Efficiency of Monte Carlo computations in very high dimensional spaces

A standard measure for comparing different Monte Carlo estimators is the efficiency, which generally thought to be declining with increasing the number of dimensions. Here we give some numerical examples, ranging from one‐hundred to one‐thousand dimensional integration problems, that contradict this belief. Monte Carlo integrations carried out in one‐thousand dimensional spaces is the other nontrivial result reported here. The examples concern the computation of the probabilities of convex sets (polyhedra and hyperellipsoids) in case of multidimensional normal probabilities.