Istituto di Matematica Applicata e Tecnologie Informatiche     
Simoncini V., Szyld D. B. Flexible Inner-Outer Krylov Subspace Methods. Preprint ercim.cnr.ian//2002-1269, 2002.
Flexible Krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next. Given a Krylov subspace method, such as CG, GMRES, QMR, etc. for the solution of a linear system $Ax=b$, instead of having a fixed preconditioner $M$ and the (right) preconditioned equation $AM^{-1} y = b$ ($Mx =y$), one may have a different matrix, say $M_k$ at each step. In this paper, the case where the preconditioner itself is a Krylov subspace method is studied. There are several papers in the literature where such situation is presented and numerical examples given. A general theory is provided encompassing many of these cases, including truncated methods. The overall space where the solution is approximated is no longer a Krylov subspace, but a subspace of a larger Krylov space. We show how this subspace keeps growing as the outer iteration progresses, thus providing a convergence theory for these inner--outer methods. Numerical tests illustrate some important implementation aspects that make the discussed inner--outer methods very appealing in practical circumstances.
Subject Flexible or inner--outer Krylov methods. Variable preconditioning. Nonsymmetric linear system. Iterative solver
65F10, 15A06

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