Optimal lebesgue constant for lagrange interpolation
SIAM Journal on Numerical Analysis
Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation
Journal of Approximation Theory
Spectral methods in MatLab
Guest Editors' Introduction: The Top 10 Algorithms
Computing in Science and Engineering
Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation
SIAM Journal on Numerical Analysis
Computers & Mathematics with Applications
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Computers & Mathematics with Applications
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
The role of the multiquadric shape parameters in solving elliptic partial differential equations
Computers & Mathematics with Applications
On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere
Journal of Computational Physics
Applied Numerical Mathematics
An SVD analysis of equispaced polynomial interpolation
Applied Numerical Mathematics
Rational radial basis function interpolation with applications to antenna design
Journal of Computational and Applied Mathematics
Error saturation in Gaussian radial basis functions on a finite interval
Journal of Computational and Applied Mathematics
A high order multivariate approximation scheme for scattered data sets
Journal of Computational Physics
Computers & Mathematics with Applications
An alternative procedure for selecting a good value for the parameter c in RBF-interpolation
Advances in Computational Mathematics
Applied Numerical Mathematics
Stabilization of RBF-generated finite difference methods for convective PDEs
Journal of Computational Physics
Spectral collocation and radial basis function methods for one-dimensional interface problems
Applied Numerical Mathematics
Optimal constant shape parameter for multiquadric based RBF-FD method
Journal of Computational Physics
Computers & Mathematics with Applications
Stable Computations with Gaussian Radial Basis Functions
SIAM Journal on Scientific Computing
Journal of Computational Physics
Journal of Scientific Computing
Stable Evaluation of Gaussian Radial Basis Function Interpolants
SIAM Journal on Scientific Computing
Vector field approximation using radial basis functions
Journal of Computational and Applied Mathematics
Journal of Approximation Theory
Stable calculation of Gaussian-based RBF-FD stencils
Computers & Mathematics with Applications
A radial basis functions method for fractional diffusion equations
Journal of Computational Physics
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Many studies, mostly empirical, have been devoted to finding an optimal shape parameter for radial basis functions (RBF). When exploring the underlying factors that determine what is a good such choice, we are led to consider the Runge phenomenon (RP; best known in cases of high order polynomial interpolation) as a key error mechanism. This observation suggests that it can be advantageous to let the shape parameter vary spatially, rather than assigning a single value to it. Benefits typically include improvements in both accuracy and numerical conditioning. Still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas. This idea is routinely used when working with splines or finite element methods. However, local refinement with RBFs may cause RP-type errors unless we use a spatially variable shape paremeter. With this enhancement, RBF approximations combine freedom from meshes with spectral accuracy on irregular domains, and furthermore permit local node clustering to improve the resolution wherever this might be needed.