The golden section search algorithm for finding a good shape parameter for meshless collocation methods

In this paper we propose to apply the golden section search algorithm to determining a good shape parameter of multiquadrics (MQ) for the solution of partial differential equations. We use two radial basis function based meshless collocation methods, the method of approximate particular solutions (MAPS) and Kansaʹs method, to solve partial differential equations. Due to the severely ill-conditioned matrix system using MQ, we also consider the truncated singular value decomposition method (TSVD) to regularize the smoothness of the error versus shape parameter curve so that a reasonably good shape parameter can be identified. We also analyze cost and accuracy for using LU decomposition and TSVD. Numerical results show that the proposed golden section search method is effective and provides a reasonable shape parameter along with acceptable accuracy of the solution.