  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.
