. Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. 1. Introduction During the past few years the idea of using so-called meshless methods for the numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community. As a few representative examples we mention Belytschko and co-worker's results [3] using the so-called element-free Galerkin method, Duarte and Oden's work [11] using h-p clouds, Babuska and Melenk 's work [1] on the partition of unity method, ...

Kernels are valuable tools in various fields of Numerical Analysis, including approximation, interpolation, meshless methods for solving partial differential equations, neural networks, and Machine Learning. This contribution explains why and how kernels are applied in these disciplines. It uncovers the links between them, as far as they are related to kernel techniques. It addresses non-expert readers and focuses on practical guidelines for using kernels in applications.

. The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation. 2000 Mathematics Subject Classification. Primary 41A25, 65N15, 65N35; Secondary 41A05, 41A30, 41A63, 58J40, 65D05. Key words and phrases. Pseudodifferential equation, collocation, zonal kernel, interpolation, approximation order, sphere, positive definite function, radial basis function. x1. Introduction Data fitting and solving differential and integral equations on the sphere are areas of growing interest with applications to physical geodesy, potential theory, oceanography, and meteorology [6,10]. As...

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Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines. Key words. spherical splines, data fitting AMS subject classifications. 65D05, 65D07, 65D17

Abstract. Methods for scattered data fitting using multivariate splines will be surveyed in this paper. Existence, uniqueness, and computational algorithms for these methods, as well as their approximation properties will be discussed. Some applications of multivariate splines for data fitting will be briefly explained. Some new research initiatives of scattered data fitting will be outlined. Given a set of scattered data, e.g., {(xi, yi, zi), i = 1, · · · , N}, we need to find a smooth function or surface S such that S(xi, yi) = zi, i = 1, · · · , N, if zi are very accurate measurements or

We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the Laplace-Beltrami operator on S n, ω is a non-zero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (S n) and τ> 2k ≥ n/2 + 2 are used to construct an approximate solution. An H 1 (S n)error estimate is derived under the assumption that the exact solution u belongs to C 2k (S n). Key words: spherical basis function, Galerkin method

Given straight lines L i , i = 1; : : : ; N , in Euclidean 3--space with associated function values f i , we study the interpolation problem of constructing a smooth real valued function F which interpolates values f i at given data lines L i . The function F shall be defined on the entire set of lines or at least on lines contained in a domain of interest in 3--space. x1.

. Positive and conditionally positive definite functions, especially radial basis functions and similar functions for spheres, tori, and even Riemannian manifolds, are of interest because of the their well-known ability to synthesize a good surface fit from scattered data. More recently, positive definite basis functions have been employed to analyze scattered data. The methods used to do this involve constructing multiresolution analyses or multilevel approximations. This paper will discuss recent developments in the synthesis and analysis problems, point out new directions in their investigation, and remark on applications. x1. Introduction Positive definite and conditionally positive definite functions and kernels are used in areas that require fitting a surface to data taken at scattered points in Euclidean space or on some surface, a sphere or torus, say. When the underlying space is Euclidean, radial basis functions (RBFs)--- e.g., Gaussians, multiquadrics, and thin-plate spline...

Abstract. In this paper we analyse a hybrid approximation of functions on the sphere S 2 ⊂ R 3 by radial basis functions combined with polynomials, with the radial basis functions assumed to be generated by a (strictly) positive definite kernel. The approximation is determined by interpolation at scattered data points, supplemented by side conditions on the coefficients to ensure a square linear system. The analysis is first carried out in the native space associated with the kernel (with no explicit polynomial component, and no side conditions). A more refined error estimate is obtained for functions in a still smaller space. Numerical calculations support the utility of this hybrid approximation. Key words. Scattered data, radial basis functions, spherical harmonics, error estimate. AMS subject classifications. 41A30, 65D30 1. Introduction. In