Numerical Stability of s-step Enlarged Krylov Subspace Conjugate Gradient Methods

Recently, enlarged Krylov subspace methods, that consists of enlarging the Krylov subspace by a maximum of t vectors per iteration based on the domain decomposition of the graph of A, were introduced in the aim of reducing communication when solving systems of linear equations Ax=b. In this paper, the s-step enlarged Krylov subspace Conjugate Gradient methods are introduced, whereby s iterations of the enlarged Conjugate Gradient methods are merged in one iteration. The numerical stability of these s-step methods is studied, and several numerically stable versions are proposed. Similarly to the enlarged Krylov subspace methods, the s-step enlarged Krylov subspace methods have a faster convergence than Krylov methods, in terms of iterations. Moreover, by computing st basis vectors of the enlarged Krylov subspace $\mathscr{K}_{k,t}(A,r_0)$ at the beginning of each s-step iteration, communication is further reduced. It is shown in this paper that the introduced methods are parallelizable with less communication, with respect to their corresponding enlarged versions and to Conjugate Gradient.