A regularizing L-curve Lanczos method for underdetermined linear systems

Abstract

Many real applications give rise to the solution of underdetermined linear systems of equations with a very ill conditioned matrix A, whose dimensions are so large as to make solution by direct methods impractical or infeasible. Image reconstruction from projections is a well-known example of 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. A well-known method to determine it is given by the L-curve approach. We present an iterative method based on the Lanczos algorithm for inexpensively evaluating an approximation of the points on the L-curve and then determine the value of the optimal regularization parameter which lets us compute an approximate solution of the regularized system of equations.