Nested Krylov methods based on GCR

Abstract

Recently the GMRESR method for the solution of linear systems of equations has been introduced by Vuik and Van der Vorst (1991). Similar methods have been proposed by Axelsson and Vassilevski (1991) and Saad (1993) (FGMRES 1 1 Since FGMRES and GMRESR are very similar, the ideas presented will be relevant for FGMRES as well. ). GMRESR involves an outer and an inner method. The outer method is GCR, which is used to compute the optimal approximation over a given set of search vectors in the sense that the residual is minimized. The inner method is GMRES, which computes a new search vector by approximately solving the residual equation. This search vector is then used by the outer algorithm to compute a new approximation. However, the optimality of the approximation over the space of search vectors is ignored in the inner GMRES algorithm. This leads to suboptimal corrections to the solution in the outer algorithm. Therefore, we propose to preserve the orthogonality relations of GCR in the inner GMRES algorithm. This gives optimal corrections to the solution and also leads to solving the residual equation in a smaller subspace and with an “improved” operator, which should also lead to faster convergence. However, this involves using Krylov methods with a singular, nonsymmetric operator. We will discuss some important properties of this. We will show by experiments that in terms of matrix-vector products, this modification (almost) always leads to better convergence. Because we do more orthogonalizations, it does not always give an improved performance in time. This depends on the costs of the matrix-vector products relative to the costs of the orthogonalizations. Of course, we can also use methods other than GMRES as the inner method. Methods with short recurrences like BiCGSTAB seem especially interesting. The experimental results indicate that, especially for such methods, it is advantageous to preserve the orthogonality in the inner method.