An order optimal regularization method for the Cauchy problem of a Laplace equation in an annulus domain

Dongdong He; Kejia Pan

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An order optimal regularization method for the Cauchy problem of a Laplace equation in an annulus domain

机译：环域中Laplace方程Cauchy问题的阶最优正则化方法。

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摘要

The detection of defects due to corrosion in oil pipeline industry is very important. Detecting corrosion by electrical field can be modeled as a Cauchy problem for a Laplace equation in an annulus domain, which is well known to be severely ill-posed. In this paper, a modified Tikhonov regularization method is proposed. And a Hoelder-type error estimate is achieved, which is order optimal according to the general regularization theory. Moreover, a Fast Fourier Transform (FFT) is used in the numerical implementation. Finally, numerical results show that the regularization method and FFT work well.

机译：在石油管道行业中，检测由于腐蚀引起的缺陷非常重要。可以将通过电场检测腐蚀建模为环域中Laplace方程的柯西问题，该问题众所周知是病态严重。本文提出了一种改进的Tikhonov正则化方法。并获得了Hoelder型误差估计，根据一般正则化理论该阶次最优。此外，在数字实现中使用了快速傅立叶变换（FFT）。最后，数值结果表明该正则化方法和FFT效果良好。

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来源
《Applied Mathematical Modelling》 |2015年第11期|3063-3074|共12页
作者
Dongdong He; Kejia Pan;
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作者单位

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China;

School of Mathematics and Statistics, Central South University, Changsha 410083, China;

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收录信息美国《科学引文索引》(SCI);美国《工程索引》(EI);
原文格式 PDF
正文语种 eng
中图分类
关键词
Tikhonov regularization; Annulus domain; Ill-posed; Laplace equation; Cauchy problem;

机译：Tikhonov正则化;环域;病态拉普拉斯方程柯西问题;

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专利

1. A regularization method for a Cauchy problem of Laplace's equation in an annular domain [J] . T.Wei, Y.G.Chen Mathematics and computers in simulation . 2012,第11期

机译：环形域中拉普拉斯方程Cauchy问题的正则化方法
2. Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points [J] . Shigeta T, Young DL Journal of Computational Physics . 2009,第6期

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
3. Mapped Regularization Methods for the Cauchy Problem of the Helmholtz and Laplace Equations [J] . Shokri Kaveh Hojjatollah, Adibi Hojjatollah Iranian journal of science and technology . 2021,第2期

机译：掌上射击正规化方法，用于亥姆霍兹和拉普拉斯方程的Cauchy问题
4. Modified regularization method for a Cauchy problem of the Laplace equation [C] . Jianfeng Mao, Guangfu Wang 2011 International Conference on Information Science and Technology . 2011

机译：Laplace方程Cauchy问题的修正正则化方法
5. Holder estimates for the solutions of the Cauchy-Riemann equations on convex domains of finite type. [D] . Solowiej, Jaroslaw Janusz. 1999

机译：有限类型凸域上Cauchy-Riemann方程解的Holder估计。
6. Approximate Analytical Solutions of the Regularized Long Wave Equation Using the Optimal Homotopy Perturbation Method [O] . Constantin Bota, Bogdan Căruntu -1

机译：使用最优同伦摄动法的正则化长波方程的近似解析解
7. Method of Fundamental Solutions with Optimal Regularization Techniques for the Cauchy Problem of the Laplace Equation with Singular Points [O] . Shigeta, Takemi, Young, D. L. 2008

机译：具有最优正则化技术的基本解法奇点拉普拉斯方程的Cauchy问题

An order optimal regularization method for the Cauchy problem of a Laplace equation in an annulus domain

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