scholarly journals A Posteriori Regularization Parameter Choice Rule for Truncation Method for Identifying the Unknown Source of the Poisson Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Xiao-Xiao Li ◽  
Dun-Gang Li
Keyword(s):  
Poisson Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Choice Rule ◽  
Conditional Stability ◽  
A Posteriori ◽  
Parameter Choice ◽  
Convergence Estimate ◽  
The Poisson Equation ◽  
Unknown Source

We consider the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation. We prove a conditional stability for this problem. Moreover, we propose a truncation regularization method combined with an a posteriori regularization parameter choice rule to deal with this problem and give the corresponding convergence estimate. Numerical results are presented to illustrate the accuracy and efficiency of this method.

Identifying an Unknown Source in the Poisson Equation with a Super Order Regularization Method

2019 ◽  
Vol 17 (07) ◽  
pp. 1950030 ◽  
Author(s):  
Zhi Li ◽  
Zhenyu Zhao ◽  
Zehong Meng ◽  
Baoqin Chen ◽  
Duan Mei
Keyword(s):  
Poisson Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
New Method ◽  
Tikhonov Regularization Method ◽  
Optimal Error ◽  
Penalty Term ◽  
The Poisson Equation ◽  
Unknown Source

This paper develops a new method to deal with the problem of identifying the unknown source in the Poisson equation. We obtain the regularization solution by the Tikhonov regularization method with a super-order penalty term. The order optimal error bounds can be obtained for various smooth conditions when we choose the regularization parameter by a discrepancy principle and the solution process of the new method is uniform. Numerical examples show that the proposed method is effective and stable.

scholarly journals Landweber Iterative Regularization Method for Identifying the Initial Value Problem of the Rayleigh–Stokes Equation

2021 ◽  
Vol 5 (4) ◽  
pp. 193
Author(s):  
Dun-Gang Li ◽  
Jun-Liang Fu ◽  
Fan Yang ◽  
Xiao-Xiao Li
Keyword(s):  
Inverse Problem ◽  
Stokes Equation ◽  
Initial Value Problem ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Choice Rule ◽  
Conditional Stability ◽  
Parameter Choice ◽  
Initial Value ◽  
Iterative Regularization

In this paper, we study an inverse problem to identify the initial value problem of the homogeneous Rayleigh–Stokes equation for a generalized second-grade fluid with the Riemann–Liouville fractional derivative model. This problem is ill posed; that is, the solution (if it exists) does not depend continuously on the data. We use the Landweber iterative regularization method to solve the inverse problem. Based on a conditional stability result, the convergent error estimates between the exact solution and the regularization solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are given. Some numerical experiments are performed to illustrate the effectiveness and stability of this method.

An a posteriori mollification method for the heat equation backward in time

2017 ◽  
Vol 25 (4) ◽  
Author(s):  
Nguyen Van Duc
Keyword(s):  
Heat Equation ◽  
Error Estimate ◽  
Regularization Method ◽  
Choice Rule ◽  
Stability Estimates ◽  
A Posteriori ◽  
Parameter Choice ◽  
A Posteriori Parameter Choice ◽  
Mollification Method ◽  
Derivatives Of

AbstractThe heat equation backward in timesubject to the constraintwhereAn a posteriori parameter choice rule for this regularization method is suggested, which yields the error estimateFurthermore, we establish stability estimates of Hölder type for all derivatives of the solutions with respect to

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

scholarly journals A Tikhonov-Type Regularization Method for Identifying the Unknown Source in the Modified Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jinghuai Gao ◽  
Dehua Wang ◽  
Jigen Peng
Keyword(s):  
Helmholtz Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Inverse Source Problem ◽  
A Posteriori ◽  
Modified Helmholtz Equation ◽  
Numerical Tests ◽  
Theoretical Frame ◽  
Set Up

An inverse source problem in the modified Helmholtz equation is considered. We give a Tikhonov-type regularization method and set up a theoretical frame to analyze the convergence of such method. A priori and a posteriori choice rules to find the regularization parameter are given. Numerical tests are presented to illustrate the effectiveness and stability of our proposed 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.

Sign in / Sign up

Export Citation Format

Share Document