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Error bounds for kernel-based numerical differentiation

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Abstract

The literature on meshless methods shows that kernel-based numerical differentiation formulae are robust and provide high accuracy at low cost. This paper analyzes the error of such formulas, using the new technique of growth functions. It allows to bypass certain technical assumptions that were needed to prove the standard error bounds on interpolants and their derivatives. Since differentiation formulas based on polynomials also have error bounds in terms of growth functions, we have a convenient way to compare kernel-based and polynomial-based formulas. It follows that kernel-based formulas are comparable in accuracy to the best possible polynomial-based formulas. A variety of examples is provided.

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Acknowledgments

The authors are grateful to the Alexander von Humboldt Foundation for the financial support that helped them to get together and work on this project for several days in September 2011.

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Authors and Affiliations

  1. Department of Mathematics, University of Giessen, Arndtstrasse 2, 35392, Giessen, Germany

    Oleg Davydov

  2. Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestraße 16-18, 37083, Göttingen, Germany

    Robert Schaback

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  1. Oleg Davydov
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  2. Robert Schaback
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Correspondence to Oleg Davydov.

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Davydov, O., Schaback, R. Error bounds for kernel-based numerical differentiation. Numer. Math. 132, 243–269 (2016). https://doi.org/10.1007/s00211-015-0722-9

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  • DOI: https://doi.org/10.1007/s00211-015-0722-9

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