Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm

An Investigation of Radial Basis Function Method for Strain Reconstruction by Energy-Resolved Neutron Imaging

“…For the precise implementation of the framework, the shape parameter evaluation is important [34]. Therefore, careful attention should be given for determining shape parameter value.…”

Certain Relations Between the Main Matrix Condition Number and Multiquadric Shape Parameter in the Non-Symmetric Kansa Method

“…Sarra [10] applied the extended precision floating-point arithmetic to improve the accuracy of RBF methods efficiently. Among other methods of determining the good value of the RBF shape parameter are the method based on the convergence analysis [11], the strategy with trigonometric, exponential and random variable shape parameter [12], the local optimization algorithm [13], the extended Rippa algorithm [14], the global genetic algorithm optimization method for the single [15] and variable [16] shape parameter value approach, the application of the principle of a minimum of the total potential energy [17] and the particle swarm optimization method [18]. As was shown above, the stability of the RBF method is often related to the ill-conditioning of the main matrix [19].…”

Determining approximate solution of partial differential equations using radial basis function model trained with parallel symbiotic organisms search algorithm

“…The key contribution of this article is as follows: To determine the approximate solution of PDEs a RBF based model is proposed. A parallel symbiotic organism search (PSOS) algorithm is proposed which is accurate and having lower computational complexity than original SOS 20 and another popular parallel algorithm based on social spider optimization 32 The proposed PSOS algorithm is used to minimize the root mean square (RMS) error between exact and approximate solution of PDEs by suitably adjusting the center of RBF model. The performance of the proposed RBF‐PSOS model is used to determine approximate solution of five real life PDEs: 2D modified Helmholtz equation, 35 Poisson problem with Dirichlet boundary conditions, 17 PDE elliptic equations, 17 convection‐diffusion equation (Example 1), 17,18 convection‐diffusion equation (Example 2) 18 Comparative performance evaluation of the proposed model is carried out with the same RBF model trained with PSSO, 32 original SOS algorithm, 20 real coded genetic algorithm (RGA), 17 and PSO 35 .…”

“…The proposed PSOS algorithm is used to minimize the root mean square (RMS) error between exact and approximate solution of PDEs by suitably adjusting the center of RBF model. The performance of the proposed RBF‐PSOS model is used to determine approximate solution of five real life PDEs: 2D modified Helmholtz equation, 35 Poisson problem with Dirichlet boundary conditions, 17 PDE elliptic equations, 17 convection‐diffusion equation (Example 1), 17,18 convection‐diffusion equation (Example 2) 18 Comparative performance evaluation of the proposed model is carried out with the same RBF model trained with PSSO, 32 original SOS algorithm, 20 real coded genetic algorithm (RGA), 17 and PSO 35 . Superior performance is reported in the form of lower RMS error values, visualization of surface of RMS error plots, and lower run time. …”