On the Choice of Subspace for Iterative Methods for Linear Discrete Ill-Posed Problems

Abstract

. Many iterative methods for the solution of linear discrete ill-posed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that it may be possible to determine a meaningful approximate solution with less computational work when this requirement is not imposed. Key words. Minimal residual method, conjugate gradient method, linear ill-posed problems. 1. Introduction. This paper is concerned with the design of iterative methods for the computation of approximate solutions of linear systems of equations Ax = b, A # R mn , x # R n , b # R m , (1.1) with a large matrix A of ill-determined rank. Thus, A has many "tiny" singular values of di#erent orders of magnitude. In particular, A is severely ill-conditioned. Some of the singular values of A may be vanishing. We allow m # n or m < n. The right-hand side vector b is not required to be in the range of A. Linear systems of equations of the fo...