Applied numerical linear algebra
Applied numerical linear algebra
A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds
Journal of Computational Physics
Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation
SIAM Journal on Numerical Analysis
Computers & Mathematics with Applications
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
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Error saturation in Gaussian radial basis functions on a finite interval
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
Preconditioning for radial basis functions with domain decomposition methods
Mathematical and Computer Modelling: An International Journal
Error saturation in Gaussian radial basis functions on a finite interval
Journal of Computational and Applied Mathematics
Vector field approximation using radial basis functions
Journal of Computational and Applied Mathematics
Computers & Mathematics with Applications
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Radial basis function (RBF) interpolation is a ''meshless'' strategy with great promise for adaptive approximation. Because it is meshless, there is no canonical grid to act as a starting point for function-adaptive grid modification. Uniform grids are therefore common with RBFs. Like polynomial interpolation, which is the limit of RBF interpolation when the RBF functions are very wide (the ''flat limit''), RBF interpolants are vulnerable to the Runge Phenomenon, whether the grid is uniform or irregular: the N-point interpolant may diverge as N-~ on the interval spanned by the interpolation points even for a function f(x) that is analytic everywhere on and near this interval. In this work, we discuss six strategies for defeating the Runge Phenomenon, specializing to a uniform grid on a finite interval and one particular species of RBFs in one space dimension: @f(x)=exp(-[@a^2/h^2]x^2) where h is the grid spacing. Three strategies fail, but three are successful. One good strategy is to vary @a with the number of interpolation points N as N^-^1^/^4. Unfortunately this gives both a subgeometric rate of convergence for the approximation (error falling as exp(-qN) for some constant q) and a matrix condition number that grows as exp(pN) for some p0. (In order to explain why fixed@a, independent of N, is not a convergent strategy, we digress to discuss error saturation, and show experimentally that the saturation error on the finite interval is about 0.6exp(-0.47/@a^2)@?f@?"~; if a user-chosen error tolerance @d is acceptable, then the optimum choice is @a"o"p"t"i"m"u"m(@d)=1/-2log(@d/0.06).) The second good strategy is RBF Extension, which uses RBF functions with centers on a interval slightly larger than the target interval to approximate f(x). A third strategy is to split the interval into two ''boundary layers'' and a large middle interval and apply separate RBF approximations on each. The slow-decrease-of-@a strategy is much cheaper, but RBF Extension is much more resistant to ill-conditioning and therefore can achieve much lower errors. The three-interval method is the least accurate, but is robust and inexpensive.