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Multigrid Reduction in Time for Chaotic Dynamical Systems

Journal Article · · SIAM Journal on Scientific Computing
DOI:https://doi.org/10.1137/22m1518335· OSTI ID:2212873
 [1];  [2];  [2];  [1]
  1. Univ. of New Mexico, Albuquerque, NM (United States)
  2. Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
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As CPU clock speeds have stagnated and high performance computers continue to have ever higher core counts, increased parallelism is needed to take advantage of these new architectures. Traditional serial time-marching schemes can be a significant bottleneck, as many types of simulations require large numbers of time-steps which must be computed sequentially. Parallel-in-time schemes, such as the Multigrid Reduction in Time (MGRIT) method, remedy this by parallelizing across time-steps and have shown promising results for parabolic problems. However, chaotic problems have proved more difficult, since chaotic initial value problems (IVPs) are inherently ill-conditioned. MGRIT relies on a hierarchy of successively coarser time-grids to iteratively correct the solution on the finest time-grid, but due to the nature of chaotic systems, small inaccuracies on the coarser levels can be greatly magnified and lead to poor coarse-grid corrections. Here we introduce a modified MGRIT algorithm based on an existing quadratically converging nonlinear extension to the multigrid Full Approximation Scheme (FAS), as well as a novel time-coarsening scheme. Together, these approaches better capture long-term chaotic behavior on coarse-grids and greatly improve convergence of MGRIT for chaotic IVPs. Further, we introduce a novel low-memory variant of the algorithm for solving chaotic PDEs with MGRIT which not only solves the IVP, but also provides estimates for the unstable Lyapunov vectors of the system. Finally, we provide supporting numerical results for the Lorenz system and demonstrate parallel speedup for the chaotic Kuramoto–Sivashinsky PDE over a significantly longer time-domain than in previous works.

Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC52-07NA27344
OSTI ID:
2212873
Report Number(s):
LLNL--JRNL-838414; 1058785
Journal Information:
SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 4 Vol. 45; ISSN 1064-8275
Publisher:
Society for Industrial and Applied Mathematics (SIAM)Copyright Statement
Country of Publication:
United States
Language:
English

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Figures / Tables (13)

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Table 1(p. 19)table Table 1
Table 2(p. 20)table Table 2
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Related Subjects

97 MATHEMATICS AND COMPUTING
Lorenz system
Lyapunov vectors
chaotic problems
multigrid
multigrid in reduction time
parallel-in-time