Stable Computations with Gaussian Radial Basis Functions

Abstract. Radial basis function (RBF) approximation is an extremely powerful tool for representing smooth functions in non-trivial geometries, since the method is meshfree and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly illconditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist, the Contour-Pad´e method and the RBF-QR method. However, the former is limited to small node sets and the latter has until now only been formulated for the surface of the sphere. This paper contains an RBF-QR formulation for planar two-dimensional problems. The algorithm is perfectly stable for arbitrarily small shape parameters and can be used for up to a thousand node points in double precision and for several thousand node points in quad precision. A sample MATLAB code is provided. Key words. RBF, radial basis function, ill-conditioning, shape parameter, stab...

• Type: Journal
• Year: 2011
• Where: SIAMSC
• Authors: Bengt Fornberg, Elisabeth Larsson, Natasha Flyer
• Keywords: Radial Basis Function, RBF Shape Parameter, Shape Parameter, SIAMSC 2011