1 Radial Basis Functions 2 1.1 Multivariate Interpolation and Positive Definiteness . . . . . . 3 1.2 Stability and Scaling . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Solving Partial Differential Equations . . . . . . . . . . . . . . 7 1.4 Comparison of Strong and Weak Problems . . . . . . . . . . . 8 1.5 Collocation Techniques . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Method ...

In this paper, we consider multivariate interpolation with radial basis functions of finite smoothness. In particular, we show that interpolants by radial basis functions in R with finite smoothness of even order converge to a polyharmonic spline interpolant as the scale parameter of the radial basis functions goes to zero, i.e., the radial basis functions become increasingly flat.

We demonstrate a relationship between the singular values of the design matrix and the discrete Fourier transform of the radial function for radial basis function networks. We then show how regularisation leads to high frequency filtering of the network output. In certain circumstances, this allows the network parameters to be chosen a priori to appropriately bias the learning process.

For radial basis function interpolation of scattered data in IR d , the approximative reproduction of high-degree polynomials is studied. Results include uniform error bounds and convergence orders on compact sets. x1. Introduction We consider interpolation of real-valued functions f deened on a set IR d ; d 1. These functions are interpolated on a set X := fx 1 ; : : : ; x N X g of N X 1 pairw...

Solving partial diierential equations by collocation with radial basis functions can be eeciently done by a technique rst proposed by E. 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 li...

Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform grids. Despite these fundamental differences, we show that the two types of representation are closely linked together . . . through fractals. First, we iden...

Radial basis functions are traditional and powerful tools for multivariate interpolation from scattered data. This self-contained talk surveys both theoretical and practical aspects of scattered data fitting by radial basis functions. To this end, basic features of the radial basis function interpolation scheme are first reviewed, such as well-posedness, numerical stability and approximation or...

| Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from three-dimensional (3-D) medical graphics. The speciic application considered is the design of cranial implants for the repair of defects, usually holes, in the skull. Radial basis functions impose few restrictions on the geometry of the interpolation centers and are su...

Radial basis function (RBFs) neural networks provide an attractive method for high dimensional nonparametric estimation for use in nonlinear control. They are faster to train than conventional feedforward networks with sigmoidal activation networks (\backpropagation nets"), and provide a model structure better suited for adaptive control. This article gives a brief survey of the use of RBFs and...

This paper proposes a Neural Network model using Generalised kernel functions for the hidden layer of a feed forward network. These functions are Generalised Radial Basis Functions (GRBF), and the architecture, weights and node topology are learned through an evolutionary algorithm. The proposed model is compared with the corresponding standard hidden-node models: Product Unit (PU) neural netwo...