Information-based complexity
Matrix computations (3rd ed.)
Scattered data fitting on the sphere
Proceedings of the international conference on Mathematical methods for curves and surfaces II Lillehammer, 1997
Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods
SIAM Journal on Scientific Computing
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
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
Convergence of Unsymmetric Kernel-Based Meshless Collocation Methods
SIAM Journal on Numerical Analysis
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Support Vector Machines
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
Computers & Mathematics with Applications
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
Stable Computations with Gaussian Radial Basis Functions
SIAM Journal on Scientific Computing
On Dimension-independent Rates of Convergence for Function Approximation with Gaussian Kernels
SIAM Journal on Numerical Analysis
On Dimension-independent Rates of Convergence for Function Approximation with Gaussian Kernels
SIAM Journal on Numerical Analysis
Stable calculation of Gaussian-based RBF-FD stencils
Computers & Mathematics with Applications
A new stable basis for radial basis function interpolation
Journal of Computational and Applied Mathematics
A meshless interpolation algorithm using a cell-based searching procedure
Computers & Mathematics with Applications
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We provide a new way to compute and evaluate Gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter, i.e., for “flat” kernels. This work is motivated by the fundamental ideas proposed earlier by Bengt Fornberg and his coworkers. However, following Mercer's theorem, an $L_2(\mathbb{R}^d, \rho)$-orthonormal expansion of the Gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by Fornberg, Larsson, and Flyer and that is applicable in arbitrary space dimensions $d$. In addition to obtaining an accurate approximation of the radial basis function interpolant (using many terms in the series expansion of the kernel), we also propose and investigate a highly accurate least-squares approximation based on early truncation of the kernel expansion.