Stability of convergence rates: kernel interpolation on non-Lipschitz domains

IF 2.3 2区 数学 Q1 MATHEMATICS, APPLIED
Tizian Wenzel, Gabriele Santin, Bernard Haasdonk
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引用次数: 0

Abstract

Error estimates for kernel interpolation in Reproducing Kernel Hilbert Spaces usually assume quite restrictive properties on the shape of the domain, especially in the case of infinitely smooth kernels like the popular Gaussian kernel. In this paper we prove that it is possible to obtain convergence results (in the number of interpolation points) for kernel interpolation for arbitrary domains $\varOmega \subset{\mathbb{R}} ^{d}$, thus allowing for non-Lipschitz domains including e.g., cusps and irregular boundaries. Especially we show that, when going to a smaller domain $\tilde{\varOmega } \subset \varOmega \subset{\mathbb{R}} ^{d}$, the convergence rate does not deteriorate—i.e., the convergence rates are stable with respect to going to a subset. We obtain this by leveraging an analysis of greedy kernel algorithms. The impact of this result is explained on the examples of kernels of finite as well as infinite smoothness. A comparison to approximation in Sobolev spaces is drawn, where the shape of the domain $\varOmega $ has an impact on the approximation properties. Numerical experiments illustrate and confirm the analysis.
收敛率的稳定性:非 Lipschitz 域上的核插值法
重现核希尔伯特空间中核插值的误差估计通常假定域的形状具有相当的限制性,特别是在无限光滑核(如常用的高斯核)的情况下。在本文中,我们证明了有可能获得任意域 $\varOmega \subset{\mathbb{R}} 的核插值的收敛结果(插值点数量),从而使我们的计算结果更加精确。^{d}$ ,从而允许非 Lipschitz 域,包括尖角和不规则边界等。我们尤其证明了,当进入一个更小的域 $\tilde{\varOmega }时\子集 \子集{mathbb{R}}}时^{d}$ 时,收敛速率不会恶化--也就是说,收敛速率在进入子集时是稳定的。我们通过对贪婪内核算法的分析得出了这一结论。我们以有限和无限光滑度的内核为例,解释了这一结果的影响。比较了索波列夫空间中的逼近,其中域 $\varOmega $ 的形状对逼近特性有影响。数值实验说明并证实了这一分析。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
IMA Journal of Numerical Analysis
IMA Journal of Numerical Analysis 数学-应用数学
CiteScore
5.30
自引率
4.80%
发文量
79
审稿时长
6-12 weeks
期刊介绍: The IMA Journal of Numerical Analysis (IMAJNA) publishes original contributions to all fields of numerical analysis; articles will be accepted which treat the theory, development or use of practical algorithms and interactions between these aspects. Occasional survey articles are also published.
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