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Efficient Spherical Designs with Good Geometric Properties

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Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

Abstract

Spherical t-designs on \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\) provide N nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most t. This paper considers the generation of efficient, where N is comparable to (1 + t)d∕d, spherical t-designs with good geometric properties as measured by their mesh ratio, the ratio of the covering radius to the packing radius. Results for \(\mathbb {S}^{2}\) include computed spherical t-designs for t = 1, …, 180 and symmetric (antipodal) t-designs for degrees up to 325, all with low mesh ratios. These point sets provide excellent points for numerical integration on the sphere. The methods can also be used to computationally explore spherical t-designs for d = 3 and higher.

Dedicated to Ian H. Sloan on the occasion of his 80th birthday in acknowledgement of his many fruitful ideas and generosity.

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Acknowledgements

This research includes extensive computations using the Linux computational cluster Katana supported by the Faculty of Science, UNSW Sydney.

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

    Robert S. Womersley

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  1. Robert S. Womersley
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Correspondence to Robert S. Womersley .

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

    Josef Dick

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

    Frances Y. Kuo

  3. Department of Computer Science, Columbia University, New York, USA

    Henryk Woźniakowski

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Womersley, R.S. (2018). Efficient Spherical Designs with Good Geometric Properties. In: Dick, J., Kuo, F., Woźniakowski, H. (eds) Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan. Springer, Cham. https://doi.org/10.1007/978-3-319-72456-0_57

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