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

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