Landweber iteration method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation

2021 ◽  
Author(s):  
Jin Wen ◽  
Zhuan-Xia Liu ◽  
Chong-Wang Yue ◽  
Shi-Juan Wang
Keyword(s):  
Initial Data ◽  
Diffusion Equation ◽  
Iteration Method ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Simultaneous Inversion ◽  
Time Fractional Diffusion Equation

Inverse source problem for a distributed-order time fractional diffusion equation

2020 ◽  
Vol 28 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Xiaoliang Cheng ◽  
Lele Yuan ◽  
Kewei Liang
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Series Representation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Time Fractional Diffusion Equation ◽  
Distributed Order ◽  
Regularized Solution

AbstractThis paper studies an inverse source problem for a time fractional diffusion equation with the distributed order Caputo derivative. The space-dependent source term is recovered from a noisy final data. The uniqueness, ill-posedness and a conditional stability for this inverse source problem are obtained. The inverse problem is formulated into a minimization functional with Tikhonov regularization method. Further, based on the series representation of the regularized solution, we give convergence rates of the regularized solution under an a-priori and an a-posteriori regularization parameter choice rule. With an adjoint technique for computing the gradient of the regularization functional, the conjugate gradient method is applied to reconstruct the space-dependent source term. Two numerical examples illustrate the effectiveness of the proposed method.

An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation

2015 ◽  
Vol 8 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Zhousheng Ruan ◽  
Zhijian Yang ◽  
Xiliang Lu
Keyword(s):  
Diffusion Equation ◽  
Optimization Problem ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Inverse Source Problem ◽  
Final Time ◽  
Elastic Net Regularization ◽  
Semismooth Newton ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, an inverse source problem for the time-fractional diffusion equation is investigated. The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure. Here the sparsity is understood with respect to the pixel basis, i.e., the source has a small support. By an elastic-net regularization method, this inverse source problem is formulated into an optimization problem and a semismooth Newton (SSN) algorithm is developed to solve it. A discretization strategy is applied in the numerical realization. Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

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