Quasi-Monte Carlo methods can be efficient for integration over products of spheres

We study the worst-case error of quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined on the product of d copies of the unit sphere S-s subset of Rs+1. The space is a tensor product of d reproducing kernel Hilbert spaces defined in terms of uniformly bounded 'weight' parameters gamma(d.j) for j = 1,2, ..., d. We prove that strong QMC tractability holds (i.e. the number of function evaluations needed to reduce the initial error by a factor of epsilon is bounded independently of d) if and only if lim sup(d ->infinity)Sigma(d)(j=1) gamma d.j < infinity; and tractability holds (i.e. the number of function evaluations crows at most polynomially in d) if and only if lim Sup(d ->infinity)Sigma(d)(j=1) gamma(d.j)/log(d + 1) < infinity. The arguments are not constructive. (c) 2004 Elsevier Inc. All rights reserved.