Some issues on interpolation matrices of locally scaled radial basis functions |
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Authors: | Mun Bae Lee |
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Institution: | a Department of Mathematics, Konkuk University, Seoul 143-701, South Korea b Department of Mathematical Sciences, KAIST, Daejeon 305-701, South Korea c Department of Mathematics and Computer Sciences, Konkuk University, Chungju, South Korea d Department of Mathematics, Ewha W. University, Seoul, South Korea |
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Abstract: | 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 × 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. |
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Keywords: | Radial basis function Singularity Conditionally positive definite function Scaling parameter |
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