Step-7: pySDC with external libraries¶
pySDC can be used with external libraries, in particular for spatial discretization, parallelization and solving of linear and/or nonlinear systems. In the following, we show a few examples of pySDC + X.
Part A: pySDC and FEniCS¶
In this example, pySDC is coupled with the FEniCS framework for using finite elements in space. This implies significant changes to the algorithm, depending on whether or not the mass matrix should be inverted. SDC, MLSDC and PFASST can be used without changes when the right-hand side of the ODE is defined with the inverse of the mass matrix. Otherwise, the mass matrix has to be used for e.g. the tau-correction. This example tests different variants of this methodology for SDC, MLSDC and PFASST.
Important things to note:
This example shows that even core routines like the BaseTransfer can be overwritten if needed.
It is also valuable to check out the data type and transfer classes required to work with FEniCS. Both can be found in the implementations folder.
Full code: pySDC/tutorial/step_7/A_pySDC_with_FEniCS.py
from pathlib import Path
import numpy as np
from pySDC.helpers.stats_helper import get_sorted
from pySDC.implementations.controller_classes.controller_nonMPI import controller_nonMPI
from pySDC.implementations.problem_classes.HeatEquation_1D_FEniCS_matrix_forced import (
fenics_heat_mass,
fenics_heat,
fenics_heat_mass_timebc,
)
from pySDC.implementations.sweeper_classes.imex_1st_order_mass import imex_1st_order_mass, imex_1st_order
from pySDC.implementations.transfer_classes.TransferFenicsMesh import mesh_to_mesh_fenics
def setup(t0=None, ml=None):
"""
Helper routine to set up parameters
Args:
t0 (float): initial time
ml (bool): use single or multiple levels
Returns:
description and controller_params parameter dictionaries
"""
# initialize level parameters
level_params = dict()
level_params['restol'] = 5e-10
level_params['dt'] = 0.2
# initialize step parameters
step_params = dict()
step_params['maxiter'] = 20
# initialize sweeper parameters
sweeper_params = dict()
sweeper_params['quad_type'] = 'RADAU-RIGHT'
if ml:
# Note that coarsening in the nodes actually HELPS MLSDC to converge (M=1 is exact on the coarse level)
sweeper_params['num_nodes'] = [3, 1]
else:
sweeper_params['num_nodes'] = [3]
problem_params = dict()
problem_params['nu'] = 0.1
problem_params['t0'] = t0 # ugly, but necessary to set up this ProblemClass
problem_params['c_nvars'] = [128]
problem_params['family'] = 'CG'
problem_params['c'] = 1.0
if ml:
# We can do rather aggressive coarsening here. As long as we have 1 node on the coarse level, all is "well" (ie.
# MLSDC does not take more iterations than SDC, but also not less). If we just coarsen in the refinement (and
# not in the nodes and order, the mass inverse approach is way better, ie. halves the number of iterations!
problem_params['order'] = [4, 1]
problem_params['refinements'] = [1, 0]
else:
problem_params['order'] = [4]
problem_params['refinements'] = [1]
# initialize controller parameters
controller_params = dict()
controller_params['logger_level'] = 30
base_transfer_params = dict()
base_transfer_params['finter'] = True
# Fill description dictionary for easy hierarchy creation
description = dict()
description['problem_class'] = None
description['problem_params'] = problem_params
description['sweeper_class'] = None
description['sweeper_params'] = sweeper_params
description['level_params'] = level_params
description['step_params'] = step_params
description['space_transfer_class'] = mesh_to_mesh_fenics
description['base_transfer_params'] = base_transfer_params
return description, controller_params
def run_variants(variant=None, ml=None, num_procs=None):
"""
Main routine to run the different implementations of the heat equation with FEniCS
Args:
variant (str): specifies the variant
ml (bool): use single or multiple levels
num_procs (int): number of processors in time
"""
Tend = 1.0
t0 = 0.0
description, controller_params = setup(t0=t0, ml=ml)
if variant == 'mass':
# Note that we need to reduce the tolerance for the residual here, since otherwise the error will be too high
description['level_params']['restol'] /= 500
description['problem_class'] = fenics_heat_mass
description['sweeper_class'] = imex_1st_order_mass
elif variant == 'mass_inv':
description['problem_class'] = fenics_heat
description['sweeper_class'] = imex_1st_order
elif variant == 'mass_timebc':
# Can increase the tolerance here, errors are higher anyway
description['level_params']['restol'] *= 20
description['problem_class'] = fenics_heat_mass_timebc
description['sweeper_class'] = imex_1st_order_mass
else:
raise NotImplementedError('Variant %s is not implemented' % variant)
# quickly generate block of steps
controller = controller_nonMPI(num_procs=num_procs, controller_params=controller_params, description=description)
# get initial values on finest level
P = controller.MS[0].levels[0].prob
uinit = P.u_exact(0.0)
# call main function to get things done...
uend, stats = controller.run(u0=uinit, t0=t0, Tend=Tend)
# compute exact solution and compare
uex = P.u_exact(Tend)
err = abs(uex - uend) / abs(uex)
Path("data").mkdir(parents=True, exist_ok=True)
f = open('data/step_7_A_out.txt', 'a')
out = f'Variant {variant} with ml={ml} and num_procs={num_procs} -- error at time {Tend}: {err}'
f.write(out + '\n')
print(out)
# filter statistics by type (number of iterations)
iter_counts = get_sorted(stats, type='niter', sortby='time')
niters = np.array([item[1] for item in iter_counts])
out = ' Mean number of iterations: %4.2f' % np.mean(niters)
f.write(out + '\n')
print(out)
out = ' Range of values for number of iterations: %2i ' % np.ptp(niters)
f.write(out + '\n')
print(out)
out = ' Position of max/min number of iterations: %2i -- %2i' % (int(np.argmax(niters)), int(np.argmin(niters)))
f.write(out + '\n')
print(out)
out = ' Std and var for number of iterations: %4.2f -- %4.2f' % (float(np.std(niters)), float(np.var(niters)))
f.write(out + '\n')
print(out)
timing = get_sorted(stats, type='timing_run', sortby='time')
out = f'Time to solution: {timing[0][1]:6.4f} sec.'
f.write(out + '\n')
print(out)
if num_procs == 1:
assert np.mean(niters) <= 6.0, 'Mean number of iterations is too high, got %s' % np.mean(niters)
if variant == 'mass' or variant == 'mass_inv':
assert err <= 1.14e-08, 'Error is too high, got %s' % err
else:
assert err <= 3.25e-07, 'Error is too high, got %s' % err
else:
assert np.mean(niters) <= 11.6, 'Mean number of iterations is too high, got %s' % np.mean(niters)
assert err <= 1.14e-08, 'Error is too high, got %s' % err
f.write('\n')
print()
f.close()
def main():
run_variants(variant='mass_inv', ml=False, num_procs=1)
run_variants(variant='mass', ml=False, num_procs=1)
run_variants(variant='mass_timebc', ml=False, num_procs=1)
run_variants(variant='mass_inv', ml=True, num_procs=1)
run_variants(variant='mass', ml=True, num_procs=1)
run_variants(variant='mass_timebc', ml=True, num_procs=1)
run_variants(variant='mass_inv', ml=True, num_procs=5)
# WARNING: all other variants do NOT work, either because of FEniCS restrictions (weak forms with different meshes
# will not work together) or because of inconsistent use of the mass matrix (locality condition for the tau
# correction is not satisfied, mass matrix does not permute with restriction).
# run_pfasst_variants(variant='mass', ml=True, num_procs=5)
if __name__ == "__main__":
main()
Results:
Variant mass_inv with ml=False and num_procs=1 -- error at time 1.0: 1.1387407230222816e-08
Mean number of iterations: 6.00
Range of values for number of iterations: 0
Position of max/min number of iterations: 0 -- 0
Std and var for number of iterations: 0.00 -- 0.00
Time to solution: 1.6104 sec.
Variant mass with ml=False and num_procs=1 -- error at time 1.0: 1.1387594756569534e-08
Mean number of iterations: 6.00
Range of values for number of iterations: 0
Position of max/min number of iterations: 0 -- 0
Std and var for number of iterations: 0.00 -- 0.00
Time to solution: 1.2121 sec.
Variant mass_timebc with ml=False and num_procs=1 -- error at time 1.0: 3.2473562155116167e-07
Mean number of iterations: 6.00
Range of values for number of iterations: 3
Position of max/min number of iterations: 3 -- 0
Std and var for number of iterations: 1.10 -- 1.20
Time to solution: 1.2093 sec.
Variant mass_inv with ml=True and num_procs=1 -- error at time 1.0: 1.138768636885694e-08
Mean number of iterations: 6.00
Range of values for number of iterations: 0
Position of max/min number of iterations: 0 -- 0
Std and var for number of iterations: 0.00 -- 0.00
Time to solution: 2.2561 sec.
Variant mass with ml=True and num_procs=1 -- error at time 1.0: 1.1387216566052821e-08
Mean number of iterations: 6.00
Range of values for number of iterations: 0
Position of max/min number of iterations: 0 -- 0
Std and var for number of iterations: 0.00 -- 0.00
Time to solution: 1.9138 sec.
Variant mass_timebc with ml=True and num_procs=1 -- error at time 1.0: 3.2473561574763597e-07
Mean number of iterations: 6.00
Range of values for number of iterations: 3
Position of max/min number of iterations: 3 -- 0
Std and var for number of iterations: 1.10 -- 1.20
Time to solution: 1.8948 sec.
Variant mass_inv with ml=True and num_procs=5 -- error at time 1.0: 1.1150087179536389e-08
Mean number of iterations: 11.60
Range of values for number of iterations: 9
Position of max/min number of iterations: 4 -- 0
Std and var for number of iterations: 3.26 -- 10.64
Time to solution: 3.6269 sec.
Part B: mpi4py-fft for parallel Fourier transforms¶
The most prominent parallel solver is, probably, the FFT. While many implementations or wrappers for Python exist, we decided to use mpi4py-fft, which provided the easiest installation, a simple API and good parallel scaling. As an example, we here test the nonlinear Schrödinger equation, using the IMEX sweeper to treat the nonlinear parts explicitly. The code allows to work both in real and spectral space, while the latter is usually faster. This example tests SDC, MLSDC and PFASST.
Important things to note:
The code runs both in serial using just python B_pySDC_with_mpi4pyfft.py and also in parallel using mpirun -np 2 python B_pySDC_with_mpi4pyfft.py.
The nonlinear Schrödinger example is not expected to work well with PFASST. In fact, SDC and MLSDC converge for larger time-steps, but PFASST does not.
Full code: pySDC/tutorial/step_7/B_pySDC_with_mpi4pyfft.py
import numpy as np
from pathlib import Path
from mpi4py import MPI
from pySDC.helpers.stats_helper import get_sorted
from pySDC.implementations.controller_classes.controller_nonMPI import controller_nonMPI
from pySDC.implementations.sweeper_classes.imex_1st_order import imex_1st_order
from pySDC.implementations.problem_classes.NonlinearSchroedinger_MPIFFT import nonlinearschroedinger_imex
from pySDC.implementations.transfer_classes.TransferMesh_MPIFFT import fft_to_fft
def run_simulation(spectral=None, ml=None, num_procs=None):
"""
A test program to do SDC, MLSDC and PFASST runs for the 2D NLS equation
Args:
spectral (bool): run in real or spectral space
ml (bool): single or multiple levels
num_procs (int): number of parallel processors
"""
comm = MPI.COMM_WORLD
rank = comm.Get_rank()
# initialize level parameters
level_params = dict()
level_params['restol'] = 1e-08
level_params['dt'] = 1e-01 / 2
level_params['nsweeps'] = [1]
# initialize sweeper parameters
sweeper_params = dict()
sweeper_params['quad_type'] = 'RADAU-RIGHT'
sweeper_params['num_nodes'] = [3]
sweeper_params['QI'] = ['LU'] # For the IMEX sweeper, the LU-trick can be activated for the implicit part
sweeper_params['initial_guess'] = 'zero'
# initialize problem parameters
problem_params = dict()
if ml:
problem_params['nvars'] = [(128, 128), (32, 32)]
else:
problem_params['nvars'] = [(128, 128)]
problem_params['spectral'] = spectral
problem_params['comm'] = comm
# initialize step parameters
step_params = dict()
step_params['maxiter'] = 50
# initialize controller parameters
controller_params = dict()
controller_params['logger_level'] = 30 if rank == 0 else 99
# controller_params['predict_type'] = 'fine_only'
# fill description dictionary for easy step instantiation
description = dict()
description['problem_params'] = problem_params # pass problem parameters
description['problem_class'] = nonlinearschroedinger_imex
description['sweeper_class'] = imex_1st_order
description['sweeper_params'] = sweeper_params # pass sweeper parameters
description['level_params'] = level_params # pass level parameters
description['step_params'] = step_params # pass step parameters
description['space_transfer_class'] = fft_to_fft
# set time parameters
t0 = 0.0
Tend = 1.0
f = None
if rank == 0:
Path("data").mkdir(parents=True, exist_ok=True)
f = open('data/step_7_B_out.txt', 'a')
out = f'Running with ml={ml} and num_procs={num_procs}...'
f.write(out + '\n')
print(out)
# instantiate controller
controller = controller_nonMPI(num_procs=num_procs, controller_params=controller_params, description=description)
# get initial values on finest level
P = controller.MS[0].levels[0].prob
uinit = P.u_exact(t0)
# call main function to get things done...
uend, stats = controller.run(u0=uinit, t0=t0, Tend=Tend)
uex = P.u_exact(Tend)
err = abs(uex - uend)
if rank == 0:
# filter statistics by type (number of iterations)
iter_counts = get_sorted(stats, type='niter', sortby='time')
niters = np.array([item[1] for item in iter_counts])
out = (
f' Min/Mean/Max number of iterations: '
f'{np.min(niters):4.2f} / {np.mean(niters):4.2f} / {np.max(niters):4.2f}'
)
f.write(out + '\n')
print(out)
out = ' Range of values for number of iterations: %2i ' % np.ptp(niters)
f.write(out + '\n')
print(out)
out = ' Position of max/min number of iterations: %2i -- %2i' % (
int(np.argmax(niters)),
int(np.argmin(niters)),
)
f.write(out + '\n')
print(out)
out = ' Std and var for number of iterations: %4.2f -- %4.2f' % (float(np.std(niters)), float(np.var(niters)))
f.write(out + '\n')
print(out)
out = f'Error: {err:6.4e}'
f.write(out + '\n')
print(out)
timing = get_sorted(stats, type='timing_run', sortby='time')
out = f'Time to solution: {timing[0][1]:6.4f} sec.'
f.write(out + '\n')
print(out)
assert err <= 1.133e-05, 'Error is too high, got %s' % err
if ml:
if num_procs > 1:
maxmean = 12.5
else:
maxmean = 6.6
else:
maxmean = 12.7
assert np.mean(niters) <= maxmean, 'Mean number of iterations is too high, got %s' % np.mean(niters)
f.write('\n')
print()
f.close()
def main():
"""
Little helper routine to run the whole thing
Note: This can also be run with "mpirun -np 2 python B_pySDC_with_mpi4pyfft.py"
"""
run_simulation(spectral=False, ml=False, num_procs=1)
run_simulation(spectral=True, ml=False, num_procs=1)
run_simulation(spectral=False, ml=True, num_procs=1)
run_simulation(spectral=True, ml=True, num_procs=1)
run_simulation(spectral=False, ml=True, num_procs=10)
run_simulation(spectral=True, ml=True, num_procs=10)
if __name__ == "__main__":
main()
Results:
Running with ml=False and num_procs=1...
Min/Mean/Max number of iterations: 9.00 / 12.70 / 16.00
Range of values for number of iterations: 7
Position of max/min number of iterations: 0 -- 19
Std and var for number of iterations: 2.10 -- 4.41
Error: 1.1321e-05
Time to solution: 1.4585 sec.
Running with ml=False and num_procs=1...
Min/Mean/Max number of iterations: 8.00 / 11.40 / 15.00
Range of values for number of iterations: 7
Position of max/min number of iterations: 0 -- 19
Std and var for number of iterations: 2.03 -- 4.14
Error: 4.1749e-06
Time to solution: 1.1842 sec.
Running with ml=True and num_procs=1...
Min/Mean/Max number of iterations: 5.00 / 6.60 / 8.00
Range of values for number of iterations: 3
Position of max/min number of iterations: 0 -- 16
Std and var for number of iterations: 1.07 -- 1.14
Error: 1.1316e-05
Time to solution: 1.4461 sec.
Running with ml=True and num_procs=1...
Min/Mean/Max number of iterations: 4.00 / 5.95 / 8.00
Range of values for number of iterations: 4
Position of max/min number of iterations: 0 -- 19
Std and var for number of iterations: 1.02 -- 1.05
Error: 4.1744e-06
Time to solution: 1.3231 sec.
Running with ml=True and num_procs=10...
Min/Mean/Max number of iterations: 7.00 / 12.45 / 18.00
Range of values for number of iterations: 11
Position of max/min number of iterations: 9 -- 10
Std and var for number of iterations: 3.11 -- 9.65
Error: 1.1306e-05
Time to solution: 3.0640 sec.
Running with ml=True and num_procs=10...
Min/Mean/Max number of iterations: 6.00 / 11.50 / 17.00
Range of values for number of iterations: 11
Position of max/min number of iterations: 9 -- 10
Std and var for number of iterations: 3.04 -- 9.25
Error: 4.1688e-06
Time to solution: 2.8564 sec.
Part C: Time-parallel pySDC with space-parallel PETSc¶
With rather unfavorable scaling properties, parallel-in-time methods are only really useful when spatial parallelization is maxed out. To work with spatial parallelization, this part shows how to (1) include and work with an external library and (2) set up space- and time-parallel runs. We use again the forced heat equation as our testbed and PETSc for the space-parallel data structures and linear solver. See implementations/datatype_classes/petsc_dmda_grid.py and implementations/problem_classes/HeatEquation_2D_PETSc_forced.py for details on the PETSc bindings.
Important things to note:
We need processors in space and time, which can be achieved by comm.Split and coloring. The space-communicator is then passed to the problem class.
Below, we run the code 3 times: with 1 and 2 processors in space as well as 4 processors (2 in time and 2 in space). Do not expect scaling due to the CI environment.
Full code: pySDC/tutorial/step_7/C_pySDC_with_PETSc.py
import sys
from pathlib import Path
import numpy as np
from mpi4py import MPI
from pySDC.helpers.stats_helper import get_sorted
from pySDC.implementations.controller_classes.controller_MPI import controller_MPI
from pySDC.implementations.problem_classes.HeatEquation_2D_PETSc_forced import heat2d_petsc_forced
from pySDC.implementations.sweeper_classes.imex_1st_order import imex_1st_order
from pySDC.implementations.transfer_classes.TransferPETScDMDA import mesh_to_mesh_petsc_dmda
def main():
"""
Program to demonstrate usage of PETSc data structures and spatial parallelization,
combined with parallelization in time.
"""
# set MPI communicator
comm = MPI.COMM_WORLD
world_rank = comm.Get_rank()
world_size = comm.Get_size()
# split world communicator to create space-communicators
if len(sys.argv) >= 2:
color = int(world_rank / int(sys.argv[1]))
else:
color = int(world_rank / 1)
space_comm = comm.Split(color=color)
space_rank = space_comm.Get_rank()
# split world communicator to create time-communicators
if len(sys.argv) >= 2:
color = int(world_rank % int(sys.argv[1]))
else:
color = int(world_rank / world_size)
time_comm = comm.Split(color=color)
time_rank = time_comm.Get_rank()
# initialize level parameters
level_params = dict()
level_params['restol'] = 1e-08
level_params['dt'] = 0.125
level_params['nsweeps'] = [1]
# initialize sweeper parameters
sweeper_params = dict()
sweeper_params['quad_type'] = 'RADAU-RIGHT'
sweeper_params['num_nodes'] = [3]
sweeper_params['QI'] = ['LU'] # For the IMEX sweeper, the LU-trick can be activated for the implicit part
sweeper_params['initial_guess'] = 'zero'
# initialize problem parameters
problem_params = dict()
problem_params['nu'] = 1.0 # diffusion coefficient
problem_params['freq'] = 2 # frequency for the test value
problem_params['cnvars'] = [(65, 65)] # number of degrees of freedom for the coarsest level
problem_params['refine'] = [1, 0] # number of refinements
problem_params['comm'] = space_comm # pass space-communicator to problem class
problem_params['sol_tol'] = 1e-12 # set tolerance to PETSc' linear solver
# initialize step parameters
step_params = dict()
step_params['maxiter'] = 50
# initialize space transfer parameters
space_transfer_params = dict()
space_transfer_params['rorder'] = 2
space_transfer_params['iorder'] = 2
space_transfer_params['periodic'] = False
# initialize controller parameters
controller_params = dict()
controller_params['logger_level'] = 20 if space_rank == 0 else 99 # set level depending on rank
controller_params['dump_setup'] = False
# fill description dictionary for easy step instantiation
description = dict()
description['problem_class'] = heat2d_petsc_forced # pass problem class
description['problem_params'] = problem_params # pass problem parameters
description['sweeper_class'] = imex_1st_order # pass sweeper (see part B)
description['sweeper_params'] = sweeper_params # pass sweeper parameters
description['level_params'] = level_params # pass level parameters
description['step_params'] = step_params # pass step parameters
description['space_transfer_class'] = mesh_to_mesh_petsc_dmda # pass spatial transfer class
description['space_transfer_params'] = space_transfer_params # pass parameters for spatial transfer
# set time parameters
t0 = 0.0
Tend = 0.25
# instantiate controller
controller = controller_MPI(controller_params=controller_params, description=description, comm=time_comm)
# get initial values on finest level
P = controller.S.levels[0].prob
uinit = P.u_exact(t0)
# call main function to get things done...
uend, stats = controller.run(u0=uinit, t0=t0, Tend=Tend)
# compute exact solution and compare
uex = P.u_exact(Tend)
err = abs(uex - uend)
# filter statistics by type (number of iterations)
iter_counts = get_sorted(stats, type='niter', sortby='time')
niters = np.array([item[1] for item in iter_counts])
# limit output to space-rank 0 (as before when setting the logger level)
if space_rank == 0:
if len(sys.argv) == 3:
fname = str(sys.argv[2])
else:
fname = 'step_7_C_out.txt'
Path("data").mkdir(parents=True, exist_ok=True)
f = open('data/' + fname, 'a+')
out = 'This is time-rank %i...' % time_rank
f.write(out + '\n')
print(out)
# compute and print statistics
for item in iter_counts:
out = 'Number of iterations for time %4.2f: %2i' % item
f.write(out + '\n')
print(out)
out = ' Mean number of iterations: %4.2f' % np.mean(niters)
f.write(out + '\n')
print(out)
out = ' Range of values for number of iterations: %2i ' % np.ptp(niters)
f.write(out + '\n')
print(out)
out = ' Position of max/min number of iterations: %2i -- %2i' % (
int(np.argmax(niters)),
int(np.argmin(niters)),
)
f.write(out + '\n')
print(out)
out = ' Std and var for number of iterations: %4.2f -- %4.2f' % (float(np.std(niters)), float(np.var(niters)))
f.write(out + '\n')
print(out)
timing = get_sorted(stats, type='timing_run', sortby='time')
out = 'Time to solution: %6.4f sec.' % timing[0][1]
f.write(out + '\n')
print(out)
out = 'Error vs. PDE solution: %6.4e' % err
f.write(out + '\n')
print(out)
f.close()
assert err < 2e-04, 'ERROR: did not match error tolerance, got %s' % err
assert np.mean(niters) <= 12, 'ERROR: number of iterations is too high, got %s' % np.mean(niters)
space_comm.Free()
time_comm.Free()
if __name__ == "__main__":
main()
Results:
1 processor in time, 1 processor in space
This is time-rank 0...
Number of iterations for time 0.00: 12
Number of iterations for time 0.12: 12
Mean number of iterations: 12.00
Range of values for number of iterations: 0
Position of max/min number of iterations: 0 -- 0
Std and var for number of iterations: 0.00 -- 0.00
Time to solution: 1.9781 sec.
Error vs. PDE solution: 1.9479e-04
1 processor in time, 2 processors in space
This is time-rank 0...
Number of iterations for time 0.00: 12
Number of iterations for time 0.12: 12
Mean number of iterations: 12.00
Range of values for number of iterations: 0
Position of max/min number of iterations: 0 -- 0
Std and var for number of iterations: 0.00 -- 0.00
Time to solution: 0.8702 sec.
Error vs. PDE solution: 1.9479e-04
2 processor in time, 2 processors in space
This is time-rank 0...
Number of iterations for time 0.00: 12
Mean number of iterations: 12.00
Range of values for number of iterations: 0
Position of max/min number of iterations: 0 -- 0
Std and var for number of iterations: 0.00 -- 0.00
Time to solution: 0.7687 sec.
Error vs. PDE solution: 1.9479e-04
This is time-rank 1...
Number of iterations for time 0.12: 12
Mean number of iterations: 12.00
Range of values for number of iterations: 0
Position of max/min number of iterations: 0 -- 0
Std and var for number of iterations: 0.00 -- 0.00
Time to solution: 0.7687 sec.
Error vs. PDE solution: 1.9479e-04