Exact and inexact breakdowns in the block GMRES method

Abstract

This paper addresses the issue of breakdowns in the block GMRES method for solving linear systems with multiple right-hand sides of the form AX = B. An exact (inexact) breakdown occurs at iteration j of this method when the block Krylov matrix ( B, AB, … , A j−1 B) is singular (almost singular). Exact breakdowns are the sign that a part of the exact solution is in the range of the Krylov matrix. They are primarily of theoretical interest. From a computational point of view, inexact breakdowns are most likely to occur. In such cases, the underlying block Arnoldi process that is used to build the block Krylov space should not be continued as usual. A natural way to continue the process is the use of deflation. However, as shown by Langou [J. Langou, Iterative Methods for Solving Linear Systems with Multiple Right-Hand Sides, Ph.D. dissertation TH/PA/03/24, CERFACS, France, 2003], deflation in block GMRES may lead to a loss of information that slows down the convergence. In this paper, instead of deflating the directions associated with almost converged solutions, these are kept and reintroduced in next iterations if necessary. Two criteria to detect inexact breakdowns are presented. One is based on the numerical rank of the generated block Krylov basis, the second on the numerical rank of the residual associated to approximate solutions. These criteria are analyzed and compared. Implementation details are discussed. Numerical results are reported.