Regularization method for an ill-posed Cauchy problem for elliptic equations

2017 ◽  
Vol 25 (3) ◽  
Author(s):  
Abderafik Benrabah ◽  
Nadjib Boussetila ◽  
Faouzia Rebbani
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Elliptic Equations ◽  
Regularization Method ◽  
Elliptic Pdes ◽  
Ill Posed

AbstractThe paper is devoted to investigating a Cauchy problem for homogeneous elliptic PDEs in the abstract Hilbert space given by

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.

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.

scholarly journals A Regularization Method for the Elliptic Equation with Inhomogeneous Source

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Tuan H. Nguyen ◽  
Binh Thanh Tran
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Boundary Conditions ◽  
Regularization Method ◽  
Dirichlet Boundary ◽  
Rectangular Domain ◽  
Dirichlet Boundary Conditions ◽  
Ill Posed ◽  
Sharp Error

We consider the following Cauchy problem for the elliptic equation with inhomogeneous source in a rectangular domain with Dirichlet boundary conditions at x=0 and x=π. The problem is ill-posed. The main aim of this paper is to introduce a regularization method and use it to solve the problem. Some sharp error estimates between the exact solution and its regularization approximation are given and a numerical example shows that the method works effectively.

A mollification method with Dirichlet kernel to solve Cauchy problem for two-dimensional Helmholtz equation

2019 ◽  
Vol 17 (05) ◽  
pp. 1950029 ◽  
Author(s):  
Shangqin He ◽  
Xiufang Feng
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Numerical Experiment ◽  
Helmholtz Equation ◽  
Regularization Method ◽  
Dirichlet Kernel ◽  
Two Dimensional ◽  
Strip Domain ◽  
Ill Posed ◽  
Mollification Method

In this paper, the ill-posed Cauchy problem for the Helmholtz equation is investigated in a strip domain. To obtain stable numerical solution, a mollification regularization method with Dirichlet kernel is proposed. Error estimate between the exact solution and its approximation is given. A numerical experiment of interest shows that our procedure is effective and stable with respect to perturbations of noise in the data.

scholarly journals Fourier Truncation Regularization Method for a Three-Dimensional Cauchy Problem of the Modified Helmholtz Equation with Perturbed Wave Number

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 705 ◽  
Author(s):  
Fan Yang ◽  
Ping Fan ◽  
Xiao-Xiao Li
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Regularization Method ◽  
Three Dimensional ◽  
Wave Number ◽  
Modified Helmholtz Equation ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Regularized Solution

In this paper, the Cauchy problem of the modified Helmholtz equation (CPMHE) with perturbed wave number is considered. In the sense of Hadamard, this problem is severely ill-posed. The Fourier truncation regularization method is used to solve this Cauchy problem. Meanwhile, the corresponding error estimate between the exact solution and the regularized solution is obtained. A numerical example is presented to illustrate the validity and effectiveness of our methods.

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 On the Cauchy problem for semilinear elliptic equations

2016 ◽  
Vol 24 (2) ◽  
Author(s):  
Nguyen Huy Tuan ◽  
Tran Thanh Binh ◽  
Tran Quoc Viet ◽  
Daniel Lesnic
Keyword(s):  
Cauchy Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Elliptic Equations ◽  
Elliptic Partial Differential Equations ◽  
Semilinear Elliptic Equations ◽  
Semilinear Elliptic ◽  
The Cauchy Problem ◽  
Ill Posed

AbstractWe study the Cauchy problem for nonlinear (semilinear) elliptic partial differential equations in Hilbert spaces. The problem is severely ill-posed in the sense of Hadamard. Under a weak

scholarly journals An iterative regularization method for variational inequalities in Hilbert spaces

2020 ◽  
Vol 36 (3) ◽  
pp. 475-482
Author(s):  
HONG-KUN XU ◽  
NAJLA ALTWAIJRY ◽  
SOUHAIL CHEBBI
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Iterative Solution ◽  
Iterative Regularization ◽  
Lipschitz Continuous ◽  
Ill Posed ◽  
Feasible Solutions ◽  
First Time

We consider an iterative method for regularization of a variational inequality (VI) defined by a Lipschitz continuous monotone operator in the case where the set of feasible solutions is decomposed to the intersection of finitely many closed convex subsets of a Hilbert space. We prove the strong convergence of the sequence generated by our algorithm. It seems that this is the first time in the literature to handle iterative solution of ill-posed VIs in the domain decomposition case.

Iterative regularization method for an abstract ill-posed generalized elliptic equation

2020 ◽  
pp. 2150069
Author(s):  
Sassane Roumaissa ◽  
Boussetila Nadjib ◽  
Rebbani Faouzia ◽  
Benrabah Abderafik
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Regularization Method ◽  
Boundary Data ◽  
Identification Problem ◽  
Infinite Domain ◽  
Iterative Regularization ◽  
Convergence Results ◽  
Ill Posed ◽  
Well Posed

A preconditioning version of the Kozlov–Maz’ya iteration method for the stable identification of missing boundary data is presented for an ill-posed problem governed by generalized elliptic equations. The ill-posed data identification problem is reformulated as a sequence of well-posed fractional elliptic equations in infinite domain. Moreover, some convergence results are established. Finally, numerical results are included showing the accuracy and efficiency of the proposed method.

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.

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