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...

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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
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WRIGHT, GRADY B.
description 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.
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source SIAM Journals Online; ABI/INFORM global; JSTOR Archival Journals and Primary Sources Collection
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
title SCATTERED DATA INTERPOLATION ON EMBEDDED SUBMANIFOLDS WITH RESTRICTED POSITIVE DEFINITE KERNELS: SOBOLEV ERROR ESTIMATES
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