Some issues on interpolation matrices of locally scaled radial basis functions

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Radial basis function interpolation on a set of scattered data is constructed from the corresponding translates of a basis function, which is conditionally positive definite of order m >= 0, with the possible addition of a polynomial term. In many applications, the translates of a basis function are scaled differently, in order to match the local features of the data such as the flat region and the data density. Then, a fundamental question is the non-singularity of the perturbed interpolation (N x N) matrix. In this paper, we provide some counter examples of the matrices which become singular for N >= 3, although the matrix is always non-singular when N = 2. One interesting feature is that a perturbed matrix can be singular with rather small perturbation of the scaling parameter. (c) 2010 Elsevier Inc. All rights reserved.
Publisher
ELSEVIER SCIENCE INC
Issue Date
2011-01
Language
English
Article Type
Article
Keywords

POSITIVE DEFINITE FUNCTIONS; DISTANCE MATRICES; SCATTERED DATA

Citation

APPLIED MATHEMATICS AND COMPUTATION, v.217, no.10, pp.5011 - 5014

ISSN
0096-3003
DOI
10.1016/j.amc.2010.11.040
URI
http://hdl.handle.net/10203/94700
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