Numerical Experiments on Optimal Shape Parameters for Radial Basis Functions

A numerical investigation on a technique for choosing an optimal shape parameter is proposed. Radial basis functions (RBFs) and their derivatives are used as interpolants in the asymmetric collocation radial basis method, for solving systems of partial differential equations. The shape parameter c in RBFs plays a major role in obtaining high quality solutions for boundary value problems. As c is a user defined value, inexperienced users may compromise the quality of the solution, often a problem of this meshless method. Here we propose a statistical technique to choose the shape parameter in radial basis functions. We use a cross-validation technique suggested by Rippa [Adv Comput Math 11 (1999), 193-210] for interpolation problems to find a cost function Cost (cl) that ideally has the same behavior as an error function. If that is the case, the parameter c that minimizes the cost function will be an optimal shape parameter, in the sense that it minimizes the error function. The form of the cost and error functions are analized for several examples. and for most cases the two functions have a similar behavior. The technique produced very accurate results, even with a small number of points and irregular grids. (C) 2009 Wiley Periodicals. Inc. Numer Methods Partial Differential Eq 26: 675-689, 2010