A regularizing Lanczos iteration method for underdetermined linear systems

Abstract

This paper is concerned with the solution of underdetermined linear systems of equations with a very ill-conditioned matrix A, whose dimensions are so large to make solution by direct methods impractical or infeasible. Image reconstruction from projections often gives rise to such systems. In order to facilitate the computation of a meaningful approximate solution, we regularize the linear system, i.e., we replace it by a nearby system that is better conditioned. The amount of regularization is determined by a regularization parameter. Its optimal value is, in most applications, not known a priori. We present a new iterative method based on the Lanczos algorithm for determining a suitable value of the regularization parameter by the discrepancy principle and an approximate solution of the regularized system of equations.