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Enlarged Krylov Subspace Conjugate Gradient Methods for Reducing Communication

Laura Grigori 1 Sophie Moufawad 2 Frédéric Nataf 2 
1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
LJLL - Laboratoire Jacques-Louis Lions, Inria Paris-Rocquencourt, INSMI - Institut National des Sciences Mathématiques et de leurs Interactions
Abstract : In this paper we introduce a new approach for reducing communication in Krylov subspace methods that consists of enlarging the Krylov subspace by a maximum of $t$ vectors per iteration, based on a domain decomposition of the graph of $A$. The obtained enlarged Krylov subspace $\mathscr{K}_{k,t}(A,r_0)$ is a superset of the Krylov subspace $\mathcal{K}_k(A,r_0)$, $\mathcal{K}_k(A,r_0) \subset \mathscr{K}_{k,t}(A,r_0)$. Thus, we search for the solution of the system $Ax=b$ in $\mathscr{K}_{k,t}(A,r_0)$ instead of $\mathcal{K}_k(A,r_0)$. Moreover, we show in this paper that the enlarged Krylov projection subspace methods lead to faster convergence in terms of iterations and parallelizable algorithms with less communication, with respect to Krylov methods.
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Submitted on : Tuesday, August 30, 2016 - 3:48:53 PM
Last modification on : Friday, July 8, 2022 - 10:07:56 AM

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Laura Grigori, Sophie Moufawad, Frédéric Nataf. Enlarged Krylov Subspace Conjugate Gradient Methods for Reducing Communication. SIAM Journal on Matrix Analysis and Applications, 2016, 37 (2), pp.744-773. ⟨10.1137/140989492⟩. ⟨hal-01357899⟩



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