## Flexible inner-outer Krylov subspace methods (2003)

Venue: | SIAM J. NUMER. ANAL |

Citations: | 21 - 2 self |

### BibTeX

@ARTICLE{Simoncini03flexibleinner-outer,

author = {Valeria Simoncini and Daniel B. Szyld},

title = {Flexible inner-outer Krylov subspace methods},

journal = {SIAM J. NUMER. ANAL},

year = {2003},

volume = {40},

number = {6},

pages = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

Flexible Krylov methodsrefersto a classof methodswhich 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 Mk, 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 a 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.