数学物理学报(英文版) ›› 2015, Vol. 35 ›› Issue (6): 1339-1348.doi: 10.1016/S0252-9602(15)30058-8

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A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION

杨帆1,2, 傅初黎2, 李晓晓1   

  1. 1. School of Science, Lanzhou University of Technology, Lanzhou 730050, China;
    2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
  • 收稿日期:2015-01-20 修回日期:2014-08-26 出版日期:2015-11-01 发布日期:2015-11-01
  • 通讯作者: Fan YANG, E-mail: yfggd114@163.com E-mail:yfggd114@163.com
  • 作者简介:Chuli FU, E-mail: fuchuli@lzu.edu.cn;Xiaoxiao LI, E-mail: lixiaoxiaogood@126.com
  • 基金资助:

    The project is supported by the National Natural Science Foundation of China (11171136, 11261032), the Distinguished Young Scholars Fund of Lan Zhou University of Technology (Q201015), the basic scientific research business expenses of Gansu province college and the Natural Science Foundation of Gansu province (1310RJYA021).

A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION

Fan YANG1,2, Chuli FU2, Xiaoxiao LI1   

  1. 1. School of Science, Lanzhou University of Technology, Lanzhou 730050, China;
    2. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
  • Received:2015-01-20 Revised:2014-08-26 Online:2015-11-01 Published:2015-11-01
  • Contact: Fan YANG, E-mail: yfggd114@163.com E-mail:yfggd114@163.com
  • Supported by:

    The project is supported by the National Natural Science Foundation of China (11171136, 11261032), the Distinguished Young Scholars Fund of Lan Zhou University of Technology (Q201015), the basic scientific research business expenses of Gansu province college and the Natural Science Foundation of Gansu province (1310RJYA021).

摘要:

In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.

关键词: Cauchy problem for Laplace equation, ill-posed problem, a posteriori parameter choice, error estimate

Abstract:

In this paper, we consider the Cauchy problem for the Laplace equation, which is severely ill-posed in the sense that the solution does not depend continuously on the data. A modified Tikhonov regularization method is proposed to solve this problem. An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained. Moreover, an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained. Numerical examples illustrate the validity and effectiveness of this method.

Key words: Cauchy problem for Laplace equation, ill-posed problem, a posteriori parameter choice, error estimate

中图分类号: 

  • 35R25