Regularized meshless method for solving the Cauchy problem

Abstract

In this paper, the Laplace problem with overspecified boundary conditions is investigated by using the regularized meshless method. The solution is represented by a distribution of the kernel functions of double-layer potentials. By using the desingularization technique of adding-back and subtracting terms to regularize the singularity and hypersingularity of the kernel functions, the source points can be located on the real boundary and the diagonal terms of influence matrices are determined. The main difficulty of the coincidence of the source and collocation points then disappears. The accompanied ill-posed problem can be remedied by using Tikhonov regularization technique, linear regularization method and truncated singular value decomposition. The optimal parameters of the Tikhonov technique and linear regularization method and truncated singular value decomposition are derived by adopting L-curve concept. The numerical evidences of the regularized meshless method are given to verify the accuracy of the solutions after comparing with the results of analytical solution. The comparison of Tikhonov regularization technique, linear regularization method and truncated singular value decomposition are also discussed in the example.本文是利用正規化無網格法求解過定邊界之拉普拉斯問題，使用雙層勢能來表示整個場解，且使用一加一減技巧來正規化處理奇異及超奇異核函數。使用提出的數值方法有別於傳統基本解法須將源點佈在虛假邊界上，可將奇異源放在真實的邊界上，並可獲得線性代數方程。配合邊界條件，即可輕易的決定出線性代數系統的未知係數。然而伴隨著的病態問題可藉由截取式奇異值分解法、Tikhonov 技術及線性正規化法來克服，在最佳化參數方面，則可用 L 曲線的觀念 來得到。所得之數值結果在與解析解作比較後可獲得滿意的結果，並對其三種克服病態問題之方法加以比較討論。