Extending the range of error estimates for radial approximation in Euclidean space and on spheres

We adapt Schaback's error doubling trick [R. Schaback. Improved error bounds for scattered data interpolation by radial basis functions. Math. Comp., 68(225):201--216, 1999.] to give error estimates for radial interpolation of functions with smoothness lying (in some sense) between that of the usual native space and the subspace with double the smoothness. We do this for both bounded subsets of R^d and spheres. As a step on the way to our ultimate goal we also show convergence of pseudoderivatives of the interpolation error.Comment: 10 page

Numerical solutions of a boundary value problem on the sphere using radial basis functions

Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this article, we construct numerical solutions to a boundary value problem defined on a spherical sub-domain (with a sufficiently smooth boundary) using radial basis functions (RBF). The error analysis between the exact solution and the approximation is provided. Numerical experiments are presented to confirm theoretical estimates

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

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

Polyharmonic approximation on the sphere

The purpose of this article is to provide new error estimates for a popular type of SBF approximation on the sphere: approximating by linear combinations of Green's functions of polyharmonic differential operators. We show that the $L_p$ approximation order for this kind of approximation is $\sigma$ for functions having $L_p$ smoothness $\sigma$ (for $\sigma$ up to the order of the underlying differential operator, just as in univariate spline theory). This is an improvement over previous error estimates, which penalized the approximation order when measuring error in $L_p$, p>2 and held only in a restrictive setting when measuring error in $L_p$, p<2.Comment: 16 pages; revised version; to appear in Constr. Appro

On Polyharmonic Interpolation

In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from $C^{\infty}$ or analytic functions in the ball $B_{R}.$ We prove two main results on the interpolation of $C^{\infty}$ or analytic functions $f$ in the ball $B_{R}$ by polyharmonic functions $h$ of a given order of polyharmonicity $p.

Zooming from Global to Local: A Multiscale RBF Approach

Because physical phenomena on Earth's surface occur on many different length scales, it makes sense when seeking an efficient approximation to start with a crude global approximation, and then make a sequence of corrections on finer and finer scales. It also makes sense eventually to seek fine scale features locally, rather than globally. In the present work, we start with a global multiscale radial basis function (RBF) approximation, based on a sequence of point sets with decreasing mesh norm, and a sequence of (spherical) radial basis functions with proportionally decreasing scale centered at the points. We then prove that we can "zoom in" on a region of particular interest, by carrying out further stages of multiscale refinement on a local region. The proof combines multiscale techniques for the sphere from Le Gia, Sloan and Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32 (2012), with those for a bounded region in $\mathbb{R}^d$ from Wendland, Numer. Math. 116 (2012). The zooming in process can be continued indefinitely, since the condition numbers of matrices at the different scales remain bounded. A numerical example illustrates the process

The Penalized Lebesgue Constant for Surface Spline Interpolation

Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity, and is far less well understood. In this article we consider the stability of surface spline interpolation (a popular type of RBF interpolation) for data with nonuniform arrangements. Using techniques similar to those recently employed by Hangelbroek, Narcowich and Ward to demonstrate the stability of interpolation from quasi-uniform data on manifolds, we show that surface spline interpolation on R^d is stable, but in a stronger, local sense. We also obtain pointwise estimates showing that the Lagrange function decays very rapidly, and at a rate determined by the local spacing of datasites. These results, in conjunction with a Lebesgue lemma, show that surface spline interpolation enjoys the same rates of convergence as those of the local approximation schemes recently developed by DeVore and Ron.Comment: 20 pages; corrected typos; to appear in Proc. Amer. Math. So

Error bound for radial basis interpolation in terms of a growth function

We suggest an improvement of Wu-Schaback local error bound for radial basis interpolation by using a polynomial growth function. The new bound is valid without any assumptions about the density of the interpolation centers. It can be useful for the localized methods of scattered data fitting and for the meshless discretization of partial differential equation