Scattered data interpolation: Strictly positive definite radial basis/cardinal functions

Abstract

Radial basis functions are a simple and accurate method for multivariate interpolation but the ill-conditioning situation due to their interpolation matrices, discourages an acceptable approximation for both large number of nodes or flat function interpolation. In current work, a new type of basis named well-conditioned RBFs (WRBFs) was created by adding the strictly positive definite Radial Basis Functions (SPD-RBFs) to cardinal functions, was introduced and applied for interpolation. To light up this manner, two classes of global cardinal functions were used for adding by SPD-RBFs. These cardinal functions are Shepard functions and Quasi-cardinal RBFs. Theoretical and numerical analyses prove that utilizing WRBFs has some advantages such as eliminating the ill-conditioning system which has arisen from the global positive definite RBFs interpolation, improving the convergence of pure cardinal functions interpolation and also working better than pure RBFs for small shape parameters.