An interval for the shape parameter in radial basis function approximation

Abstract

The radial basis functions (RBFs) depend on an auxiliary parameter, called the shape parameter. Great theoretical and numerical efforts have been made to find the relationship between the accuracy of the RBF-approximations and the value of the shape parameter. In many cases, the numerical approaches are based on minimization of an estimation of the error function, such as Rippa's approach and its modifications. These approaches determine a value for the shape parameter. In this paper, we propose a practical approach to determine an interval, instead of a value, without any minimization and estimation of error function. The idea is based on adding a loop on the shape parameter, but not to minimize an error norm. Suitable values of the shape parameter are determined by take into account the practical convergence behavior of the problem. The proposed method is applied to some illustrative examples, including one and two-dimensional interpolation and partial differential equations. The results of the method are compared with those of some other approaches to confirm the reliability of the method. Furthermore, numerical stability of the method of line respect to the shape parameter is considered for time-dependent PDEs. The results show that, the proposed method can be used as an independent method to find a shape parameter, and also as an alternative for investigating the validity of the values of the other methods.