Convergence in Backward Error of Relaxed GMRES

This work is the follow-up of the experimental study presented in [A. Bouras and V. Fraysse´, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 660-678]. It is based on and extends some theoretical results in [V. Simoncini and D. B. Szyld, SIAM J. Sci. Comput., 25 (2003), pp. 454-477; J. van den Eshof and G. L. G. Sleijpen, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 125-153]. In a backward error framework we study the convergence of GMRES when the matrix-vector products are performed inaccurately. This inaccuracy is modeled by a perturbation of the original matrix. We prove the convergence of GMRES when the perturbation size is proportional to the inverse of the computed residual norm; this implies that the accuracy can be relaxed as the method proceeds which gives rise to the terminology “relaxed GMRES.” As for the exact GMRES we show under proper assumptions that only happy breakdowns can occur. Furthermore, the convergence can be detected using a byproduct of the algorithm. We explore the links between relaxed right-preconditioned GMRES and flexible GMRES (FGMRES). In particular, this enables us to derive a proof of convergence of FGMRES. Finally, we report results of numerical experiments to illustrate the behavior of the relaxed GMRES monitored by the proposed relaxation strategies.

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