Volume 8, Issue 1
An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation

Zhousheng Ruan, Zhijian Yang & Xiliang Lu

Adv. Appl. Math. Mech., 8 (2016), pp. 1-18.

Published online: 2018-05

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  • Abstract

In 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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@Article{AAMM-8-1, author = {Zhousheng Ruan, Zhijian Yang and Xiliang Lu}, title = {An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {8}, number = {1}, pages = {1--18}, abstract = {

In 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2014.m722}, url = {http://global-sci.org/intro/article_detail/aamm/12073.html} }
TY - JOUR T1 - An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation AU - Zhousheng Ruan, Zhijian Yang & Xiliang Lu JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 1 EP - 18 PY - 2018 DA - 2018/05 SN - 8 DO - http://dor.org/10.4208/aamm.2014.m722 UR - https://global-sci.org/intro/aamm/12073.html KW - AB -

In 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.

Zhousheng Ruan, Zhijian Yang & Xiliang Lu. (1970). An Inverse Source Problem with Sparsity Constraint for the Time-Fractional Diffusion Equation. Advances in Applied Mathematics and Mechanics. 8 (1). 1-18. doi:10.4208/aamm.2014.m722
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