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# Varying the s in Your s-step GMRES

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 subspace methods are commonly used iterative methods for solving large sparse linear systems, however they suffer from communication bottlenecks on parallel computers. Therefore, $s$-step methods have been developed where the Krylov subspace is built block by block, so that $s$ matrix-vector multiplications can be done before orthonormalizing the block. Then Communication-Avoiding algorithms can be used for both kernels. This paper introduces a new variation on $s$-step GMRES in order to reduce the number of iterations necessary to ensure convergence, with a small overhead in the number of communications. Namely, we develop a $s$-step GMRES algorithm, where the block size is variable and increases gradually. Our numerical experiments show a good agreement with our analysis of condition numbers and demonstrate the efficiency of our variable $s$-step approach.
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https://hal.inria.fr/hal-01299652
Contributor : Jocelyne Erhel <>
Submitted on : Thursday, December 7, 2017 - 10:18:31 AM
Last modification on : Thursday, January 7, 2021 - 4:24:16 PM

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FibGMRES.pdf
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• HAL Id : hal-01299652, version 3

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David Imberti, Jocelyne Erhel. Varying the s in Your s-step GMRES. Electronic Transactions on Numerical Analysis, Kent State University Library, 2017, 47, pp.206-230. ⟨hal-01299652v3⟩

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