Holger Wendland

Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree

We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind.

Meshless Galerkin methods using radial basis functions

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.

Multivariate interpolation for fluid-structure-interaction problems using radial basis functions

A multivariate interpolation scheme for coupling fluid (CFD) and structural models (FE) in three-dimensional space is presented using radial basis functions. For the purpose of numerical aeroelastic computations, a selection of applicable functions is chosen: a classical without compact support, and some recently presented smooth compactly supported radial basis functions. The scheme is applied to a typical static aeroelastic problem, the prediction of the equilibrium of an elastic wing model in transonic fluid flow. The resulting coupled field problem containing the fluid and the structural state equations is solved by applying a partitioned solution procedure. The structure is represented by finite elements and the related equations are solved using a commercial FE analysis code. The transonic fluid flow is described by the three-dimensional Euler equations, solved by an upwind scheme procedure.

Kernel Techniques: From Machine Learning to Meshless Methods

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, in so far as they are related to kernel techniques. It addresses non-expert readers and focuses on practical guidelines for using kernels in applications.

Adaptive greedy techniques for approximate solution of large RBF systems

For the solution of large sparse linear systems arising from interpolation problems using compactly supported radial basis functions, a class of efficient numerical algorithms is presented. They iteratively select small subsets of the interpolation points and refine the current approximative solution there. Convergence turns out to be linear, and the technique can be generalized to positive definite linear systems in general. A major feature is that the approximations tend to have only a small number of nonzero coefficients, and in this sense the technique is related to greedy algorithms and best n-term approximation.

Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does not belong to the native space of the RBF.

Computational aspects of radial basis function approximation

This paper gives an overview on numerical aspects of multivariate interpolation and approximation by radial basis functions. It comments on the correct choice of the basis function. It discusses the reduction of complexity by dier ent methods as well as the problem of ill-conditioning. It is aimed to be a user’s guide to an ecient employment of radial basis functions for the reconstruction of multivariate functions.

Near-optimal data-independent point locations for radial basis function interpolation

Abstract The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples.

Approximate Interpolation with Applications to Selecting Smoothing Parameters

In this paper, we study the global behavior of a function that is known to be small at a given discrete data set. Such a function might be interpreted as the error function between an unknown function and a given approximant. We will show that a small error on the discrete data set leads under mild assumptions automatically to a small error on a larger region. We will apply these results to spline smoothing and show that a specific, a priori choice of the smoothing parameter is possible and leads to the same approximation order as the classical interpolant. This has also a surprising application in stabilizing the interpolation process by splines and positive definite kernels.

Direct and Inverse Sobolev Error Estimates for Scattered Data Interpolation via Spherical Basis Functions

The purpose of this paper is to get error estimates for spherical basis function (SBF) interpolation and approximation for target functions in Sobolev spaces less smooth than the SBFs, and to show that the rates achieved are, in a sense, best possible. In addition, we establish a Bernstein-type theorem, where the smallest separation between data sites plays the role of a Nyquist frequency. We then use these Berstein-type estimates to derive inverse estimates for interpolation via SBFs.