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

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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 γd,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→∞ ∑j=1γd,jd < ∞; and tractability holds (i.e. the number of function evaluations grows at most polynomially in d) if and only if lim supd→∞ ∑j=1γd,j/log (d + 1) < ∞. The arguments are not constructive. © 2004 Elsevier Inc. All rights reserved.

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Kuo, F. Y., & Sloan, I. H. (2005). Quasi-Monte Carlo methods can be efficient for integration over products of spheres. Journal of Complexity, 21(2), 196–210. https://doi.org/10.1016/j.jco.2004.07.001

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