scholarly journals The Tikhonov Regularization Method in Hilbert Scales for Determining the Unknown Source for the Modified Helmholtz Equation

2016 ◽  
Vol 04 (01) ◽  
pp. 140-148 ◽  
Author(s):  
Lei You ◽  
Zhi Li ◽  
Juang Huang ◽  
Aihua Du
Keyword(s):  
Helmholtz Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Tikhonov Regularization Method ◽  
Modified Helmholtz Equation ◽  
Unknown Source

scholarly journals The Simplified Tikhonov Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Fan Yang ◽  
HengZhen Guo ◽  
XiaoXiao Li
Keyword(s):  
Exact Solution ◽  
Helmholtz Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Numerical Results ◽  
Tikhonov Regularization Method ◽  
Convergence Estimate ◽  
Modified Helmholtz Equation ◽  
Unknown Source ◽  
Ill Posed

This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the simplified Tikhonov regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

scholarly journals Landweber iterative regularization method for reconstructing the unknown source of the modified Helmholtz equation

2021 ◽  
Vol 6 (9) ◽  
pp. 10327-10342
Author(s):  
Dun-Gang Li ◽  
◽  
Fan Yang ◽  
Ping Fan ◽  
Xiao-Xiao Li ◽  
...  
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Iterative Regularization ◽  
Modified Helmholtz Equation ◽  
Unknown Source

scholarly journals The Quasireversibility Regularization Method for Identifying the Unknown Source for the Modified Helmholtz Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xiao-Xiao Li ◽  
Fan Yang ◽  
Jie Liu ◽  
Lan Wang
Keyword(s):  
Exact Solution ◽  
Helmholtz Equation ◽  
Regularization Method ◽  
Numerical Results ◽  
Convergence Estimate ◽  
Modified Helmholtz Equation ◽  
Unknown Source ◽  
Ill Posed

This paper discusses the problem of determining an unknown source which depends only on one variable for the modified Helmholtz equation. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The regularization solution is obtained by the quasireversibility regularization method. Convergence estimate is presented between the exact solution and the regularization solution. Moreover, numerical results are presented to illustrate the accuracy and efficiency of this method.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

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