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

  • Authors:

  • Frances Y. Kuo;Ian H. Sloan

  • Affiliations:

  • School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia;School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia

  • Venue:
  • Journal of Complexity
  • Year:
  • 2005

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Abstract

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 Ss ⊆ Rs+1. The space is a tensor product of d reproducing kernel Hilbert spaces defined in terms of uniformly bounded 'weight' parameters λd,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 ε is bounded independently of d) if and only if lim supd →∞Σj=1dγd,j d) if and only if lim supd→∞Σj=1d=γd,j/log(d + 1)