Computing Discrepancies of Smolyak Quadrature Rules

AbstractIn recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as possible competition to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadrature formulas consists in computing theirL2-discrepancy. Especially for larger dimensions, such computations are a highly complex task. In this paper we develop a fast recursive algorithm for computing theL2-discrepancy (and related quality measures) of general Smolyak quadratures. We carry out numerical comparisons between the discrepancies of certain Smolyak rules and Hammersley and Monte Carlo sequences