VARYING THE S IN YOUR S-STEP GMRES

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. Key words. Communication-Avoiding, s-step Krylov subspace method, GMRES algorithm, variable s-step AMS subject classifications. 65F10, 65N22