69 research outputs found
A multigrid perspective on the parallel full approximation scheme in space and time
For the numerical solution of time-dependent partial differential equations,
time-parallel methods have recently 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
many more is work in progress. 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 block-wise local Fourier analysis are taken. The
theoretical results are applied to examples of diffusive and advective type
Multilevel convergence analysis of multigrid-reduction-in-time
This paper presents a multilevel convergence framework for
multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid
estimates. The framework provides a priori upper bounds on the convergence of
MGRIT V- and F-cycles, with different relaxation schemes, by deriving the
respective residual and error propagation operators. The residual and error
operators are functions of the time stepping operator, analyzed directly and
bounded in norm, both numerically and analytically. We present various upper
bounds of different computational cost and varying sharpness. These upper
bounds are complemented by proposing analytic formulae for the approximate
convergence factor of V-cycle algorithms that take the number of fine grid time
points, the temporal coarsening factors, and the eigenvalues of the time
stepping operator as parameters.
The paper concludes with supporting numerical investigations of parabolic
(anisotropic diffusion) and hyperbolic (wave equation) model problems. We
assess the sharpness of the bounds and the quality of the approximate
convergence factors. Observations from these numerical investigations
demonstrate the value of the proposed multilevel convergence framework for
estimating MGRIT convergence a priori and for the design of a convergent
algorithm. We further highlight that observations in the literature are
captured by the theory, including that two-level Parareal and multilevel MGRIT
with F-relaxation do not yield scalable algorithms and the benefit of a
stronger relaxation scheme. An important observation is that with increasing
numbers of levels MGRIT convergence deteriorates for the hyperbolic model
problem, while constant convergence factors can be achieved for the diffusion
equation. The theory also indicates that L-stable Runge-Kutta schemes are more
amendable to multilevel parallel-in-time integration with MGRIT than A-stable
Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material
Parareal in time 3D numerical solver for the LWR Benchmark neutron diffusion transient model
We present a parareal in time algorithm for the simulation of neutron
diffusion transient model. The method is made efficient by means of a coarse
solver defined with large time steps and steady control rods model. Using
finite element for the space discretization, our implementation provides a good
scalability of the algorithm. Numerical results show the efficiency of the
parareal method on large light water reactor transient model corresponding to
the Langenbuch-Maurer-Werner (LMW) benchmark [1]
A numerical study of a semi-Lagrangian Parareal method applied to the viscous Burgers equation
This is the author accepted manuscript. The final version is available from Springer via the DOI in this record.This work focuses on the Parareal parallelin-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 semiLagrangian
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.Schmitt: The work of this author is supported by the ’Excellence
Initiative’ of the German Federal and State Governments
and the Graduate School of Computational Engineering
at Technische Universit¨at Darmstadt
Peixoto: Acknowledges the Sao Paulo Research Foundation
(FAPESP) under the grant number 2016/18445-7 and the National
Science and Technology Development Council (CNPq)
under grant number 441328/2014-
Time Parallel Gravitational Collapse Simulation
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 eld. 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
Numerical simulation of skin transport using Parareal
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
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