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Numerische Mathematik

, Volume 121, Issue 3, pp 473–502 | Cite as

Quasi–Monte Carlo rules for numerical integration over the unit sphere \({\mathbb{S}^2}\)

  • Johann S. Brauchart
  • Josef Dick
Article

Abstract

We study numerical integration on the unit sphere \({\mathbb{S}^2 \subseteq\mathbb{R}^3}\) using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a (0, m, 2)-net given in the unit square [0, 1]2 to the sphere \({\mathbb{S}^2}\) by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J Sci Comput 18(2):595–609, 1997]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on \({\mathbb{S}^2}\) . And finally, we prove an upper bound on the spherical cap L 2-discrepancy of order N −1/2(log N)1/2 (where N denotes the number of points). This improves upon the bound on the spherical cap L 2-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Commun Pure Appl Math 39(S, suppl):S149–S186, 1986] by a factor of \({\sqrt{\log N}}\) . Numerical results suggest that the (0, m, 2)-nets lifted to the sphere \({\mathbb{S}^2}\) have spherical cap L 2-discrepancy converging with the optimal order of N −3/4.

Mathematics Subject Classification (2000)

65D30 65D32 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

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