Radial Basis Functions

Abstract

In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing, it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a powerful tool which work well in very general circumstances and so are becoming of widespread use as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent. The author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.

Tensor Field Interpolation with PDEs

“…Inspired from spline interpolation, a suitable n-dimensional generalisation of (3) would use the Laplacian operator (also called harmonic or linear diffusion operator) Lu := ∆u for m = 1, the biharmonic operator Lu := −∆ 2 u for m = 2, or the triharmonic operator Lu := ∆ 3 u for m = 3. These linear operators correspond to interpolation with radial basis functions [3]. From the theory of nonlinear diffusion filtering, on the other side, it would be interesting to use the isotropic nonlinear operator Lu := div (g(|∇u| 2 ) ∇u) or its anisotropic counterpart 1 Lu := div (g(∇u σ ∇u σ ) ∇u), where u σ is a Gaussian-smoothed version of u, and g is a decreasing positive diffusivity function.…”

Reduction Strategies for Shape Dependent Inverse Problems in Haemodynamics

“…Both the settings deal with p = 6 parameters, given by the vertical displacements of some selected control points; in the FFD case we introduce a 6 × 8 lattice of control points, while in the RBF case we introduce in total 12 control points close to the bifurcation and at the extrema (see Fig. 3), using the thin-plate spline (TPS) and the Gaussian shape functions [2]. In this last case, we deal with the displacement of the six control points located at the center of the configuration.…”

Automated synthesis of action selection policies for unmanned vehicles operating in adverse environments

“…However, the potential number of modes needed to be evolved can be very high for some tasks. Another work (Kohl and Miikkulainen 2008) extends NEAT to use radial basis function (RBF) (Buhmann 2001) nodes to evolve controllers with better performance for complex problems. The above mentioned neuroevolutionary approaches make localized changes to the policy by directly modifying the structure of a neural network.…”