Solving linear systems of the form $(A + \gamma UU^T)\, {\bf x} = {\bf b}$ by preconditioned iterative methods

TL;DR

This paper develops and analyzes preconditioned iterative methods for solving large linear systems where the coefficient matrix is a sum of a sparse matrix and a rank-deficient matrix, with applications across various scientific fields.

Contribution

It introduces a novel preconditioning strategy combining alternating splitting and Sherman-Morrison-Woodbury identity for such systems.

Findings

Effective preconditioning improves convergence

Applicable to large, sparse, and rank-deficient systems

Demonstrated success in diverse application scenarios

Abstract

We consider the iterative solution of large linear systems of equations in which the coefficient matrix is the sum of two terms, a sparse matrix $A$ and a possibly dense, rank deficient matrix of the form $\gamma UU^T$, where $\gamma > 0$ is a parameter which in some applications may be taken to be 1. The matrix $A$ itself can be singular, but we assume that the symmetric part of $A$ is positive semidefinite and that $A+\gamma UU^T$ is nonsingular. Linear systems of this form arise frequently in fields like optimization, fluid mechanics, computational statistics, and others. We investigate preconditioning strategies based on an alternating splitting approach combined with the use of the Sherman-Morrison-Woodbury matrix identity. The potential of the proposed approach is demonstrated by means of numerical experiments on linear systems from different application areas.