TY - JOUR

T1 - Some issues on interpolation matrices of locally scaled radial basis functions

AU - Lee, Mun Bae

AU - Lee, Yeon Ju

AU - Sunwoo, Hasik

AU - Yoon, Jungho

PY - 2011/1/15

Y1 - 2011/1/15

N2 - 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.

AB - 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.

KW - Conditionally positive definite function

KW - Radial basis function

KW - Scaling parameter

KW - Singularity

UR - http://www.scopus.com/inward/record.url?scp=78651314407&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2010.11.040

DO - 10.1016/j.amc.2010.11.040

M3 - Article

AN - SCOPUS:78651314407

SN - 0096-3003

VL - 217

SP - 5011

EP - 5014

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

IS - 10

ER -