A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation

In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics -root c(2)+parallel to x parallel to(2) when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments demonstrate that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c.

Automated summary

In this paper, we propose a method of directly choosing the shape parameter c in the multiquadrics using MN-curve theory for the RBF collocation method. Experiments show that the quality of the obtained c value is very close to the best approximation error among all possible choices.