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

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Abstract

Multivariate interpolation of smooth data using smooth radial basis functions is considered. The behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically. Explicit criteria for different types of limits are given. Using the results for the limits, the dependence of the error on the shape parameter of the radial basis function is investigated. The mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error.

Keywords

Radial basis function
RBF
Interpolation
Polynomial unisolvency

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The work was supported by a postdoctoral grant from STINT, The Swedish Foundation for International Cooperation in Research and Higher Education and by a grant from The Swedish Research Council.

The work was supported by NSF Grants DMS-9810751 (VIGRE) and DMS-0309803.