Support vector machine adapted Tikhonov regularization method to solve Dirichlet problem

Numerical solutions of partial differential equations are traditional topics that have been studied by many researchers. During the last decade, support vector machine (SVM) has been widely used for approximation problems. The contribution of this paper is two folds. One is to combine the reproducing kernel-SVM method with the Tikhonov regularization method, called the SVM-Tik methods, in which the kernels KλKλ and Kλσ (see below) are newly developed. In the paper they are respectively phrased as the SVM-Tik-KλKλ and SVM-Tik-Kλσ methods. The second contribution is to use the two models, SVM-Tik-KλKλ and SVM-Tik-Kλσ, to solve the Dirichlet problem. The methods are meshless. They produce sparse representations in the linear combination form of specific functions (the KλKλ and Kλσ kernels). The generalization bound result in learning theory is used to give an estimation of the approximation errors. With the illustrative examples the sparseness and robustness properties, as well as the effectiveness of the methods are presented. The proposed methods are compared with currently the most commonly used finite difference method (FDM) showing promising results.