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.

The Regularization Method Based on Complex Collinearity Diagnostics and Metrics

2013 ◽  
Vol 416-417 ◽  
pp. 1393-1398
Author(s):  
Chao Zhong Ma ◽  
Yong Wei Gu ◽  
Ji Fu ◽  
Yuan Lu Du ◽  
Qing Ming Gui
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Measurement Data ◽  
Tikhonov Regularization Method ◽  
Parameter Determination ◽  
Selection Methods ◽  
Ill Posed ◽  
Collinearity Diagnostics ◽  
Regularization Matrix

In a large number of measurement data processing, the ill-posed problem is widespread. For such problems, this paper introduces the solution of ill-posed problem of the unity of expression and Tikhonov regularization method, and then to re-collinearity diagnostics and metrics based on proposed based on complex collinearity diagnostics and the metric regularization method is given regularization matrix selection methods and regularization parameter determination formulas. Finally, it uses a simulation example to verify the effectiveness of the method.

scholarly journals A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Songshu Liu ◽  
Lixin Feng
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Tikhonov Regularization Method ◽  
Two Dimensional ◽  
Convergence Estimate ◽  
Ill Posed

In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.

scholarly journals Identical Approximation Operator and Regularization Method for the Cauchy problem of 2D Heat Conduction Equation

2020 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Regularization Method ◽  
Heat Conduction Equation ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
Numerical Results ◽  
Conduction Equation ◽  
The Cauchy Problem ◽  
Ill Posed

In this paper, an identical approximate regularization method is extended to the Cauchy problem of two-dimensional heat conduction equation, this kind of problem is severely ill-posed. The convergence rates are obtained under a priori regularization parameter choice rule. Numerical results are presented for two examples with smooth and continuous but not smooth boundaries, and compared the identical approximate regularization solutions which are displayed in paper. The numerical results show that our method is effective, accurate and stable to solve the ill-posed Cauchy problems.

scholarly journals An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

2011 ◽  
Vol 9 (4) ◽  
pp. 878-896 ◽  
Author(s):  
Houde Han ◽  
Leevan Ling ◽  
Tomoya Takeuchi
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Regularization Method ◽  
Boundary Data ◽  
Outer Boundary ◽  
Cauchy Problems ◽  
Electrical Field ◽  
Numerical Examples ◽  
The Cauchy Problem ◽  
Ill Posed

AbstractDetecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

scholarly journals EXTRAPOLATION OF TIKHONOV REGULARIZATION METHOD

2010 ◽  
Vol 15 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Uno Hämarik ◽  
Reimo Palm ◽  
Toomas Raus
Keyword(s):  
Approximate Solution ◽  
Tikhonov Regularization ◽  
Noise Level ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Noisy Data ◽  
Tikhonov Regularization Method ◽  
Guaranteed Accuracy ◽  
Numerical Examples ◽  
Ill Posed

We consider regularization of linear ill‐posed problem Au = f with noisy data fδ, ¦fδ - f¦≤ δ . The approximate solution is computed as the extrapolated Tikhonov approximation, which is a linear combination of n ≥ 2 Tikhonov approximations with different parameters. If the solution u* belongs to R((A*A) n ), then the maximal guaranteed accuracy of Tikhonov approximation is O(δ 2/3) versus accuracy O(δ 2n/(2n+1)) of corresponding extrapolated approximation. We propose several rules for choice of the regularization parameter, some of these are also good in case of moderate over‐ and underestimation of the noise level. Numerical examples are given.

scholarly journals Regularization and Error Estimate for the Poisson Equation with Discrete Data

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 422
Author(s):  
Nguyen Anh Triet ◽  
Nguyen Duc Phuong ◽  
Van Thinh Nguyen ◽  
Can Nguyen-Huu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Poisson Equation ◽  
Regularization Method ◽  
Random Noise ◽  
Two Dimensional ◽  
The Poisson Equation ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Dimensional Domain

In this work, we focus on the Cauchy problem for the Poisson equation in the two dimensional domain, where the initial data is disturbed by random noise. In general, the problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. To regularize the instable solution of the problem, we have applied a nonparametric regression associated with the truncation method. Eventually, a numerical example has been carried out, the result shows that our regularization method is converged; and the error has been enhanced once the number of observation points is increased.

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