Robert Schaback

Error estimates and condition numbers for radial basis function interpolation

For interpolation of scattered multivariate data by radial basis functions, an “uncertainty relation” between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the Narcowich-Ward theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use.

Solving partial differential equations by collocation using radial basis functions

After a series of application papers have proven the approach to be numerically effective, this paper gives the first theoretical foundation for methods solving partial differential equations by collocation with (possibly radial) basis functions.

On unsymmetric collocation by radial basis functions

Solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by Kansa in 1990. It rewrites the problem as a generalized interpolation problem, and the solution is obtained by solving a (possibly large) linear system. The method has been used successfully in a variety of applications, but a proof of nonsingularity of the linear system was still missing. This paper shows that a general proof of this fact is impossible. However, numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort.

Convergence order estimates of meshless collocation methods using radial basis functions

We study meshless collocation methods using radial basis functions to approximate regular solutions of systems of equations with linear differential or integral operators. Our method can be interpreted as one of the emerging meshless methods, cf. T. Belytschko et al. (1996). Its range of application is not confined to elliptic problems. However, the application to the boundary value problem for an elliptic operator, connected with an integral equation, is given as an example. Although the method has been used for special cases for about ten years, cf. E.J. Kansa (1990), there are no error bounds known. We put the main emphasis on detailed proofs of such error bounds, following the general outline described in C. Franke and R. Schaback (preprint).

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.

SHAPE PRESERVING PROPERTIES AND CONVERGENCE OF UNIVARIATE MULTIQUADRIC QUASI-INTERPOLATION

With a suitable modiication at the endpoints of the range, quasi{interpolation with univariate multiquadrics (x) = p c 2 + x 2 is shown to preserve convexity and monotonic-ity. If h is the maximum distance of centres, convergence of the quasi{interpolant is of order O(h 2 j log hj) if c = O(h). The log term can not be removed by introducing diierent boundary conditions or special placements of the centres.

Improved error bounds for scattered data interpolation by radial basis functions

If additional smoothness requirements and boundary conditions are met, the well-known approximation orders of scattered data interpolants by radial functions can roughly be doubled.

Native Hilbert Spaces for Radial Basis Functions I

This contribution gives a partial survey over the native spaces associated to (not necessarily radial) basis functions. Starting from reproducing kernel Hilbert spaces and invariance properties, the general construction of native spaces is carried out for both the unconditionally and the conditionally positive definite case. The definitions of the latter are based on finitely supported functional only. Fourier or other transforms are not required. The dependence of native spaces on the domain is studied, and criteria for functions and functionals to be in the native space are given. Basic facts on optimal recovery, power functions, and error bounds are included.

Stable and Convergent Unsymmetric Meshless Collocation Methods

In the theoretical part of this paper, we introduce a simplified proof technique for error bounds and convergence of a variation of Kansas well-known unsymmetric meshless collocation method. For a numerical implementation of the convergent variation, a previously proposed greedy technique is coupled with linear optimization. This algorithm allows a fully adaptive on-the-fly data-dependent meshless selection of test and trial spaces. The new method satisfies the assumptions of the background theory, and numerical experiments demonstrate its stability.