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Radial basis function approximation of noisy scattered data on the sphere

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Abstract

In this paper we consider the approximation of noisy scattered data on the sphere by radial basis functions generated by a strictly positive definite kernel. The approximation is the minimizer in the native space for that kernel of a quadratic functional in which the smoothing term is a multiple of the square of the native space norm. The balance between data fitting and smoothness is controlled by a smoothing parameter, the choice of which should depend on the nature and magnitude of the noise. The main results concern the choice of that smoothing parameter, under the assumption that the noise is deterministic rather than random. Four strategies for choosing the smoothing parameter are considered: Morozov’s discrepancy principle, and three a priori strategies. For each of these strategies we derive an \(L_2\) error bound. The error bounds are similar, with the discrepancy principle giving marginally the best bound. A numerical example supports the theoretical results.

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Acknowledgements

The authors are particularly grateful to one of the anonymous referees who pointed out that Theorem 4.1 and its proof could be simplified and improved. We also owe this referee the a priori parameter choice (iii) in Theorem 4.1.

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Authors and Affiliations

  1. Department of Mathematics, Paderborn University, Warburger Straße 100, 33098, Paderborn, Germany

    Kerstin Hesse

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Ian H. Sloan & 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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Correspondence to Kerstin Hesse.

Additional information

The first author gratefully acknowledges the support of the Engineering and Physical Sciences Research Council under the EPSRC Overseas Travel Grant Scheme (Reference: EP/G038724/1). The second and third authors gratefully acknowledge support from the Australian Research Council.

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Hesse, K., Sloan, I.H. & Womersley, R.S. Radial basis function approximation of noisy scattered data on the sphere. Numer. Math. 137, 579–605 (2017). https://doi.org/10.1007/s00211-017-0886-6

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  • DOI: https://doi.org/10.1007/s00211-017-0886-6

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