Expected absolute value estimators for a spatially adapted regularization parameter choice rule in
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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.

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

International Journal of Partial Differential Equations ◽

10.1155/2013/590737 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 1

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.

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Recovering the space source term for the fractional-diffusion equation with Caputo–Fabrizio derivative

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02557-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Le Nhat Huynh ◽

Nguyen Hoang Luc ◽

Dumitru Baleanu ◽

Le Dinh Long

Keyword(s):

Diffusion Equation ◽

Source Term ◽

Regularization Parameter ◽

A Priori ◽

Fractional Diffusion ◽

Fractional Diffusion Equation ◽

Choice Rule ◽

Parameter Choice ◽

Landweber Method ◽

Regularized Solution

AbstractThis article is devoted to the study of the source function for the Caputo–Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method.

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

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0062 ◽

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.

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