Interpolation of scattered Data by radial Functions

Abstract The problem of interpolation of scattered data by radial functions is considered. Several families of radial functions including the “surface splines” and the “Hardy multiquadrics” are discussed. Results by Micchelli, which allow us to determine the wellposedness of the interpolation problem in any IRd, based on a given radial function, are reviewed. The optimality of the interpolant and the identification of the corresponding variational principle, based on the work of Madych and Nelson, are presented. The variational principle is then used to devise a method for preconditioning the linear system determining the radial interpolant, which is known to be ill-conditioned for large data sets. This approach extends the method of preconditioning for “surface splines” developed by Dyn, Levin, and Rippa, to a wide class of radial functions.

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