Bryan Lewis

On the regularizing properties of the GMRES method

Summary. The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. However, little is known about the behavior of this method when it is applied to the solution of nonsymmetric linear ill-posed problems with a right-hand side that is contaminated by errors. We show that when the associated error-free right-hand side lies in a finite-dimensional Krylov subspace, the GMRES method is a regularization method. The iterations are terminated by a stopping rule based on the discrepancy principle.

GMRES-type methods for inconsistent systems

Abstract The behavior of iterative methods of GMRES-type when applied to singular, possibly inconsistent, linear systems is discussed and conditions under which these methods converge to the least-squares solution of minimal norm are presented. Error bounds for the computed iterates are shown. This paper complements previous work by Brown and Walker [P.N. Brown, H.F. Walker, SIAM J. Matrix Anal. Appl. 18 (1997) 37–51].

GMRES, L-Curves, and Discrete Ill-Posed Problems

The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. This paper discusses application of the GMRES method to the solution of large linear systems of equations that arise from the discretization of linear ill-posed problems. These linear systems are severely ill-conditioned and are referred to as discrete ill-posed problems. We are concerned with the situation when the right-hand side vector is contaminated by measurement errors, and we discuss how a meaningful approximate solution of the discrete ill-posed problem can be determined by early termination of the iterations with the GMRES method. We propose a termination criterion based on the condition number of the projected matrices defined by the GMRES method. Under certain conditions on the linear system, the termination index corresponds to the “vertex” of an L-shaped curve.

On the solution of large Sylvester‐observer equations

The design of a Luenberger observer for large control systems is an important problem in Control Theory. Recently, several computational methods have been proposed by Datta and collaborators. The present paper discusses numerical aspects of one of these methods, described by Datta and Saad (1991). Copyright

Krylov subspace iterative methods for nonsymmetric discrete ill-posed problems in image restoration

The BiCG and QMR methods are well-known Krylov subspace iterative methods for the solution of linear systems of equations with a large nonsymmetric, nonsingular matrix. However, little is known of the performance of these methods when they are applied to the computation of approximate solutions of linear systems of equations with a matrix of ill-determined rank. Such linear systems are known as linear discrete ill-posed problems. We describe an application of the BiCG and QMR methods to the solution of linear discrete ill-posed problems that arise in image restoration, and compare these methods to the conjugate gradient method applied to the associated normal equations and to total variation-penalized Tikhonov regularization.

On the selection of poles in the single-input pole placement problem

Abstract It is well known that the single-input pole placement problem can be quite sensitive to perturbations in the data. Recent results by Mehrmann and Xu show how this sensitivity depends on the location of the poles. In many applications it suffices to prescribe a set K in the complex plane that contains the poles. Mehrmann and Xu formulated a minimization problem for allocating the poles in a given set K so that the sensitivity to perturbations is reduced. The present paper uses methods of potential theory to derive simple algorithms that yield approximate solutions of this minimization problem.

A hybrid GMRES and TV-norm-based method for image restoration

Total variation-penalized Tikhonov regularization is a popular method for the restoration of images that have been degraded by noise and blur. The method is particularly effective, when the desired noise- and blur-free image has edges between smooth surfaces. The method, however, is computationally expensive. We describe a hybrid regularization method that combines a few steps of the GMRES iterative method with total variation-penalized Tikhonov regularization on a space of small dimension. This hybrid method requires much less computational work than available methods for total variation-penalized Tikhonov regularization and can produce restorations of similar quality.

L-curve for the MINRES method

A variant of the MINRES method, often referred to as the MR-II method, has in the last few years become a popular iterative scheme for computing approximate solutions of large linear discrete ill- posed problems with a symmetric matrix. It is important to terminate the iterations sufficiently early in order to avoid severe amplification of measurement and round-off errors. We present a new L-curve for determining when to terminate the iterations with the MINRES and MR-II method.

Parallel deconvolution methods for three dimensional image restoration

Restoration by deconvolution of three-dimensional images that have been contaminated by noise and spatially invariant blur is computationally demanding. We describe efficient parallel implementations of iterative methods for image deconvolution on a distributed memory computing cluster.

Smooth or abrupt: a comparison of regularization methods

In this paper we compare a new regularizing scheme based on the exponential filter function with two classical regularizing methods: Tikhonov regularization and a variant of truncated singular value regularization. The filter functions for the former methods are smooth, but for the latter discontinuous. These regularization methods are applied to the restoration of images degraded by blur and noise. The norm of the noise is assumed to be known, and this allows application of the Morozov discrepancy principle to determine the amount of regularization. We compare the restored images produced by the three regularization methods with optimal values of the regularization parameter. This comparison sheds light on how these different approaches are related.