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Numerical Integration on the Sphere

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Abstract

This chapter is concerned with numerical integration over the unit sphere \(\mathbb{S}^{2} \subset \mathbb{R}^{3}\). We first discuss basic facts about numerical integration rules with positive weights. Then some important types of rules are discussed in detail: rules with a specified polynomial degree of precision, including the important case of longitude-latitude rules; rules using scattered data points; rules based on equal-area partitions; and rules for numerical integration over subsets of the sphere. Finally we show that for numerical integration over the whole sphere and for functions with an appropriate degree of smoothness, an optimal rate of convergence can be achieved by positive-weight rules with polynomial precision and also by rules obtained by integrating a suitable radial basis function interpolant.

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Acknowledgements

The support of the Australian Research Council is gratefully acknowledged. IHS and RSW acknowledge the support of the Hong Kong Polytechnic University, where much of this work was carried out. The authors also thank Ronald Cools for helpful advice.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Paderborn, Paderborn, Germany

    Kerstin Hesse

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Ian H. Sloan

  3. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Robert S. Womersley

Authors
  1. Kerstin Hesse
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  2. Ian H. Sloan
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  3. Robert S. Womersley
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Corresponding author

Correspondence to Kerstin Hesse .

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Editors and Affiliations

  1. Geomathematics Group, University of Kaiserslautern, Kaiserslautern, Germany

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    M. Zuhair Nashed

  3. Computational Mathematics, Technische Universität Braunschweig, Braunschweig, Germany

    Thomas Sonar

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Hesse, K., Sloan, I.H., Womersley, R.S. (2015). Numerical Integration on the Sphere. In: Freeden, W., Nashed, M., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54551-1_40

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