Polynomial Approximation to the Inverse of a Large Matrix

TL;DR

This paper explores polynomial methods for approximating the inverse of large matrices, analyzing their accuracy, stability, and practical applications in linear systems and Monte Carlo methods.

Contribution

It provides theoretical and experimental insights into polynomial inverse approximations, including stability improvements and alternative approaches like polynomial preconditioning.

Findings

GMRES-generated polynomials effectively approximate matrix inverses

Adding polynomial terms enhances stability of the approximation

Polynomial approximations facilitate efficient solutions for multiple linear systems

Abstract

The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its approximation to the inverse often seems to track the accuracy of the GMRES iteration. We investigate the quality of this approximation through theory and experiment, noting the practical need to add copies of some polynomial terms to improve stability. To mitigate storage and orthogonalization costs, other approaches have appeal, such as polynomial preconditioned GMRES and deflation of problematic eigenvalues. Applications of such polynomial approximations include solving systems of linear equations with multiple right-hand sides (where the solutions to subsequent problems come simply by multiplying the polynomial against the new right-hand sides) and variance reduction in multilevel Monte Carlo methods.