Inexact Krylov subspace methods for linear systems

There is a class of linear problems for which the computation of the matrix-vector product is very expensive since a time consuming method is necessary to approximate it with some prescribed relative precision. In this paper we investigate the impact of approximately computed matrix-vector products on the convergence and attainable accuracy of several Krylov subspace solvers. We will argue that the sensitivity towards perturbations is mainly determined by the underlying way the Krylov subspace is constructed and does not depend on the optimality properties of the particular method. The obtained insight is used to tune the precision of the matrix-vector product in every iteration step in such a way that an overall efficient process is obtained. Our analysis confirms the empirically found relaxation strategy of Bouras and Fraysse for the GMRES method proposed in [A Relaxation Strategy for Inexact Matrix-Vector Products for Krylov Methods, Technical Report TR/PA/00/15, CERFACS, France, 2000]. Furthermore, we give an improved version of a strategy for the conjugate gradient method of Bouras, Fraysse, and Giraud used in [A Relaxation Strategy for Inner-Outer Linear Solvers in Domain Decomposition Methods, Technical Report TR/PA/00/17, CERFACS, France, 2000].