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

Published online by Cambridge University Press:  21 December 2015

Zhousheng Ruan
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China School of Science, East China Institute of Technology, Nanchang 330013, China
Zhijian Yang
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Xiliang Lu*
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
*Corresponding author. (Z. S. Ruan), (Z. J. Yang), (X. L. Lu)
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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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