Iterative SVD-Based Methods for Ill-Posed Problems

Abstract

Very large matrices with rapidly decaying singular values commonly arise in the numerical solution of ill-posed problems. The singular value decomposition (SVD) is a basic tool for both the analysis and computation of solutions to such problems. In most applications, it suffices to obtain a partial SVD consisting of only the largest singular values and their corresponding singular vectors. In this paper, two separate approaches—one based on subspace iteration and the other based on the Lanczos method—are considered for the efficient iterative computation of partial SVDs. In the context of ill-posed problems, an analytical and numerical comparison of these two methods is made and the role of the regularization operator in convergence acceleration is explored.