Stability of linear GMRES convergence with respect to compact perturbations

Suppose that a linear bounded operator $B$ on a Hilbert space exhibits at least linear GMRES convergence, i.e., there exists $M_B<1$ such that the GMRES residuals fulfill $\|r_k\|\leq M_B\|r_{k-1}\|$ for every initial residual $r_0$ and step $k\in\mathbb{N}$. We prove that GMRES with a compactly perturbed operator $A=B+C$ admits the bound $\|r_k\|/\|r_0\|\leq\prod_{j=1}^k\bigl(M_B+(1+M_B)\,\|A^{-1}\|\,\sigma_j(C)\bigr)$, i.e., the singular values $\sigma_j(C)$ control the departure from the bound for the unperturbed problem. This result can be seen as an extension of [I. Moret, A note on the superlinear convergence of GMRES, SIAM J. Numer. Anal., 34 (1997), pp. 513-516, https://doi.org/10.1137/S0036142993259792], where only the case $B=\lambda I$ is considered. In this special case $M_B=0$ and the resulting convergence is superlinear.Comment: 11 pages; this revision adds merely funding informatio

On choice of preconditioner for minimum residual methods for nonsymmetric matrices

Existing convergence bounds for Krylov subspace methods such as GMRES for nonsymmetric linear systems give little mathematical guidance for the choice of preconditioner. Here, we establish a desirable mathematical property of a preconditioner which guarantees that convergence of a minimum residual method will essentially depend only on the eigenvalues of the preconditioned system, as is true in the symmetric case. Our theory covers only a subset of nonsymmetric coefficient matrices but computations indicate that it might be more generally applicable

70 years of Krylov subspace methods: The journey continues

Using computed examples for the Conjugate Gradient method and GMRES, we recall important building blocks in the understanding of Krylov subspace methods over the last 70 years. Each example consists of a description of the setup and the numerical observations, followed by an explanation of the observed phenomena, where we keep technical details as small as possible. Our goal is to show the mathematical beauty and hidden intricacies of the methods, and to point out some persistent misunderstandings as well as important open problems. We hope that this work initiates further investigations of Krylov subspace methods, which are efficient computational tools and exciting mathematical objects that are far from being fully understood.Comment: 38 page

Preconditioning for nonsymmetry and time-dependence

In this short paper, we decribe at least one simple and frequently arising situation |that of nonsymmetric real Toeplitz (constant diagonal) matrices| where we can guarantee rapid convergence of the appropriate iterative method by manipulating the problem into a symmetric form without recourse to the normal equations. This trick can be applied regardless of the nonnormality of the Toeplitz matrix. We also propose a symmetric and positive definite preconditioner for this situation which is proved to cluster eigenvalues and is by consequence guaranteed to ensure convergence in a number of iterations independent of the matrix dimension

GMRES ON A TRIDIAGONAL TOEPLITZ LINEAR SYSTEM

The Generalized Minimal Residual method (GMRES) is often used to solve a nonsymmetric linear system Ax = b. But its convergence analysis is a rather difficult task in general. A commonly used approach is to diagonalize A = XΛX-1 and then separate the study of GMRES convergence behavior into optimizing the condition number of X and a polynomial minimization problem over As spectrum. This artificial separation could greatly overestimate GMRES residuals and likely yields error bounds that are too far from the actual ones. On the other hand, considering the effects of both As spectrum and the conditioning of X at the same time poses a difficult challenge, perhaps impossible to deal with in general but only possible for certain particular linear systems. This thesis will do so for a (nonsymmetric) tridiagonal Toeplitz system. Sharp error bounds on and sometimes exact expressions for residuals are obtained. These expressions and/or bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first kind or the second kind