Weighted Relaxation for Multigrid Reduction in Time

TL;DR

This paper introduces weighted relaxation into multigrid-reduction-in-time (MGRIT) methods, deriving convergence bounds and demonstrating that non-unit weights improve convergence speed and robustness in parallel time-integration.

Contribution

The work extends MGRIT by incorporating weighted relaxation, providing theoretical convergence analysis and empirical evidence of improved efficiency over unweighted methods.

Findings

Weighted relaxation improves convergence rates by 10-20%.

Non-unit weights enable faster iteration counts.

Under-relaxation restores convergence in some cases.

Abstract

Based on current trends in computer architectures, faster compute speeds must come from increased parallelism rather than increased clock speeds, which are currently 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 multigrid-reduction-in-time (MGRIT), a multilevel method applied to the time dimension that 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. All current MGRIT implementations are based on unweighted-Jacobi relaxation; here we introduce the concept of weighted relaxation to MGRIT. We derive new convergence bounds for weighted relaxation, and use this analysis to guide the selection of relaxation weights. Numerical results then demonstrate that non-unitary relaxation weights consistently yield faster convergence rates and lower iteration counts for MGRIT when compared with unweighted relaxation. In most cases, weighted relaxation yields a 10%-20% saving in iterations. For A-stable integration schemes, results also illustrate that under-relaxation can restore convergence in some cases where unweighted relaxation is not convergent.