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Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

机译：带奇异点的拉普拉斯方程的柯西问题的最优正则化基本解法

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摘要

The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of the MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of the MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises. (C) 2008 Elsevier Inc. All rights reserved.
机译：本研究的目的是为拉普拉斯方程的柯西问题提出一种高精度，快速的数值方法。我们的问题通过基本解决方案（MFS）直接离散化。对于给定的高噪声柯西数据，Tikhonov正则化方法可以稳定该问题的数值解。数值解的精度取决于Tikhonov正则化技术的正则化参数和MFS的某些参数。 L曲线确定合适的正则化参数以获得准确的解。数值实验表明，这种合适的正则化参数与最优参数吻合。而且，在数值上观察到了对MFS参数的更好选择。值得注意的是，其解决方案具有奇异点的问题可以成功解决。结论是，本文提出的数值方法对于具有不规则域，奇异点和高噪声柯西数据的问题是有效的。 （C）2008 Elsevier Inc.保留所有权利。

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