Weighted relaxation for multigrid reduction in time

Journal Article · Tue Sep 06 00:00:00 EDT 2022 · Numerical Linear Algebra with Applications

DOI:https://doi.org/10.1002/nla.2465· OSTI ID:1889975

Sugiyama, Masumi ^[1]; Schroder, Jacob B. ^[2]; Southworth, Ben S. ^[3]; Friedhoff, Stephanie ^[4]

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1. University of Tennessee, Chattanoga, TN (United States)
2. University of New Mexico, Albuquerque, NM (United States)
3. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
4. University of Wuppertal (Germany)

Current trends in computer architectures now mean that faster computation speed must come primarily from increased concurrency, not faster clock speeds, which are stagnating. Thus, this situation creates bottlenecks for serial algorithms, including 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. Here, 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 by choosing appropriate non-unitary relaxation weights, one can achieve 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, which is significant when using large high-performance computers. For A-stable integration schemes, results also illustrate that under-relaxation can restore convergence in some cases where unweighted relaxation is not convergent.

• Research Organization: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
• Sponsoring Organization: USDOE; USDOE Office of Science (SC)
• Grant/Contract Number: 89233218CNA000001
• OSTI ID: 1889975
• Alternate ID(s): OSTI ID: 1996041
• Report Number(s): LA-UR-21-26114
• Journal Information: Numerical Linear Algebra with Applications, Journal Name: Numerical Linear Algebra with Applications Journal Issue: 1 Vol. 30; ISSN 1070-5325
• Publisher: Wiley
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
multigrid
multigrid-reduction-in-time
parallel-in-time
polynomial relaxation
weighted relaxation