VARIANTS OF THE KOKSMA-HLAWKA INEQUALITY FOR VERTEX-MODIFIED QUASI-MONTE CARLO INTEGRATION RULES

Vertex-modified rules have recently been introduced by the authors as a way of improving the performance of quasi-Monte Carlo methods for nu merical integration. In this paper, we establish variants of the Koksm a-Hlawka inequality for vertex-modified rules, and we show that there are choices for the vertex weights which, in general, yield smaller er ror bounds than the classical Koksma-Hlawka bound. Low-discrepancy poi nt sets for which the local discrepancy has constant sign emerge as in teresting objects of study.