SCATTERED DATA INTERPOLATION ON EMBEDDED SUBMANIFOLDS WITH RESTRICTED POSITIVE DEFINITE KERNELS: SOBOLEV ERROR ESTIMATES

In this paper we present error estimates for kernel interpolation at scattered sites on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on R d , such as radial basis functions, to a smooth, compact embedded submanifold M C R d with no boundary. For...

Full description

Saved in:
Bibliographic Details
Published in:SIAM journal on numerical analysis 2012-01, Vol.50 (3), p.1753-1776
Main Authors: FUSELIER, EDWARD, WRIGHT, GRADY B.
Format: Article
Language:English
Subjects:
Applied mathematics
Approximation
CAD
Computer aided design
Euclidean space
Fourier transformations
Interpolation
Lie groups
Mathematical functions
Mathematical manifolds
Mathematical surfaces
Mathematics
Riemann manifold
Sobolev spaces
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:In this paper we present error estimates for kernel interpolation at scattered sites on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on R d , such as radial basis functions, to a smooth, compact embedded submanifold M C R d with no boundary. 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 for smooth and nonsmooth kernels. In the case of nonsmooth kernels, we provide error estimates for target functions too rough to be within the native space of the kernel. Numerical results verifying the theory are also presented for a one-dimensional curve embedded in R³ and a two-dimensional torus.
ISSN:0036-1429
1095-7170
DOI:10.1137/110821846