Eigenvalues of Euclidean distance matrices
Journal of Approximation Theory
Journal of Approximation Theory
Norm estimates for inverses of Toeplitz distance matrices
Journal of Approximation Theory
Lower bounds for norms of inverses of interpolation matrices for radial basis functions
Journal of Approximation Theory
On shifted cardinal interpolation by Gaussians and multiquadrics
Journal of Approximation Theory
Applied Mathematics and Computation
Applied numerical linear algebra
Applied numerical linear algebra
Solving partial differential equations by collocation using radial basis functions
Applied Mathematics and Computation
Meshless Galerkin methods using radial basis functions
Mathematics of Computation
Compactly supported radial basis functions for shallow water equations
Applied Mathematics and Computation
Radial Basis Functions
Multiresolution Methods in Scattered Data Modelling
Multiresolution Methods in Scattered Data Modelling
Numerical methods for high dimensional Hamilton-Jacobi equations using 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
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
Computing eigenmodes ofelliptic operators using radial basis functions
Computers & Mathematics with Applications
Global field interpolation for particle methods
Journal of Computational Physics
On the optimal Halton sequence
Mathematics and Computers in Simulation
Applied Numerical Mathematics
The uselessness of the Fast Gauss Transform for summing Gaussian radial basis function series
Journal of Computational Physics
Improved Scaling for Quantum Monte Carlo on Insulators
SIAM Journal on Scientific Computing
Computers & Mathematics with Applications
Computers & Mathematics with Applications
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Through numerical experiments, we examine the condition numbers of the interpolation matrix for many species of radial basis functions (RBFs), mostly on uniform grids. For most RBF species that give infinite order accuracy when interpolating smooth f(x)-Gaussians, sech's and Inverse Quadratics-the condition number @k(@a,N) rapidly asymptotes to a limit @k"a"s"y"m"p(@a) that is independent of N and depends only on @a, the inverse width relative to the grid spacing. Multiquadrics are an exception in that the condition number for fixed @a grows as N^2. For all four, there is growth proportional to an exponential of 1/@a (1/@a^2 for Gaussians). For splines and thin-plate splines, which contain no width parameter, the condition numbers grows asymptotically as a power of N-a large power as the order of the RBF increases. Random grids typically increase the condition number (for fixed RBF width) by orders of magnitude. The quasi-random, low discrepancy Halton grid may, however, have a lower condition number than a uniform grid of the same size.