Limit problems for interpolation by analytic radial basis functions

Authors:
Robert Schaback
Affiliations:
Institut für Numerische und Angewandte Mathematik, Lotzestraíe 16-18, D-37083 Göttingen, Germany
Venue:
Journal of Computational and Applied Mathematics
Year:
2008

Citing 5
Cited 4

Solving partial differential equations by collocation using radial basis functions

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Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions

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Stable computation of multiquadric interpolants for all values of the shape parameter

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A new class of oscillatory radial basis functions

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Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval

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Multivariate interpolation with increasingly flat radial basis functions of finite smoothness

Advances in Computational Mathematics
Stable Evaluation of Gaussian Radial Basis Function Interpolants

SIAM Journal on Scientific Computing
Full length article: On collocation matrices for interpolation and approximation

Journal of Approximation Theory

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Abstract

Interpolation problems for analytic radial basis functions like the Gaussian and inverse multiquadrics can degenerate in two ways: the radial basis functions can be scaled to become increasingly flat, or the data points coalesce in the limit while the radial basis functions stay fixed. Both cases call for a careful regularization, which, if carried out explicitly, yields a preconditioning technique for the degenerating linear systems behind these interpolation problems. This paper deals with both cases. For the increasingly flat limit, we recover results by Larsson and Fornberg together with Lee, Yoon, and Yoon concerning convergence of interpolants towards polynomials. With slight modifications, the same technique can also handle scenarios with coalescing data points for fixed radial basis functions. The results show that the degenerating local Lagrange interpolation problems converge towards certain Hermite-Birkhoff problems. This is an important prerequisite for dealing with approximation by radial basis functions adaptively, using freely varying data sites.