Fast Evaluation of Radial Basis Functions: A Multivariate Momentary Evaluation Scheme

Abstract : This paper presents a scheme for fast evaluation of splines, or 'radial' basis functions, of the form s(.) = p(.) + summation from i = 1 to i = N of function (lambda(i)Phi(. - zi)). Here p is a low degree polynomial and Phi from R(exp n) to R is a function that need not be radial, This multivariate momentary evaluation scheme is a generalization of the fast multipole method in which calculations with far field expansions are replaced by calculations involving moments of the data. The primary advantage of this new algorithm is that it is highly adaptive to changes in Phi.