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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,
Zhijian Yang and
Xiliang Lu
Zhousheng Ruan
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaSchool of Science, East China Institute of Technology, Nanchang 330013, China
Zhijian Yang
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Xiliang Lu*
Affiliation:
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
*
*Corresponding author. Email:zhshruan@whu.edu.cn (Z. S. Ruan), zjyang.math@whu.edu.cn (Z. J. Yang), xllv.math@whu.edu.cn (X. L. Lu)

Abstract

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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.

Keywords

65N2149M15Inverse source problemtime-fractional diffusion equationsparse constraintelastic-net regularization methodsemismooth Newton method

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 1 , February 2016, pp. 1 - 18
Copyright
Copyright © Global-Science Press 2016 

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