Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation
Journal of Approximation Theory
Conversion of a power to a series of Chebyshev polynomials
Communications of the ACM
Scattered node compact finite difference-type formulas generated from radial basis functions
Journal of Computational Physics
The Runge phenomenon and spatially variable shape parameters in RBF interpolation
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
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Local radial basis function based gridfree scheme for unsteady incompressible viscous flows
Journal of Computational Physics
Computers & Mathematics with Applications
Stable computation of multiquadric interpolants for all values of the shape parameter
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
Meshfree explicit local radial basis function collocation method for diffusion problems
Computers & Mathematics with Applications
Error saturation in Gaussian radial basis functions on a finite interval
Journal of Computational and Applied Mathematics
Stabilization of RBF-generated finite difference methods for convective PDEs
Journal of Computational Physics
Preconditioning for radial basis functions with domain decomposition methods
Mathematical and Computer Modelling: An International Journal
Stabilization of RBF-generated finite difference methods for convective PDEs
Journal of Computational Physics
Computers & Mathematics with Applications
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
International Journal of Numerical Modelling: Electronic Networks, Devices and Fields
Vector field approximation using radial basis functions
Journal of Computational and Applied Mathematics
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
A new stable basis for radial basis function interpolation
Journal of Computational and Applied Mathematics
Full length article: On collocation matrices for interpolation and approximation
Journal of Approximation Theory
A meshless interpolation algorithm using a cell-based searching procedure
Computers & Mathematics with Applications
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Radial basis function (RBF) approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh-free and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist: the Contour-Padé method and the RBF-QR method. However, the former is limited to small node sets, and the latter has until now been formulated only for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in one, two, and three dimensions. The algorithm is stable for arbitrarily small shape parameters. It can be used for thousands of node points in two dimensions and still more in three dimensions. A sample MATLAB code for the two-dimensional case is provided.