Low Discrepancy Sequences in High Dimensions: How Well Are Their Projections Distributed?

Quasi-Monte Carlo (QMC) methods have been successfully used to compute high-dimensional integrals arising in many applications, especially in finance. To understand this success and the possible potential limitations of QMC, this paper focuses on quantitative measures of the quality of a point set in high dimensions. We introduce the order-`, superposition and truncation discrepancies, which measure the quality of selected projections of the point set on lower-dimensional spaces. These measures are more informative than the classical ones. We study their relationships with the integration errors and discuss the tractability issues. We present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the Sobol nets with that of Latin hypercube sampling and random point sets. The numerical results show that in high dimensions the superiority of the Sobol nets mainly derives from the order-1 (i.e., one-dimensional) projections and the projections associated with the earlier dimensions; for order-2 and higher-order projections all these point sets have similar behavior (on the average). In weighted cases with fast decaying weights, the Sobol nets have a better performance than the other two kinds of point sets. The investigation enables us to better understand the inherent properties of QMC and throws new light on when and why QMC can have a better (or no better) performance than Monte Carlo for multivariate integration.