Schultz matrix iteration based method for stable solution of discrete ill-posed problems

Abstract

In this paper, we propose an iterative method for solving discrete ill-posed problems based on matrix iterations generated by Schultz method and known to converge to the Moore–Penrose pseudoinverse of a matrix. Practically, letting the Schultz matrix iteration be Xk, we construct the vector xk=Xkb where b is a data vector. Hence, by construction, the iterates converge to the minimum 2-norm solution of a least squares problem with coefficient matrix A and data vector b. We derive theoretical properties of the sequence xk and show that it is quadratically convergent. In the case of corrupted data, we analyze the semi-convergence behavior of the iterates and conclude that the iteration must be truncated to control the propagation of the noise error. As a result, we derive an error estimate for the case where the truncation parameter is chosen by the discrepancy principle. In addition, combining a projected approach with the new method, we propose variants of the method that are well suited for large-scale problems. Several numerical results are presented to illustrate the effectiveness of the method on well known test problems.