Flexible Variants of Block Restarted GMRES Methods with Application to Geophysics

Abstract

In a wide number of applications in computational science and engineering the solution of large linear systems of equations with several right-hand sides given at once is required. Direct methods based on Gaussian elimination are known to be especially appealing in that setting. Nevertheless when the dimension of the problem is very large, preconditioned block Krylov space solvers are often considered as the method of choice. The purpose of this paper is thus to present iterative methods based on block restarted GMRES that allow variable preconditioning for the solution of linear systems with multiple right-hand sides. The use of flexible methods is especially of interest when approximate possibly iterative solvers are considered in the preconditioning phase. First we introduce a new variant of block flexible restarted GMRES that includes a strategy for detecting when a linear combination of the systems has approximately converged. This explicit block size reduction is often called deflation. We analyze the main properties of this flexible method based on deflation and notably prove that the Frobenius norm of the block residual is always nonincreasing. We also present a flexible variant based on both deflation and truncation to especially be used in case of limited memory. Finally we illustrate the numerical behavior of these flexible block methods on large industrial simulations arising in geophysics, where indefinite linear systems of size up to one billion of unknowns with multiple right-hand sides have been successfully solved in a parallel distributed memory environment.

GMRES with multiple preconditioners

“…The new iterate is optimal in this subspace in the minimum residual least-squares sense. We mention that Calandra et al [7] have recently described a block version of FGMRES which-while sharing some similarities with the work described here-differs both in scope and details with the GMRES algorithm with multiple preconditioners we propose.…”

“…It was studied, e.g., in [9], [25], [34], [45], and applied recently, e.g., in [7]; see also [18] and references therein. We also mention block Arnoldi as a necessary important ingredient; see, e.g., [40], [41], [38, §6.12].…”

Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

“…A first study of the Block GMRES was proposed in the thesis of Vital [33], followed by new theoretical results [17]. Plenty of applications of these methods have been proposed since then [34], [35], mainly for the simulation of wave propagation phenomena [36], [37] or lattice quantum chromodynamics [38] with multiple sources. Since each of these sources may yield a different right-hand side, efficient block methods are needed to handle demanding simulations with sequences of tall and skinny right-hand sides B i .…”

A parallel evolution strategy for an earth imaging problem in geophysics

“…Due to their indefiniteness, these systems are known to be very challenging for iterative methods [18]. Based on previous studies [11,12,26], we consider a recently proposed block flexible Krylov subspace method (BFGMRES-S [11, Algorithm 3]) for the solution of the linear system with multiple right-hand sides (2). In [11] the authors have shown the relevance of the BFGMRES-S algorithm combined with a variable two-level preconditioner to address the solution of such large-scale acoustic forward problems in a distributed memory parallel environment.…”

“…We refer the reader to [12,Algorithm 5] for a complete description of the geometric two-grid preconditioner and to [33] for additional theoretical properties in relation with Krylov subspace methods. Applications to large-scale forward acoustic problems have been considered in detail in [26,Chapter 4].…”