Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates

TL;DR

This paper studies how kernel-based interpolation methods perform on embedded manifolds, providing Sobolev error estimates and numerical validation for RBF kernels restricted to submanifolds.

Contribution

It characterizes the native space of restricted kernels on manifolds and derives Sobolev-type error estimates for kernel interpolation on embedded submanifolds.

Findings

Complete characterization of native spaces for restricted kernels.

Sobolev error estimates for kernel interpolation on manifolds.

Numerical validation on curves and tori.

Abstract

In this paper we investigate the approximation properties of kernel interpolants on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on $\R^d$, such as radial basis functions (RBFs), to a smooth, compact embedded submanifold $\M\subset \R^d$. For restricted kernels having finite smoothness, we provide a complete characterization of the native space on $\M$. After this and some preliminary setup, we present Sobolev-type error estimates for the interpolation problem. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in $\R^3$ and a two-dimensional torus.