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 -