Solving Ill-Posed Linear Systems with GMRES and a Singular Preconditioner

Almost singular linear systems arise in discrete ill-posed problems. Either because of the intrinsic structure of the problem or because of preconditioning, the spectrum of the coefficient matrix is often characterized by a sizable gap between a large group of numerically zero eigenvalues and the rest of the spectrum. Correspondingly, the right-hand side has leading eigencomponents associated with the eigenvalues away from zero. In this paper the effect of this setting in the convergence of the Generalized Minimal RESidual (GMRES) method is considered. It is shown that in the initial phase of the iterative algorithm, the residual components corresponding to the large eigenvalues are reduced in norm, and these can be monitored without extra computation. In particular ill-posed Cauchy problems for partial differential equations with variable coefficients are considered, where the preconditioner is a fast, low-rank solver for the corresponding problem with constant coefficients. The analysis is supported by numerical experiments. Our results, both theoretical and experimental, demonstrate that the preconditioned GMRES method converges within few iterations to the accuracy that is required for such ill-posed problems.