Multigrid Reduction in Time for Chaotic Dynamical Systems

TL;DR

This paper enhances the Multigrid Reduction in Time (MGRIT) method to better handle chaotic dynamical systems by introducing a modified algorithm and a low memory variant, enabling efficient parallel simulation and Lyapunov vector estimation.

Contribution

It develops a new MGRIT algorithm with improved coarse-grid correction for chaotic systems and introduces a low memory variant that estimates Lyapunov vectors.

Findings

Improved convergence of MGRIT for chaotic IVPs.

Successful parallel speedup on Lorenz and Kuramoto-Sivashinsky systems.

Enhanced long-term simulation accuracy for chaotic PDEs.

Abstract

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. We provide supporting numerical results for the Lorenz system and demonstrate parallel speedup for the chaotic Kuramoto- Sivashinsky partial differential equation over a significantly longer time-domain than in previous works.