Abstract. A hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support. A nested sequence of subsets of the data is computed efficiently using successive Delaunay triangulations. The scale of the basis function at each level is determined from the current density of the points using information from the triangulation. The method is rotationally invariant and has good reproduction properties. Moreover the solution can be calculated and evaluated in acceptable computing time. During the last two decades radial basis functions have become a well established tool for multivariate interpolation of both scattered and gridded data; see [2,7,8,22,25] for some surveys. The major part

Most literature on Support Vector Machines (SVMs) concentrate on the dual optimization problem. In this paper, we would like to point out that the primal problem can also be solved efficiently, both for linear and non-linear SVMs, and that there is no reason for ignoring this possibilty. On the contrary, from the primal point of view new families of algorithms for large scale SVM training can be investigated.

A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can therefore be more efficient than the usual finite element methods. An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for a-posteriori error estimation for this new method are also proved. Key words: Finite element method, meshless finite element method, finite element methods for highly oscillatory solutions TICAM, The University of Texas at Austin, Austin, TX 78712. Research was partially supported by US Office of Naval Research under grant N00014--90--J1030 y Seminar for Applied Mathematics, ETH Zurich, CH--8092 Zurich, Switzerland....

. This contribution will touch the following topics: ffl Short introduction into the theory of multivariate interpolation and approximation by finitely many (irregular) translates of a (not necessarily radial) basis function, motivated by optimal recovery of functions from discrete samples. ffl Native spaces of functions associated to conditionally positive definite functions, and relations between such spaces. ffl Error bounds and condition numbers for interpolation of functions from native spaces. ffl Uncertainty Relation: Why are good error bounds always tied to bad condition numbers? ffl Shift and Scale: How to cope with the Uncertainty Relation? x1. Introduction and Overview This contribution contains the author's view of a certain area of multivariate interpolation and approximation. It is not intended to be a complete survey of a larger area of research, and it will not account for the history of the theory it deals with. Related surveys are [15, 21, 22, 27, 30, 47, 48, 58...

. Motivated by [5] we describe a method related to scattered Hermite interpolation for which the solution of elliptic partial differential equations by collocation is well-posed. We compare the method of [5] with our method. x1. Introduction In this paper we discuss the numerical solution of elliptic partial differential equations using a collocation approach based on radial basis functions. To make the discussion transparent we will focus on the case of a time independent linear elliptic partial differential equation in IR 2 . In the following we assume we are given a set of nodes \Xi = f ~ ¸ 1 ; : : : ; ~ ¸ N g ae IR d , along with a continuous function ' : [0; 1) ! IR. We then refer to ~x 7! '(k~x\Gamma ~ ¸ k k 2 ), ~x 2 IR d , k 2 f1; : : : ; Ng, as radial basis functions centered at ~ ¸ k . Some of the most commonly used radial basis functions are the (reciprocal) multiquadrics '(r) = (r 2 + c 2 ) \Sigma1=2 , the Gaussians '(r) = e \Gammacr 2 , and the thin pla...

We consider error estimates for the interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated "native" Hilbert spaces are shown to be norm-equivalent to Sobolev spaces. Thus we can derive approximation orders for functions from Sobolev spaces which are comparable to those of thin-plate-spline interpolation. Finally, we investigate the numerical stability of the interpolation process.

. We discuss several approaches to the problem of interpolating or approximating data given at scattered points lying on the surface of the sphere. These include methods based on spherical harmonics, tensorproduct spaces on a rectangular map of the sphere, functions defined over spherical triangulations, spherical splines, spherical radial basis functions, and some associated multi-resolution methods. In addition, we briefly discuss sphere-like surfaces, visualization, and methods for more general surfaces. The paper includes a total of 206 references. x1. Introduction Let S be the unit sphere in IR 3 , and suppose that fv i g n i=1 is a set of scattered points lying on S. In this paper we are interested in the following problem: Problem 1. Given real numbers fr i g n i=1 , find a (smooth) function s defined on S which interpolates the data in the sense that s(v i ) = r i ; i = 1; : : : ; n; (1) or approximates it in the sense that s(v i ) ß r i ; i = 1; : : : ; n: (2) Data f...

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.

Abstract. We combine the theory of radial basis functions with the field of Galerkin methods to solve partial differential equations. After a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions. 1.

Abstract: Multistep interpolation of scattered data by compactly supported radial basis functions requires hierarchical subsets of the data. This paper analyzes thinning algorithms for generating evenly distributed subsets of scattered data in a given domain in IR d.