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Solving large sparse linear systems with a variable s-step GMRES preconditioned by DD

David Imberti 1 Jocelyne Erhel 1 
1 FLUMINANCE - Fluid Flow Analysis, Description and Control from Image Sequences
IRMAR - Institut de Recherche Mathématique de Rennes, IRSTEA - Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture, Inria Rennes – Bretagne Atlantique
Abstract : Krylov methods such as GMRES are efficient iterative methods to solve large sparse linear systems, with only a few key kernel operations: the matrix-vector product, solving a preconditioning system, and building the orthonormal Krylov basis. Domain Decomposition methods allow parallel computations for both the matrix-vector products and preconditioning by using a Schwarz approach combined with deflation (similar to a coarse-grid correction). However, building the orthonormal Krylov basis involves scalar products, which in turn have a communication overhead. In order to avoid this communication, it is possible to build the basis by a block of vectors at a time, sometimes at the price of a loss of orthogonality. We define a sequence of such blocks with a variable size. We show through some theoretical results and some numerical experiments that increasing the block size as a Fibonacci sequence improves stability and convergence.
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Submitted on : Thursday, November 23, 2017 - 1:56:57 PM
Last modification on : Friday, May 20, 2022 - 9:04:52 AM


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  • HAL Id : hal-01528636, version 1


David Imberti, Jocelyne Erhel. Solving large sparse linear systems with a variable s-step GMRES preconditioned by DD. DD24 - International Conference on Domain Decomposition Methods, Feb 2017, Longyearbyen, Norway. ⟨hal-01528636⟩



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