Making conditionally negative definite radial basis function interpolation well-conditioned by adding cardinal basis functions

Abstract

A class of basis functions so called well-conditioned RBF (WRBFs) has been introduced. This basis has been manipulated by adding cardinal functions to the conditionally negative definite RBFs of order 1, such as Multiquadric functions 1+(∊r)2 (MQ) and log(1+(∊r)2) (LOG). The condition number of the interpolation matrix arising from this basis is of O(N), where N is the number of center nodes. This order is independent of shape parameter and therefore applying this basis functions would recover the ill–posed linear system associated with the order 1 conditionally negative definite RBFs interpolation. Keywords: Radial basis functions, Cardinal functions, Multiquadrics (MQ), AMS subject classifications: 65M20, 65M06