A regularizing L-curve Lanczos method for underdetermined linear systems

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 infeasib le. Image reconstruction from projections is a well-known example of such s ystems. 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 determin ed by a regularization parameter. Its optimal value is, in most application s, 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 algor ithm for inexpensively evaluating an approximation of the points on the L-c urve and then determine the value of the optimal regularization parameter w hich lets us compute an approximate solution of the regularized system of e quations. (C) 2001 Elsevier Science Inc. All rights reserved.