Stable Computations with Gaussian Radial Basis Functions

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...

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Vydáno v:SIAM journal on scientific computing Ročník 33; číslo 2; s. 869 - 892
Hlavní autoři: Fornberg, Bengt, Larsson, Elisabeth, Flyer, Natasha
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Témata:
Algebra
Algorithms
Approximation
Approximations and expansions
Computation
Dimensional analysis
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Matlab
Methods of scientific computing (including symbolic computation, algebraic computation)
Normal distribution
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Obstacles
Sciences and techniques of general use
Studies
Three dimensional
Two dimensional
Three-dimensional calculations
ill-conditioning
shape parameter
Numerical linear algebra
Algorithm
Surface
Numerical approximation
Geometry
Sphere
Interpolation
radial basis function
Numerical analysis
Scientific computation
stable
Smooth function
Condition number
65D05
Two-dimensional calculations
65D15
65F35
Mesh method
ISSN:1064-8275, 1095-7197, 1095-7197
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Popis
Shrnutí: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-Pade 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. [PUBLICATION ABSTRACT]
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ISSN:1064-8275
1095-7197
1095-7197
DOI:10.1137/09076756X