## Varying the s in your s-step GMRES

David Imberti and Jocelyne Erhel

### 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 the $s$-step GMRES method in order to reduce the number of iterations necessary to ensure convergence with a small overhead in the number of communications. Namely, we develop an $s$-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.

Full Text (PDF) [723 KB]

### Key words

Communication-Avoiding, $s$-step Krylov subspace method, GMRES algorithm, variable $s$-step

65F10, 65N22

### Links to the cited ETNA articles

 [13] Vol. 3 (1995), pp. 160-176 Jocelyne Erhel: A parallel GMRES version for general sparse matrices [27] Vol. 40 (2013), pp. 381-406 Desire Nuentsa Wakam and Jocelyne Erhel: Parallelism and robustness in GMRES with a Newton basis and deflated restarting

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