  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.
