Generalized Approximate Inverse Preconditioners for Least Squares Problems

Abstract

This paper is concerned with a new approach for preconditioning large sparse least squares problems. Based on the idea of the approximate inverse preconditioner, which was originally developed for square matrices, we construct a generalized approximate inverse (GAINV) $M$ which approximately minimizes $\|I-MA\|_{\mathrm{F}}$ or $\|I-AM\|_{\mathrm{F}}$. Then, we also discuss the theoretical issues such as the equivalence between the original least squares problem and the preconditioned problem. Finally, numerical experiments on problems from Matrix Market collection and random matrices show that although the preconditioning is expensive, it pays off in certain cases.