RBF collocation path-following approach: optimal choice for shape parameter based on genetic algorithm

Abstract

This paper presents a new method to solve a challenging problem and a topic of current research namely the selection of optimal shape parameters for the Radial Basis Function (RBF) collocation methods in both interpolation and nonlinear Partial Differential Equations (PDEs) problems. To this intent, a compromise must be made to achieve the conflict between accuracy and stability referred to as the trade-off or uncertainty principle. The use of genetic algorithm and path-following continuation allows us on the one hand to avoid the local optimum issue associated with RBF interpolation matrices, which are inherently ill-conditioned and on the other side, to map the original optimization problem of defining a shape parameter into a root-finding problem. Our computational experiments applied on nonlinear problems in structural calculations using our proposed adaptive algorithm based on genetic optimization with automatic selection of the shape parameter can yield more accuracy and a good precision compared to the same state of the art algorithm from literature with a fixed and given shape parameter and Finite Element Method (FEM).