Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation

Authors:
Rodrigo B. Platte;Tobin A. Driscoll
Affiliations:
-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2005

Citing 0
Cited 15

Adaptive residual subsampling methods for radial basis function interpolation and collocation problems

Computers & Mathematics with Applications
The Runge phenomenon and spatially variable shape parameters in RBF interpolation

Computers & Mathematics with Applications
A Hybrid Fourier---Chebyshev Method for Partial Differential Equations

Journal of Scientific Computing
Eigenvalue stability of radial basis function discretizations for time-dependent problems

Computers & Mathematics with Applications
The uselessness of the Fast Gauss Transform for summing Gaussian radial basis function series

Journal of Computational Physics
Error saturation in Gaussian radial basis functions on a finite interval

Journal of Computational and Applied Mathematics
Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems

Journal of Scientific Computing
Six strategies for defeating the Runge Phenomenon in Gaussian radial basis functions on a finite interval

Computers & Mathematics with Applications
A Simple Regularization of the Polynomial Interpolation for the Resolution of the Runge Phenomenon

Journal of Scientific Computing
Spectral collocation and radial basis function methods for one-dimensional interface problems

Applied Numerical Mathematics
Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples

SIAM Review
Matrix Stability of Multiquadric Radial Basis Function Methods for Hyperbolic Equations with Uniform Centers

Journal of Scientific Computing
A Fast Treecode for Multiquadric Interpolation with Varying Shape Parameters

SIAM Journal on Scientific Computing
Vector field approximation using radial basis functions

Journal of Computational and Applied Mathematics
Full length article: Univariate interpolation by exponential functions and Gaussian RBFs for generic sets of nodes

Journal of Approximation Theory

Quantified Score

Hi-index	0.01

Visualization

Abstract

We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these radial functions are susceptible to the Runge phenomenon, not only in the limit of increasingly flat functions, but also in the finite shape parameter case. We show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations. Using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices, while not imposing restrictions on the shape parameter or number of points.