Iterative Solution Methods for Large Linear Discrete Ill-posed Problems

This paper discusses iterative methods for the solution of very large severely ill-conditioned linear systems of equations that arise from the discretization of linear ill-posed problems. The right-hand side vector represents the given data and is assumed to be contaminated by errors. Solution methods proposed in the literature employ some form of ltering to reduce the in uence of the error in the right-hand side on the computed approximate solution. The amount of ltering is determined by a parameter, often referred to as the regularization parameter. We discuss how the ltering a ects the computed approximate solution and consider the selection of regularization parameter. Methods in which a suitable value of the regularization parameter is determined during the computation, without user intervention, are emphasized. New iterative solution methods based on expanding explicitly chosen lter functions in terms of Chebyshev polynomials are presented. The properties of these methods are illustrated with applications to image restoration.