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Order-preserving derivative approximation with periodic radial basis functions

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Abstract

In this exploratory paper we study the convergence rates of an iterated method for approximating derivatives of periodic functions using radial basis function (RBF) interpolation. Given a target function sampled on some node set, an approximation of the m th derivative is obtained by m successive applications of the operator “interpolate, then differentiate” - this process is known in the spline community as successive splines or iterated splines. For uniformly spaced nodes on the circle, we give a sufficient condition on the RBF kernel to guarantee that, when the error is measured only at the nodes, this iterated method approximates all derivatives with the same rate of convergence. We show that thin-plate spline, power function, and Matérn kernels restricted to the circle all satisfy this condition, and numerical evidence is provided to show that this phenomena occurs for some other popular RBF kernels. Finally, we consider possible extensions to higher-dimensional periodic domains by numerically studying the convergence of an iterated method for approximating the surface Laplace (Laplace-Beltrami) operator using RBF interpolation on the unit sphere and a torus.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, High Point University, High Point, NC, 27262, USA

    Edward Fuselier

  2. Department of Mathematics, Boise State University, Boise, ID, 83725-1555, USA

    Grady B. Wright

Authors
  1. Edward Fuselier
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  2. Grady B. Wright
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Correspondence to Edward Fuselier.

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Communicated by: Robert Schaback

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Fuselier, E., Wright, G.B. Order-preserving derivative approximation with periodic radial basis functions. Adv Comput Math 41, 23–53 (2015). https://doi.org/10.1007/s10444-014-9348-1

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  • DOI: https://doi.org/10.1007/s10444-014-9348-1

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