Optimal Relaxation Weights for Multigrid Reduction In Time (MGRIT)

Masumi Sugiyama (University of New Mexico)

Based on current trends in computer architectures, faster compute speeds must come from increased parallelism rather than increased clock speeds, which are stagnate. This situation has created the well-known bottleneck for sequential time-integration, where each individual time-value (i.e., time-step) is computed sequentially. One approach to alleviate this and achieve parallelism in time is with multigrid. In this work, we consider the scheme known as multigrid-reduction-in-time (MGRIT), but note that there exist other parallel-in-time methods such as parareal and the parallel full approximation scheme in space and time (PFASST). MGRIT is a full multi-level method applied to the time dimension and computes multiple time-steps in parallel. Like all multigrid methods, MGRIT relies on the complementary relationship between relaxation on a fine-grid and a correction from the coarse grid to solve the problem. In this work, we analyze and select relaxation weights for MGRIT using a convergence analysis and find that this is beneficial since it improves the convergence rate and consequently improves the efficiency of computation. We note that choosing appropriate weights for relaxation (here weighted-Jacobi) has a long history for improving the convergence of spatial multigrid methods, and thus it is no surprise that such weight selection can be beneficial for MGRIT, too. Our numerical results demonstrate an improved convergence rate and lower iteration count for MGRIT when non-unitary weights are used for weighted-Jacobi.