How fast do radial basis function interpolants of analytic functions converge?
How fast do radial basis function interpolants of analytic functions converge?
复制标题
解析函数的径向基函数插值收敛的速度有多快？
DOI：
10.1093/imanum/drq020
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发表时间：
2011
影响因子：
2.1
通讯作者：
Platte R
中科院分区：
文献类型：
--
作者：
Platte R
标题中的问题是用势理论的工具来回答的。分析了单位区间上解析函数插值的敛散率。径向基函数插值的出发点是求余项的复变轮廓积分公式。我们研究了广义龙格现象，并探讨了中心的位置如何影响收敛。特别注意高斯和逆二次径向函数，但一些结果可以推广到其他光滑基函数。除此之外，我们证明了，在温和的条件下，逆二次RBF插值的功能，是分析内带|Im（z）|<（1/2 λ），其中λ是形状参数，指数收敛。
The question in the title is answered using tools of potential theory. Convergence and divergence rates of interpolants of analytic functions on the unit interval are analysed. The starting point is a complex variable contour integral formula for the remainder in radial basis function (RBF) interpolation. We study a generalized Runge phenomenon and explore how the location of centres affects convergence. Special attention is given to Gaussian and inverse quadratic radial functions, but some of the results can be extended to other smooth basis functions. Among other things, we prove that, under mild conditions, inverse quadratic RBF interpolants of functions that are analytic inside the strip |Im (z)| < (1/2ϵ), whereϵis the shape parameter, converge exponentially.