Enlarged Krylov Subspace Conjugate Gradient Methods for Reducing Communication

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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.