Multivariate Algorithms and Information-Based Complexity

3 RBF-based penalized least-squares approximation of noisy scattered data on the sphere

Kerstin Hesse

Institute of Mathematics, Paderborn University, Paderborn, Germany

ORCID:https://orcid.org/0000-0003-4125-1941

Abstract

This paper reports on some recent developments in global and local RBF-based penalized least-squares approximation on the sphere from noisy scattered data. Given is noisy scattered data $\mathit{f}\mathrm{(}{\mathit{x}}_{\mathit{i}}\mathrm{)}$, $\mathit{i}=1\mathrm{,}2\mathrm{,}\dots \mathrm{,}\mathit{N}$, at N distinct points ${\mathit{x}}_1\mathrm{,}{\mathit{x}}_2\mathrm{,}\dots \mathrm{,}{\mathit{x}}_{\mathit{N}}\in \bar{\mathrm{\Omega }}$ from a function f in the Sobolev (Hilbert) space ${\mathit{W}}_2^{\mathit{s}}\mathrm{(}\mathrm{\Omega }\mathrm{)}$, $\mathit{s}\mathrm{>}\mathit{d}\mathrm{/}2$, where Ω is either the whole unit sphere ${\mathbb{S}}^{\mathit{d}}$ in ${\mathbb{R}}^{\mathit{d}+1}$ or a nonempty connected open subset of ${\mathbb{S}}^{\mathit{d}}$ with Lipschitz-continuous boundary, contained in an open hemisphere. The penalized least-squares approximation of f is defined ...