NDLI: Two-Level Convergence Theory for Multigrid Reduction in Time (MGRIT)

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Petersson, N. A. Kolev, Tz. Dobrev, V. A. Schroder, J. B.
Copyright Year	2017
Abstract	In this paper we develop a two-grid convergence theory for the parallel-in-time scheme known as multigrid reduction in time (MGRIT), as it is implemented in the open-source package [XBraid: Parallel Multigrid in Time, http://llnl.gov/casc/xbraid]. MGRIT is a scalable and multilevel approach to parallel-in-time simulations that nonintrusively uses existing time-stepping schemes, and in a specific two-level setting it is equivalent to the widely known parareal algorithm. The goal of this paper is twofold. First, we present a two-level MGRIT convergence analysis for linear problems where the spatial discretization matrix can be diagonalized, and then apply this analysis to our two basic model problems, the heat equation and the advection equation. One important assumption is that the coarse and fine time-grid propagators can be diagaonalized by the same set of eigenvectors, which is often the case when the same spatial discretization operator is used on the coarse and fine time grids. In many cases, the MGRIT algorithm is guaranteed to converge, and we demonstrate numerically that the theoretically predicted convergence rates are sharp in practice for our model problems. Second, we explore how the convergence of MGRIT compares to the stability of the chosen time-stepping scheme. In particular, we demonstrate that a stable time-stepping scheme does not necessarily imply convergence of MGRIT, although MGRIT with FCF-relaxation always converges for the diffusion dominated problems considered here.
Sponsorship	Lawrence Livermore National Laboratory
Starting Page	S501
Ending Page	S527
Page Count	27
File Format	PDF
ISSN	10648275
DOI	10.1137/16M1074096
e-ISSN	10957197
Issue Number	5
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-10-26
Access Restriction	Subscribed
Subject Keyword	Solution of discretized equations parallel-in-time Iterative methods for linear systems multigrid multigrid-in-time convergence theory high performance computing Multigrid methods; domain decomposition
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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