Quasi-Monte Carlo tractability of high dimensional integration over products of simplices

Abstract

Quasi-Monte Carlo (QMC) methods for high dimensional integrals over unit cubes and products of spheres are well-studied in the literature. We study QMC tractability of integrals of functions defined over the product of m copies of the simplex Td⊂Rd. The domain is a tensor product of m reproducing kernel Hilbert spaces defined by ‘weights’ γm,j, for j=1,2,…,m. Similar to the results on the unit cube in m dimensions, and the product of m copies of the d-dimensional sphere, we prove that strong polynomial tractability holds iff lim supm→∞∑j=1mγm,j<∞ and polynomial tractability holds iff lim supm→∞∑j=1mγm,jlog(m+1)<∞. We also show that weak tractability holds iff limm→∞∑j=1mγm,jm=0. The proofs employ Sobolev space techniques and weighted reproducing kernel Hilbert space techniques for the simplex and products of simplices as domain. Properties of orthogonal polynomials on a simplex are also used extensively.