2011/01/01 by Bengt Fornberg, Elisabeth Larsson, Natasha Flyer · 421 citations
Engineering · Physics and Astronomy · Mathematics · #Soil, Finite Element Methods #Electromagnetic Scattering and Analysis #Advanced Numerical Analysis Techniques #Radial basis function #Mathematics #Interpolation (computer graphics) #Computation #Algorithm #Node (physics) #Basis function #Gaussian #Hierarchical RBF #Applied mathematics #Mathematical analysis #Computer science #Artificial intelligence #Artificial neural network
paper · doi:10.1137/09076756x
published in SIAM Journal on Scientific Computing 33(2), 869-892 (Society for Industrial and Applied Mathematics)
openalex publication_date 2011/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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-Padé 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.