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 approximation method is necessary to compute it with some prescribed relative precision. In this paper we investigate the effect of an approximately computed matrix-vector product on the convergence and accuracy of several Krylov subspace solvers. The obtained insight is used to tune the precision of the matrix-vector product in every iteration so that an overall efficient process is obtained. This gives the empirical relaxation strategy of Bouras and Frayssé proposed in [2]. These strategies can lead to considerable savings over the standard approach of using a fixed relative precision for the matrix-vector product in every step. We will argue that the success of a relaxation strategy depends on the underlying way the Krylov subspace is constructed and not on the optimality properties for the residuals. Our analysis leads to an improved version of a strategy of Bouras, Frayssé, and Giraud [3] for the Conjugate Gradient method in case of Hermitian indefinite matrices.