Rapid evaluation of radial basis functions

Authors:
George Roussos;Brad J. C. Baxter
Affiliations:
School of Computer Science and Information Systems, Birkbeck College, University of London, Malet Street, London WC1E 7HX, UK;School of Economics, Mathematics and Statistics, Birkbeck College, University of London, Malet Street, London WC1E 7HX, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2005

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Abstract

Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example, the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(MN) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating-point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail.