Stable computations with Gaussian radial basis functions

AMS Citation:
Fornberg, B., E. Larrson, and N. Flyer, 2011: Stable computations with Gaussian radial basis functions. Siam Journal on Scientific Computing, 33, 869-892,.
Resource Type:article
Title:Stable computations with Gaussian radial basis functions
Abstract: Radial basis function (RBF) approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh-free and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist: the Contour-PadĂŠ method and the RBF-QR method. However, the former is limited to small node sets, and the latter has until now been formulated only for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in one, two, and three dimensions. The algorithm is stable for arbitrarily small shape parameters. It can be used for thousands of node points in two dimensions and still more in three dimensions. A sample MATLAB code for the two-dimensional case is provided.
Subject(s):Gaussian, radial basis functions, ill-conditioning, shape parameter, stable
Peer Review:Refereed
Copyright Information:Copyright 2011 Society for Industrial and Applied Mathematics.
OpenSky citable URL: ark:/85065/d7zp47fz
  • B. Fornberg
  • E. Larrson
  • Natasha Flyer - NCAR/UCAR
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