45–68, 1993.

C. Oosterlee and P. Wesseling, “Multigrid schemes for time-dependent incompressible Navier-Stokes equations,” IMPACT of Computing in Science and Engineering, vol. 5, no. 3, pp. 153–175, 1993, doi: 10.1006/icse.1993.1007. [Online]. Available at: http://dx.doi.org/10.1006/icse.1993.1007

ScalaBose1993

M. La Scala and A. Bose, “Relaxation/Newton methods for concurrent time step solution of differential-algebraic equations in power system dynamic simulations,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 40, no. 5, pp. 317–330, 1993, doi: 10.1109/81.232576. [Online]. Available at: http://dx.doi.org/10.1109/81.232576
BibTeX entry ScalaBose1993

@article{ScalaBose1993, author = {{La Scala}, M. and {Bose}, A.}, doi = {10.1109/81.232576}, journal = {IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications}, number = {5}, pages = {317--330}, title = {{Relaxation/Newton methods for concurrent time step solution of differential-algebraic equations in power system dynamic simulations}}, url = {http://dx.doi.org/10.1109/81.232576}, volume = {40}, year = {1993} }
Abstract for BibTeX entry ScalaBose1993

A class of algorithms that exploits the concurrent solution of many time steps is presented. By applying a stable integration method, the overall algebraic-differential set of equations can be transformed into a unique algebraic problem at each time step. The dynamic behavior of the system can be obtained by solving an enlarged set of algebraic equations relative to the simultaneous solution of many time steps. A class of relaxation/Newton algorithms can be used to solve this problem efficiently. This formulation permits easy implementation of multigrid techniques. The convergence rates and computational complexity of the algorithms are discussed. Test results for realistic power systems confirm theoretical expectations and show the promise of a several-fold increase in speed over that obtainable by traditional parallel-in-space approaches. The synergism obtainable by parallelism in time and in space can provide speed-up adequate for online implementations of transient stability analysis
Sommeijer1993

B. P. Sommeijer, “Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations,” Journal of Computational and Applied Mathematics, vol. 45, no. 1, pp. 151–168, 1993, doi: 10.1016/0377-0427(93)90271-C. [Online]. Available at: http://dx.doi.org/10.1016/0377-0427(93)90271-C
BibTeX entry Sommeijer1993

@article{Sommeijer1993, author = {Sommeijer, B.P.}, doi = {10.1016/0377-0427(93)90271-C}, journal = {{Journal of Computational and Applied Mathematics}}, number = {1}, pages = {151--168}, title = {{Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations}}, url = {http://dx.doi.org/10.1016/0377-0427(93)90271-C}, volume = {45}, year = {1993} }
Abstract for BibTeX entry Sommeijer1993

For the numerical integration of a stiff ordinary differential equation, fully implicit Runge-Kutta methods offer nice properties, like a high classical order and high stage order as well as an excellent stability behaviour. However, such methods need the solution of a set of highly coupled equations for the stage values and this is a considerable computational task. This paper discusses an iteration scheme to tackle this problem. By means of a suitable choice of the iteration parameters, the implicit relations for the stage values, as they occur in each iteration, can be uncoupled so that they can be solved in parallel. The resulting scheme can be cast into the class of Diagonally Implicit Runge-Kutta (DIRK) methods and, similar to these methods, requires only one LU factorization per step (per processor). The stability as well as the computational efficiency of the process strongly depends on the particular choice of the iteration parameters and on the number of iterations performed. We discuss several choices to obtain good stability and fast convergence. Based on these approaches, we wrote two codes possessing local error control and stepsize variation. We have implemented both codes on an ALLIANT FX/4 machine (four parallel vector processors and shared memory) and measured their speedup factors for a number of test problems. Furthermore, the performance of these codes is compared with the performance of the best stiff ODE codes for sequential computers, like SIMPLE, LSODE and RADAU5.
VanderHouwen1993

P. J. van der Houwen and B. P. Sommeijer, “Analysis of parallel diagonally implicit iteration of Runge-Kutta methods,” Applied Numerical Mathematics, vol. 11, no. 1, pp. 169–188, 1993, doi: 10.1016/0168-9274(93)90047-U. [Online]. Available at: http://dx.doi.org/10.1016/0168-9274(93)90047-U
BibTeX entry VanderHouwen1993

@article{VanderHouwen1993, author = {van der Houwen, P.J. and Sommeijer, B.P.}, doi = {10.1016/0168-9274(93)90047-U}, journal = {Applied Numerical Mathematics}, number = {1}, pages = {169--188}, title = {{Analysis of parallel diagonally implicit iteration of Runge-Kutta methods}}, url = {http://dx.doi.org/10.1016/0168-9274(93)90047-U}, volume = {11}, year = {1993} }
Abstract for BibTeX entry VanderHouwen1993

In this paper, we analyze parallel, diagonally implicit iteration of Runge-Kutta methods (PDIRK methods) for solving large systems of stiff equations on parallel computers. Like Newton-iterated backward differentiation formulas (BDFs), these PDIRK methods are such that in each step the (sequential) costs consist of solving a number of linear systems with the same matrix of coefficients and with the same dimension as the system of differential equations. Although for PDIRK methods the number of linear systems is usually higher than for Newton iteration of BDFs, the more computationally intensive work of computing the matrix of coefficients and its LU-decomposition are identical. The advantage of PDIRK methods over Newton-iterated BDFs is their unconditional stability (A-stability for Gauss-based methods and L-stability for Radau-based methods) for any order of accuracy. Special characteristics of the PDIRK methods will be studied, such as the rate of convergence, the influence of particular predictors on the resulting stability properties, and the stiff error constants in the global error.

H. G. Bock, W. Hackbusch, and W. Rannacher, Eds., Parallel multigrid waveform relaxation for parabolic problems. Stuttgart: B. G. Teubner, 1993 [Online]. Available at: http://dx.doi.org/10.1007/978-3-322-94761-1

J. Janssen and S. Vandewalle, “Multigrid waveform relaxation on spatial finite element meshes,” in Contributions to multigrid (Amsterdam, 1993), vol. 103, Amsterdam: Math. Centrum Centrum Wisk. Inform., 1994, pp. 75–86.

M. Kiehl, “Parallel multiple shooting for the solution of initial value problems,” Parallel Computing, vol. 20, no. 3, pp. 275–295, 1994, doi: 10.1016/S0167-8191(06)80013-X. [Online]. Available at: http://dx.doi.org/10.1016/S0167-8191(06)80013-X

N. Toomarian, A. Fijany, and J. Barmen, “Time-parallel solution of linear partial differential equations on the Intel Touchstone Delta supercomputer,” Concurrency: Practice and Experience, vol. 6, no. 8, pp. 641–652, 1994, doi: 10.1002/cpe.4330060803. [Online]. Available at: http://dx.doi.org/10.1002/cpe.4330060803

S. Vandewalle and G. Horton, “Multicomputer-Multigrid Solution of Parabolic Partial Differential Equations,” in Multigrid Methods IV, vol. 116, P. W. Hemker and P. Wesseling, Eds. Birkhäuser Basel, 1994, pp. 97–109 [Online]. Available at: http://dx.doi.org/10.1007/978-3-0348-8524-9_7

S. G. Vandewalle and E. F. Van de Velde, “Space-time concurrent multigrid waveform relaxation,” Annals of Numerical Mathematics, vol. 1, no. 1-4, pp. 347–360, 1994, doi: 10.13140/2.1.1146.1761. [Online]. Available at: http://dx.doi.org/10.13140/2.1.1146.1761

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Pre 1990

J. Nievergelt, “Parallel methods for integrating ordinary differential equations,” Commun. ACM, vol. 7, no. 12, pp. 731–733, 1964, doi: 10.1145/355588.365137. [Online]. Available at: http://dx.doi.org/10.1145/355588.365137

W. L. Miranker and W. Liniger, “Parallel methods for the numerical integration of ordinary differential equations,” Mathematics of Computation, vol. 21, no. 99, pp. 303–320, 1967, doi: 10.1090/S0025-5718-1967-0223106-8. [Online]. Available at: http://dx.doi.org/10.1090/S0025-5718-1967-0223106-8

P. B. Worland, “Parallel Methods for the Numerical Solution of Ordinary Differential Equations,” Computers, IEEE Transactions on, vol. C-25, no. 10, pp. 1045–1048, 1976, doi: 10.1109/TC.1976.1674545. [Online]. Available at: http://dx.doi.org/10.1109/TC.1976.1674545

Franklin1978

M. A. Franklin, “Parallel Solution of Ordinary Differential Equations,” IEEE Transactions on Computers, vol. C-27, no. 5, pp. 413–420, 1978, doi: 10.1109/TC.1978.1675121. [Online]. Available at: http://dx.doi.org/10.1109/TC.1978.1675121
BibTeX entry Franklin1978

@article{Franklin1978, author = {Franklin, M. A.}, doi = {10.1109/TC.1978.1675121}, journal = {IEEE Transactions on Computers}, number = {5}, pages = {413--420}, title = {Parallel Solution of Ordinary Differential Equations}, url = {http://dx.doi.org/10.1109/TC.1978.1675121}, volume = {C-27}, year = {1978} }
Abstract for BibTeX entry Franklin1978

This paper presents a performance comparison of three parallel algorithms for the solution of sets of coupled first-order differential equations. A general format for comparison of the algorithms is given, and performance equations for the two processor cases are developed. The equations take into account both the computational and intercommunications requirements of the processors. These equations are applied to a benchmark of six problems and a simple, single bus multiprocessor architecture. The more than two processor case is considered for one of the parallel algorithms.

W. Hackbusch, “Parabolic multi-grid methods,” Computing Methods in Applied Sciences and Engineering, VI, pp. 189–197, 1984 [Online]. Available at: http://dl.acm.org/citation.cfm?id=4673.4714

M. Chu and H. Hamilton, “Parallel Solution of ODE’s by Multiblock Methods,” SIAM Journal on Scientific and Statistical Computing, vol. 8, no. 3, pp. 342–353, 1987, doi: 10.1137/0908039. [Online]. Available at: http://dx.doi.org/10.1137/0908039

C. Lubich and A. Ostermann, “Multi-grid dynamic iteration for parabolic equations,” BIT Numerical Mathematics, vol. 27, no. 2, pp. 216–234, 1987, doi: 10.1007/BF01934186. [Online]. Available at: http://dx.doi.org/10.1007/BF01934186

C. W. Gear, “Parallel methods for ordinary differential equations,” CALCOLO, vol. 25, no. 1-2, pp. 1–20, 1988, doi: 10.1007/BF02575744. [Online]. Available at: http://dx.doi.org/10.1007/BF02575744

A. Bellen and M. Zennaro, “Parallel algorithms for initial-value problems for difference and differential equations,” Journal of Computational and Applied Mathematics, vol. 25, no. 3, pp. 341–350, 1989, doi: 10.1016/0377-0427(89)90037-X. [Online]. Available at: http://dx.doi.org/10.1016/0377-0427(89)90037-X

GallopoulosEtAl1989

E. Gallopoulos and Y. Saad, “On the Parallel Solution of Parabolic Equations,” in Proceedings of the 3rd International Conference on Supercomputing, New York, NY, USA, 1989, pp. 17–28, doi: 10.1145/318789.318793 [Online]. Available at: http://doi.acm.org/10.1145/318789.318793
BibTeX entry GallopoulosEtAl1989

@inproceedings{GallopoulosEtAl1989, acmid = {318793}, address = {New York, NY, USA}, author = {Gallopoulos, E. and Saad, Y.}, booktitle = {Proceedings of the 3rd International Conference on Supercomputing}, doi = {10.1145/318789.318793}, numpages = {12}, pages = {17--28}, publisher = {ACM}, series = {ICS '89}, title = {On the Parallel Solution of Parabolic Equations}, url = {http://doi.acm.org/10.1145/318789.318793}, year = {1989} }
Abstract for BibTeX entry GallopoulosEtAl1989

We propose new parallel algorithms for the solution of linear parabolic problems. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Padé and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. We also present experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors.

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