Accelerating Krylov solvers with low rank correction

In this paper we present some aspects of recent works we have been developping on preconditioning techniques for accelerating Krylov solvers that are based on low rank corrections of a prescribed preconditioner. For SPD linear systems, we investigate the behaviour of some techniques based on spectral approaches when the eigenelements are only known approximately. We use the first-order perturbation theory for eigenvalues and eigenvectors to investigate the behaviour of the spectrum of the preconditioned systems using first order approximation. For unsymmetric linear system, we present a similar technique suited for the solution of sequences of linear systems is described. This technique is a combination of a low rank update spectral preconditioner and a Krylov solver that computes on the fly approximations of the eigenvectors associated with the smallest eigenvalues. We illustrate the interest of this approach in large parallel calculations for electromagnetic simulations. In this latter context, the solution technique enables the reduction of the simulation times by a factor of up to eight; these simulation times previously exceeded several hours of computation on a modern high performance computer