A proportional-derivative control strategy for restarting the GMRES(m) algorithm

Restarted GMRES (or GMRES(m)) is normally used for solving large linear systems Ax=b with a general, possibly nonsymmetric, matrix A. Although, the restarted GMRES consumes less computational time than its counterpart full GMRES, if the restarting parameter is not correctly chosen its convergence cannot be guaranteed and the method may converge slowly. Unfortunately, it is difficult to know how to choose this parameter a priori. In this article, we regard the GMRES(m) method as a control problem, in which the parameter m is the controlled variable and propose a new control-inspired strategy for choosing the parameter m adaptively at each iteration. The advantage of this control strategy method is that only a few additional vectors need to be stored and the controller has the capacity to modify the dimension of the Krylov subspace whenever any convergence problem is detected. Numerical experiments, based on benchmark problems, show that the proposed control strategy accelerates the convergence of GMRES(m).