A Multilevel Approach to Variance Reduction in the Stochastic Estimation of the Trace of a Matrix

TL;DR

This paper introduces a multilevel Monte Carlo method leveraging multigrid hierarchies to significantly reduce variance in stochastic trace estimation, enabling higher precision with less computational effort.

Contribution

It proposes a novel multilevel approach for variance reduction in stochastic trace estimation using multigrid hierarchies, improving efficiency over traditional methods.

Findings

Substantial variance reduction achieved

Higher precision estimates require less computational effort

Effective across different matrix classes

Abstract

The trace of a matrix function f(A), most notably of the matrix inverse, can be estimated stochastically using samples< x,f(A)x> if the components of the random vectors x obey an appropriate probability distribution. However such a Monte-Carlo sampling suffers from the fact that the accuracy depends quadratically of the samples to use, thus making higher precision estimation very costly. In this paper we suggest and investigate a multilevel Monte-Carlo approach which uses a multigrid hierarchy to stochastically estimate the trace. This results in a substantial reduction of the variance, so that higher precision can be obtained at much less effort. We illustrate this for the trace of the inverse using three different classes of matrices.