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
The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems
Journal of Computational Physics
Three ways to solve the Poisson equation on a sphere with Gaussian forcing
Journal of Computational Physics
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
Error saturation in Gaussian radial basis functions on a finite interval
Journal of Computational and Applied Mathematics
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
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
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
Vector field approximation using radial basis functions
Journal of Computational and Applied Mathematics
A study of different modeling choices for simulating platelets within the immersed boundary method
Applied Numerical 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
Journal of Computational Physics
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When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Runge-type oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstable algorithm for a stable problem in the case of near-flat basis functions, there will typically also be a separate point at which disastrous ill-conditioning enters. We introduce here a new method, RBF-QR, which entirely eliminates such ill-conditioning, and we apply it in the special case when the data points are distributed over the surface of a sphere. This algorithm works even for thousands of node points, and it allows the RBF shape parameter to be optimized without the limitations imposed by stability concerns. Since interpolation in the flat RBF limit on a sphere is found to coincide with spherical harmonics interpolation, new insights are gained as to why the RBF approach (with nonflat basis functions) often is the more accurate of the two methods.