Quasi-Monte Carlo, quasi-random numbers and quasi-error estimates

Abstract

We discuss quasi-random number sequences as a basis for numerical integration with potentially better convergence properties than standard Monte Carlo. The importance of the discrepancy as both a measure of smoothness of distribution and an ingredient in the error estimate is reviewed. It is argued that the classical Koksma-Hlawka inequality is not relevant for error estimates in realistic cases, and a new class of error estimates is presented, based on a generalization of the Woźniakowski lemma.