Matrix-free stochastic calculation of operator norms without using adjoints

TL;DR

This paper introduces a stochastic, matrix-free method for computing the operator norm of a linear map using only evaluations, with proven convergence and demonstrated effectiveness through numerical experiments.

Contribution

It presents a novel stochastic algorithm that maximizes the Rayleigh quotient without requiring matrix adjoints or extensive storage, ensuring convergence to the operator norm.

Findings

Algorithm converges to the global maximum almost surely

Method exhibits sublinear convergence behavior

Numerical experiments validate the approach's effectiveness

Abstract

This paper considers the problem of computing the operator norm of a linear map between finite dimensional Hilbert spaces when only evaluations of the linear map are available and under restrictive storage assumptions. We propose a stochastic method of random search type to maximize the Rayleigh quotient and employ an exact line search in the random search directions. Moreover, we show that the proposed algorithm converges to the global maximum (the operator norm) almost surely and a sublinear convergence behavior for the corresponding eigenvector and eigenvalue equation. Finally, we illustrate the performance of the method with numerical experiments.