Flexible inner-outer Krylov subspace methods

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.