Quasi-Random Methods for Estimating Integrals Using Relatively Small Samples

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Abstract
Summary:Many low-discrepancy sets are suitable for quasi-Monte Carlo integration. Skriganov showed that the intersections of suitable lattices with the unit cube have low discrepancy. We introduce an algorithm based on linear programming which scales any given lattice appropriately and computes its intersection with the unit cube. We compare the quality of numerical integration using these low-discrepancy lattice sets with approximations using other known (quasi-)Monte Carlo methods. The comparison is based on several numerical experiments, where we consider both the precision of the approximation and the speed of generating the sets. We conclude that up to dimensions about 15, low-discrepancy lattices yield fairly good results. In higher dimensions, our implementation of the computation of the intersection takes too long and ceases to be feasible

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