Rapid evaluation of radial basis functions

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(M N) 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

Fast evaluation of radial basis functions : moment based methods

In this paper we introduce a new algorithm for fast evaluation of univariate radial basis functions of the form s(x) = Σᶰn₌₁ dn(⃒x - xn⃒) to within accuracy . The algorithm has a setup cost of (N⃒log⃒log⃒log⃒) operations and an incremental cost per evaluation of s(x) of (⃒log⃒) operations. It is based on a hierarchical subdivision of the unit interval, the adaptive construction of a corresponding hierarchy of polynomial approximations, and the fast accumulation of moments. It can be applied in any case where the basic function smooth on (0, 1], and on any grid of centres ｛Xn｝. The algorithm does not require that be analytic at infinity, nor that the user specify new polynomial approximations or modify the data structures for each new , nor that the points Xn form any sort of regular array. Furthermore the algorithm can be extended to problems in higher dimensions