Interpolation in the limit of increasingly flat radial basis functions

被引:247
作者
Driscoll, TA
Fornberg, B
机构
[1] Univ Delaware, Dept Math Sci, Newark, DE 19716 USA
[2] Univ Colorado, Dept Appl Math, Boulder, CO 80309 USA
基金
美国国家科学基金会;
关键词
radial basis functions; RBF; PDEs; singular limit; interpolation; Lagrange polynomial; ill-conditioning;
D O I
10.1016/S0898-1221(01)00295-4
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Many types of radial basis functions, such as multiquadrics, contain a free parameter. In the limit where the basis functions become increasingly flat, the linear system to solve becomes highly ill-conditioned, and the expansion coefficients diverge. Nevertheless, we find in this study that limiting interpolants often exist and take the form of polynomials. In the 1-D case, we prove that with simple conditions on the basis function, the interpolants converge to the Lagrange interpolating polynomial. Hence, differentiation of this limit is equivalent to the standard finite difference method. We also summarize some preliminary observations regarding the limit in 2-D. (C) 2002 Elsevier Science Ltd. All rights reserved.
引用
收藏
页码:413 / 422
页数:10
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