Shape parameter estimation in RBF function approximation

The radial basis function (RBF) collocation method is applied for the approximation of functions in two variables. When the RBFs employed include a shape parameter, the determination of an appropriate value for it is a major issue. In this work, this is addressed by including the value of the shape parameter in the unknowns along with the coefficients of the RBFs in the approximation. The variable shape parameter case when a different shape parameter is associated with each RBF in the approximation is also considered. Both approaches yield nonlinear systems of equations, which are solved by a standard non-linear solver. The results of several numerical experiments are presented.