Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions

Authors:
E. Larsson;B. Fornberg
Affiliations:
-;-
Venue:
Computers & Mathematics with Applications
Year:
2005

Citing 8
Cited 19

Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation

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Solving partial differential equations by collocation using radial basis functions

Applied Mathematics and Computation
Improved error bounds for scattered data interpolation by radial basis functions

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Local accuracy for radial basis function interpolation on finite uniform grids

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Fast Solution of the Radial Basis Function Interpolation Equations: Domain Decomposition Methods

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Fast Evaluation of Radial Basis Functions: Methods for Generalized Multiquadrics in $\RR^\protectn$

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Stable computation of multiquadric interpolants for all values of the shape parameter

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A new class of oscillatory radial basis functions

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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

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Limit problems for interpolation by analytic radial basis functions

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On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere

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Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform

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Improved radial basis function methods for multi-dimensional option pricing

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Design of embedded differential equation solver

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A new class of oscillatory radial basis functions

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Integrated multiquadric radial basis function approximation methods

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The role of the multiquadric shape parameters in solving elliptic partial differential equations

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Application of the RBF meshless method to the solution of the radiative transport equation

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RBF-FD formulas and convergence properties

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An alternative procedure for selecting a good value for the parameter c in RBF-interpolation

Advances in Computational Mathematics
Optimal constant shape parameter for multiquadric based RBF-FD method

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Stable Computations with Gaussian Radial Basis Functions

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Multivariate interpolation with increasingly flat radial basis functions of finite smoothness

Advances in Computational Mathematics
Stable Evaluation of Gaussian Radial Basis Function Interpolants

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Stable calculation of Gaussian-based RBF-FD stencils

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Full length article: On collocation matrices for interpolation and approximation

Journal of Approximation Theory

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Abstract

Multivariate interpolation of smooth data using smooth radial basis functions is considered. The behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically. Explicit criteria for different types of limits are given. Using the results for the limits, the dependence of the error on the shape parameter of the radial basis function is investigated. The mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error.