Determination of shape parameter in RBF approximation

Abstract

We apply a radial basis function (RBF) collocation method for the approximation of functions in two dimensions. The solution is approximated by a linear combination of radial basis functions. The issue of determining the optimal value of the shape parameter is tackled by including it in the unknowns along with the coefficients of the RBFs in the approximation. The resulting nonlinear system of equations is solved by directly applying a standard non-linear solver. The results of some numerical experiments are presented and analysed.