Jülich Supercomputing Centre is providing travel support for PhD students working in the field of parallel-in-time integration methods.

With this commitment, JSC is emphasizing the importance of regular community meetings, in particular for young scientists working on their PhD thesis. Up to three students per workshop will receive up to EUR 1.500 each for covering parts of their travel expenses.

The Scientific Committee of the PinT Workshop Series will select the students based on applications. These should include:

  • a description of their thesis topic, stating the connection to parallel-in-time methods (“This is what I’m working on”, 1-2 pages)
  • a letter of motivation, containing a short CV (“This is why I should receive the grant”, 1-2 pages)
  • a short description of the group and institute they are affiliated to (“This is where I work”, up to 1 page)

Also, students traveling to one of the meetings via this grant are expected to present their work either on a poster or during a talk. This grant will be awarded only once per student.

Details on the application procedure will be posted for each workshop separately on the respective local websites.

Previous Grantees

8th PinT Workshop, Bielefeld, Germany

Abram Ellison, University of Colorado at Boulder, USA

Talk: “A parallel-in-time approach for wave-type PDEs”

Numerical solutions to wave-type PDEs utilizing method-of-lines requires the ODE solver’s stability domain to include a quite large stretch of the imaginary axis surrounding the origin. We show here that extrapolation based solvers of Gragg-Bulirsch-Stoer (GBS) type can meet this requirement. Extrapolation methods utilize several independent time stepping sequences, making them highly suited for parallel execution. Traditional extrapolation schemes use all time stepping sequences to maximize the method’s order of accuracy. The present method instead maintains a desired order of accuracy while employing additional time stepping sequences to shape the resulting stability domain. We present an optimization method to maximize the stability domain’s imaginary axis coverage. This yields an explicit scheme with maximal time step size for wave propagation problems. In a parallel implementation we achieve both high order and fast time to solution compared with traditional ODE integrators.

Wisdom C. Agboh, University of Leeds, United Kingdom

Talk: “Combining Coarse and Fine Physics for Manipulation using Parallel-in-Time Integration”

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.

Hieu Nguyen, University of Texas at Austin, USA

Talk: “A Stable Parareal by Energy-Preserving Boost for the Second Order Wave Equation”

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 shorter time steps and finer spatial grid, and is run in parallel for shorter time intervals. The two propagators are coupled in a way that resembles the parareal method. The computed data gathered during the iterations are used to stabilize the iterations by bridging the gap in the energy of the solutions computed by the two propagators. Some numerical examples in one and two dimensions are provided to demonstrate the convergence and stability property of the proposed method.

7th PinT Workshop, Roscoff, France

Andrew Clarke, University of Leeds, United Kingdom

Talk: “Parallel in time integration of the Kinematic Dynamo”

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 spaces 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. This talk will investigate Parareal’s ability to speed up simulations of the kinematic dynamo, where the velocity field is a prescribed field in the induction equation. We will describe an implementation of Parareal into the open-source Python software Dedalus, which implements spectral methods with implicit-explicit (IMEX) time-stepping. Good performance in Parareal depends on an efficient coarse propagator, which should be faster than the fine propagator, whilst still giving adequate accuracy for good convergence. In this case, reduced spatial resolution is used for the coarse solver. This also allows larger time step sizes, further increasing the speed of the coarse propagator. Convergence properties, speed-up, and efficiency of the Parareal algorithm for the Roberts flow and other types of dynamos will be presented.

Asad Anees, Clausthal University of Technology, Germany

Talk: “Time Domain Finite Element Methods for Maxwell’s Equations”

In this presentation, we discuss a time-domain finite element methods to the solution of linear and nonlinear Maxwell’s equations. A weak formulation is derived for the electric and magnetic field with appropriate initial and boundary conditions, and the problem is discretised both in space and time. Nédeléc curl conforming and Raviart Thomas divergence conforming finite elements are used to discretized the electric and magnetic fields respectively in space. In time domain, problems are discretized by symplectic and backward Euler methods, and provide some numerical results to validate our simulation for fully discretized problems. The proposed methods are accurate in time up to order 4, in case of symplectic time integration and they are conditionally stable. Our methods also allow to treat a complex geometries of various physical system coupled to electromagnetics fields. Electric and magnetic fields are also visualized at each time step on tetrahedron meshes. Our proposed methods are also parallel, but time integration is not parallel. We are working to add parallel time integrations in our proposed mixed finite element methods for electromagnetic using XBraid.

Charles Murray, Durham University, United Kingdom

Talk: “Full space-time multigrid using time stepping specific multigrid restriction and prolongation”

Classic multigrid methods operate on a grid discretised in space and use a complementing method (time stepping) to advance a solution in time. This synchronises/sequentialises the time steps. A space-time grid can be used to discretise both space and time requiring a stencil extended in the time dimension. Traditional multigrid methods on space-time grids use multigrid exclusively in space and couple space and time on the finest level, while traditional space parallelisation couples whole time slices with each other. We propose (i) to use multigid on the global space-time grid which is well-known to comprise both classic spatial multigrid and parallel-in-time methods. We propose (ii) to use classic Galerkin operator construction to come up with coarse time stepping rules. The time stepping is read as limit of a convection operator. We propose (ii) to extend the algebraic multigrid technique of Black Box Multigrid (BoxMG) to compute operator- and time stepping-specific multigrid restriction and prolongation. Such an approach is doomed to fail if the proposed realisation is not fast and scaling. Our approach relies on three pillars. First, we apply dynamically adaptive meshes in space in time and we use full multigrid cycles in space and time, this is combined it with Full Approximation Storage (FAS). The latter allows for a straightforward realisation of arbitrary dynamic mesh refinement (AMR), combined with the space-time solver this gives dynamic time-stepping effectively for free. Second, we observe that classic multigrid methods require all stencils, i.e. matrix entries, to be computed prior to the solve, at the same time, the first few iterations determine approximations of limited accuracy only. To be able to exploit massively parallel computers without a ramp up (assembly) phase we propose to kick off multigrid with very rough stencil approximations but to use idle cores to determine better and better stencil integrations on-the-fly. Finally, we note that multigrid suffers from limited arithmetic intensity and concurrency, while the gap between compute power and memory bandwidth is widening. It is thus important to read each unknown as rarely as possible. Additive multigrid variants are well-known to have advantageous scaling properties but tend to be less stable and efficient than their multiplicative counterparts. Asynchronous Fast Adaptive Composite (AFAC) candidates to stabilise additive multigrid as each correction term is damped by an additional multilevel coarse grid solve. We propose a pipelined, single-touch realisation of AFAC in space and time where each degree of freedom is loaded only once per multilevel smoothing step. It is memory-access optimal.

6th Conference on Parallel-in-Time Integration, Ascona, Switzerland

Ben S. Southworth, University of Colorado at Boulder, USA

Talk: “Solving Space-time Discretizations of the Wave Equation with Algebraic Multigrid”

Algebraic multigrid (AMG) is an iterative solver for large sparse linear systems that, for many applications, scales linearly in complexity with the number of degrees of freedom. It also scales in parallel to hundreds of thousands of cores, making it a key component of many high-performance simulation codes. For the symmetric positive definite case, often resulting from the discretization of elliptic partial differential equations (PDEs), convergence of AMG is well motivated and AMG is among the fastest numerical solvers available. However, highly nonsymmetric matrices pose difficulties for AMG in theory and in practice. Although a number of efforts have been made to develop AMG for nonsymmetric linear systems, few have considered highly nonsymmetric cases, and those that have demonstrated success on nonsymmetric systems typically use a multilevel strategy that does not scale well in parallel (K-cycles or W-cycles). Each of these issues are likely part of the reason that AMG has yet to be demonstrated as an effective parallel-in-time solver. Although a number of works have looked at geometric multigrid and parabolic PDEs, there is a large gap in the parallel-intime literature in terms of algebraic solvers and/or space-time discretizations of hyperbolic PDEs.

Although hyperbolic PDEs arise frequently in physical simulation, their solution is often constrained to sequential, explicit time stepping schemes, or implicit schemes with relatively slow (not O(n)) linear solvers. Recently, we developed a reduction-based algebraic multigrid (AMG) solver, AMGir, and its generalization LAIR, to solve upwind discretizations of the steady- state transport equation. AMGir/LAIR proved to be a fast and robust solver for the steady-state transport equation, even on unstructured meshes and with high-order finite elements. Due to the success of AMGir/LAIR applied to upwind discretizations of gradients, a natural research direction to consider is further developing the method with a focus on parallel-in-time applications. Full spacetime discretizations often use some form of upwind or semi-upwind discretization in time, which leads to a global space-time matrix amenable to AMGir/LAIR.

Here, we consider a discretization of the 2d-space, 1d-time, wave equation written in first-order form, corresponding to a system with two variables. A modification of AMGir/LAIR is developed to account for the system structure, and the method proves very effective. In particular, the reduction aspect of AMGir/LAIR applied to the wave equation is striking – initial iterations appear to diverge, but convergence factors then plummet to ρ ~ 1E−10. AMGir/LAIR is able to seamlessly handle the system structure and solve the space-time wave equation, a well-known challenging parallel-in-time problem, showing promise as a robust framework for parallel-in-time.

Federico Danieli, Mathematical Institute, University of Oxford, UK

Title: “An Alternative to the Coarse Solver for the Parareal Algorithm”

The Parareal algorithm is one of the simplest and most widely spread techniques to achieve parallelisation in the computation of the solution of ODEs and PDEs by splitting their time domain. However, ensuring its stability can be a challenging task, which for the largest part revolves around the choice of the most apt pair of fine and coarse solvers for the problem at hand. Stability is also an issue in the case of advection-dominated equations, where the algorithm has often been shown to perform poorly. In the attempt to overcome these problems, an alternative formulation of Parareal is presented. Starting from an interpretation of the algorithm as a Newton method, we notice how the sensitivity of the solution from the application of the fine solver, with respect to variations on the initial conditions, appears in the update formula. Rather than resorting to the application of a coarse solver in order to approximate this term and consequently propagate the update along the time domain, we aim to estimate this sensitivity in a direct manner. The approach chosen is suitable for systems of ODEs of small size and some simple PDEs, and extensions to general cases are not trivial and still remain object of further work. However, the first experiments on model problems show the potential of this method to overcome some of the limitations of Parareal, as well as to boost its convergence speed.

Marc Olm, Technical University of Catalonia & CIMNE, Spain

Title: “Nonlinear parallel-in-time multilevel Schur complement solvers for ordinary differential equations”

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.

5th PinT Workshop, Banff, Canada

Thibaut Lunet, ISAE-Supaero & CERFACS, Toulouse, France

Talk: “On the time-parallelization of the solution of Navier-Stokes equations using Parareal

Unsteady turbulent flow simulations using the Navier Stokes equations require larger and larger problem sizes. On an other side, new supercomputer architectures will be available in the next decade, with computational power based on a larger number of cores rather than significantly increased CPU frequency. Hence most of the current generation CFD software will face critical efficiency issues if bounded to massive spatial parallelization and we consider time parallelization as an attractive alternative to enhance efficiency on multi-cores architectures. Several algorithms developed in the last decades (Parareal, PFASST) may be straightforwardly applied to the Navier-Stokes equations, but the Parareal algorithm remains one of the simplest solutions in the case of explicit time stepping, compressible flow Based on an optimized implementation of Parareal we modelize the speed-up obtained when combining both space and time parallelizations. This modelization takes into account the speedup of an actual structured, massively parallel CFD solver and the cost of time communications, both measured on two different supercomputers. Some preliminary requirements for a worthy time-parallel integration will be then derived, in terms of both Parareal iteration count and size of the time subdomain window. We then study within this framework, possible enhancements of the well-known convergence difficulties for Parareal encountered for advection dominated problems. The proposed approach is based on the representation of Parareal as an algebraic system of nonlinear equations solved by a preconditioned Newton’s method. The new formulation targets the reduction of the degree of non-normality of its Jacobian by slightly modifying the Parareal iteration. Performance on examples related to canonical linear problems, like the Dahlquist and the one-dimensional advection equation, is analysed. To conclude we comment on the extension of this method to nonlinear problems.

Stephanie Günther, TU Kaiserslautern, AG Scientific Computing, Kaiserslautern, Germany

Talk: “Adjoint Sensitivity Computation for the Parallel Multigrid Reduction in Time Software Library XBraid

In this paper we present an adjoint solver for the multigrid in time software library XBraid. 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. However, in many engineering applications not only the primal unsteady flow computation is of interest but also the ability to compute sensitivities that determine the influence of design changes to some output quantity. In recent years, adjoint solvers have widely been developed which propagate sensitivity information backwards through the time domain. We develop an adjoint solver for XBraid that enhances the primal iterations by an iteration for computing adjoint sensitivities. In each iteration, the adjoint code runs backwards through the primal XBraid actions and computes the consistent discrete adjoint sensitivities parallel in time. It is highly non-intrusive as existing adjoint time propagators can easily be integrated through the adjoint interface. We validate the adjoint code by applying it to an unsteady partial differential equation that mimics the behavior of separated flows past bluff bodies. In our 1D model, the near wake is mimicked by a nonlinear ODE, namely the Lorenz attractor which exhibits self-excited oscillations. The far wake is modeled by an advection - diffusion equation whose upstream boundary condition is determined by the ODE mimicking the near wake. We demonstrate the integration of a serial time stepping algorithm, that solves the PDE forward in time, into the parallel-in-time XBraid framework as well as the development of the corresponding adjoint interface. The resulting sensitivities are in good agreement with those computed from finite differences. Nevertheless, there is still great potential for optimizing the performance of the adjoint code using advanced algorithmic differentiation techniques such as reverse accumulation and checkpointing. Due to the iterative nature of the primal and the adjoint flow computation, the method is very well suited for simultaneous optimization algorithms like the One-shot method which solve the optimization problem in the full space. They have proven to be very efficient for optimization with steady-state PDE constraints while its application to unsteady PDE is still under development. The non-intrusive adjoint XBraid solver is therefore highly desirable and will extend its application range from pure simulation to optimization.