Adaptive radial basis function methods for time dependent partial differential equations
Applied Numerical Mathematics
Scattered node compact finite difference-type formulas generated from radial basis functions
Journal of Computational Physics
Computers & Mathematics with Applications
The Runge phenomenon and spatially variable shape parameters in RBF interpolation
Computers & Mathematics with Applications
Limit problems for interpolation by analytic radial basis functions
Journal of Computational and Applied Mathematics
On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere
Journal of Computational Physics
Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform
Journal of Computational and Applied Mathematics
Improved radial basis function methods for multi-dimensional option pricing
Journal of Computational and Applied Mathematics
The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems
Journal of Computational Physics
Adaptive radial basis function methods for time dependent partial differential equations
Applied Numerical Mathematics
Computers & Mathematics with Applications
A new class of oscillatory radial basis functions
Computers & Mathematics with Applications
Exact polynomial reproduction for oscillatory radial basis functions on infinite lattices
Computers & Mathematics with Applications
Integrated multiquadric radial basis function approximation methods
Computers & Mathematics with Applications
The uselessness of the Fast Gauss Transform for summing Gaussian radial basis function series
Journal of Computational Physics
Application of the RBF meshless method to the solution of the radiative transport equation
Journal of Computational Physics
A high order multivariate approximation scheme for scattered data sets
Journal of Computational Physics
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
Compact local integrated-RBF approximations for second-order elliptic differential problems
Journal of Computational Physics
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
Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode
SIAM Journal on Scientific Computing
CAD and mesh repair with Radial Basis Functions
Journal of Computational Physics
Journal of Computational Physics
Multivariate interpolation with increasingly flat radial basis functions of finite smoothness
Advances in Computational Mathematics
Journal of Computational Physics
Stable Evaluation of Gaussian Radial Basis Function Interpolants
SIAM Journal on Scientific Computing
Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs
Journal of Computational Physics
A compact five-point stencil based on integrated RBFs for 2D second-order differential problems
Journal of Computational Physics
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
Computers & Mathematics with Applications
Full length article: On collocation matrices for interpolation and approximation
Journal of Approximation Theory
Journal of Computational Physics
A meshless interpolation algorithm using a cell-based searching procedure
Computers & Mathematics with Applications
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Abstract-Spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries. Since radial basis function (RBF) approxima-ions permit this even for multivariate scattered data, there has been much recent interest in practical algorithms to compute these approximations effectively. Several types of RBFs feature a free parameter (e.g., c in the multiquadric (MQ) case @f(r) = @/r^2 + c^2). The limit of c - ~ (increasingly flat basis functions) has not received much attention because it leads to a severely ill-conditioned problem. We present here an algorithm which avoids this difficulty, and which allows numerically stable computations of MQ RBF interpolants for all parameter values. We then find that the accuracy of the resulting approximations, in some cases, becomes orders of magnitude higher than was the case within the previously available parameter range. Our new method provides the first tool for the numerical exploration of MQ RBF interpolants in the limit of c - ~. The method is in no way specific to MQ basis functions and can-without any change-be applied to many other cases as well.