GMRES Convergence for Perturbed Coefficient Matrices, with Application to Approximate Deflation Preconditioning

Abstract

Abstract. How does GMRES convergence change when the coefficient matrix is perturbed? Using spectral perturbation theory and resolvent estimates, we develop simple, general bounds that quantify the lag in convergence such a perturbation can induce. This analysis is particularly relevant to preconditioned systems, where an ideal preconditioner is only approximately applied in practical computations. To illustrate the utility of this approach, we combine our analysis with Stewart's invariant subspace perturbation theory to develop rigorous bounds on the performance of approximate deflation preconditioning using Ritz vectors.

Preconditioning

“…In particular, when restarting gmres, one can use information from previous iterates via deflation. There are many papers describing various approaches: for symmetric and positive definite matrices, see for example Frank and Vuik (2001), and for non-symmetric matrices, see for example Erhel, Burrage and Pohl (1995), Sifuentes, Embree and Morgan (2013) and Giraud, Gratton, Pinel and Vasseur (2010). When solving sequences of linear systems, the related idea of 'recycling' can be considered (Parks et al 2006).…”

Pseudoeigenvector bases and deflated GMRES for highly nonnormal matrices

“…Using 30 pseudoeigenvectors is effective. It seems that GMRES‐Proj may need more pseudoeigenvectors than GMRES‐DR.Example The next test matrix is a SUPG matrix from previous work , with n = 2500 and ν = .01. Figure shows the eigenvalues of the matrix and the pseudoeigenvalues using perturbations with three different values of ε .…”

“…Next, the Butterfly matrix of size n = 500 is considered. The condition number for the matrix of eigenvectors is now 2.5*10 19 . This is shown on Figure 1 with the horizontal solid line.…”

“…Example 13. The next test matrix is a SUPG matrix from previous work 19,34,35 with n = 2500 and = .01. Figure 20 shows the eigenvalues of the matrix and the pseudoeigenvalues using perturbations with three different values of .…”

On uniform connectivity of algebraic matrix sets

“…The main sources of motivation for the reserach reported in this document came from structure preserving perturbation theory in the sense of [16], specially by the effect that preconditioning in the sense of [6,7,11,17,18,19], has on the numerical solution of linear systems of equations and eigenvalue/diagonalization problems, which are two of the main problems in numerical linear algebra. Another sources of motivation were, the perturbation theory of matrix polynomials in the sense of [10], and the simultaneous block-diagonalization of matrices in the sense of [14].…”