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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, China School 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)
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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.

Keywords

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

[1]Cheng, J., Nakagawa, J., Yamamoto, M. and Yamazaki, T., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25(11) (2009), 115002.CrossRefGoogle Scholar
[2]Clason, C. and Kunisch, K., A duality-based approach to elliptic control problems in non-reflexive banach spaces, ESAIM: Control, Optimisation and Calculus of Variations, 17(01) (2011), pp. 243–266.Google Scholar
[3]Debnath, L., Recent applications of fractional calculus to science and engineering, Int. Math. Math. Sci., 54 (2003), pp. 3413–3442.Google Scholar
[4]Eidelman, S. D. and Kochubei, A. N., Cauchy problem for fractional diffusion equations, J. Differential Equations, 199(2) (2004), pp. 211–255.CrossRefGoogle Scholar
[5]Ekeland, I. and Temam, R., Convex Analysis and Variational Problems, volume 28, SIAM, 1999.CrossRefGoogle Scholar
[6]Fan, Q., Jiao, Y. and Lu, X., A primal dual active set algorithm with continuation for compressed sensing, Signal Process. IEEE Trans., 62 (2014), pp. 6276–6285.CrossRefGoogle Scholar
[7]Fan, Q., Jiao, Y., Lu, X. and Sun, Z., L q-regularization for the inverse robin problem, J. Inverse Ill-Posed Problems, (2014).Google Scholar
[8]Griesse, R. and Lorenz, D. A., A semismooth newton method for tikhonov functionals with sparsity constraints, Inverse Problems, 24(3) (2008), 035007.CrossRefGoogle Scholar
[9]Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semismooth newton method, SIAM J. Optim., 13(3) (2002), pp. 865–888.CrossRefGoogle Scholar
[10]Jiang, Y. and Ma, J., High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235(11) (2011), pp. 3285–3290.CrossRefGoogle Scholar
[11]Jiao, Y., Jin, B. and Lu, X., A primal dual active set with continuation algorithm for the l0-regularized optimization problem , Appl. Comput. Harmonic Anal., in press, (2014).Google Scholar
[12]Jin, B., Lazarov, R., Pasciak, J. and Zhou, Z., Galerkin fem for fractional order parabolic equations with initial data in h–s, 0<s≤1, Numer. Anal. Appl., (2013), pp. 24–37.Google Scholar
[13]Jin, B., Lorenz, D. A. and Schiffler, S., Elastic-net regularization: error estimates and active set methods, Inverse Problems, 25(11) (2009), 115022.Google Scholar
[14]Jin, B. and Rundell, W., An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems, 28(7) (2012), 075010.Google Scholar
[15]Kilbas, A. A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, 2006.Google Scholar
[16]Kunisch, K. and Lu, X., Optimal control for an elliptic system with convex polygonal control constraints, IMA J. Numer. Anal., 33 (2013), pp. 875–897.CrossRefGoogle Scholar
[17]Kunisch, K. and Lu, X., Optimal control for elliptic systems with pointwise euclidean norm constraints on the controls, Math. Program., 142(1-2) (2013), pp. 461–483.Google Scholar
[18]Li, X. and Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47(3) (2009), pp. 2108–2131.Google Scholar
[19]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225(2) (2007), pp. 1533–1552.CrossRefGoogle Scholar
[20]Liu, J. and Yamamoto, M., A backward problem for the time-fractional diffusion equation, Appl. Anal., 89(11) (2010), pp. 1769–1788.Google Scholar
[21]Lu, S., Pereverzev, S. V., Shao, Y. and Tautenhahn, U., Discrepancy curves for multiparameter regularization, J. Inverse Ill-Posed Problems, 18(6) (2010), pp. 655–676.Google Scholar
[22]Metzler, R. and Klafter, J., Boundary value problems for fractional diffusion equations, Phys. A Statist. Mech. Appl., 278(1) (2000), pp. 107–125.Google Scholar
[23]Nakagawa, J., Sakamoto, K. and Yamamoto, M., Overview to mathematical analysis for fractional diffusion equations-new mathematical aspects motivated by industrial collaboration, J. Math. Industry, 2(10) (2010), pp. 99–108.Google Scholar
[24]Nigmatullin, R., The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status Solid B, 133(1) (1986), pp. 425–430.CrossRefGoogle Scholar
[25]Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 198, Academic Press, 1998.Google Scholar
[26]Pollard, H., The completely monotonic character of the mittag-leffler function eα(–x), Bulletin American Math. Society, 54(12) (1948), pp. 1115–1116.Google Scholar
[27]Roman, H. and Alemany, P., Continuous-time random walks and the fractional diffusion equation, J. Phys. A Math. General, 27(10) (1994), 3407.Google Scholar
[28]Sakamoto, K. and Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382(1) (2011), pp. 426–447.Google Scholar
[29]Wachsmuth, G. and Wachsmuth, D., Convergence and regularization results for optimal control problems with sparsity functional, ESAIM: Control, Optimisation and Calculus of Variations, 17(3) (2011), pp. 858–886.Google Scholar
[30]Stadler, G., Elliptic optimal control problems with l1-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44(2) (2009), pp. 159–181.Google Scholar
[31]Sun, Z., Jiao, Y., Jin, B. and Lu, X., Numerical identification of a sparse robin coefficient, Adv. Comput. Math., (2014), pp. 1–18.Google Scholar
[32]Tikhonov, A. N. and Glasko, V., Use of the regularization method in non-linear problems, USSR Comput. Math. Math. Phys., 5(3) (1965), pp. 93–107.Google Scholar
[33]Wang, J., Zhou, Y. and Wei, T., Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl. Numer. Math., 68 (2013), pp. 39–57.Google Scholar
[34]Wang, L. and Liu, J., Data regularization for a backward time-fractional diffusion problem, Comput. Math. Appl., 64(11) (2012), pp. 3613–3626.Google Scholar
[35]Wang, Z., Qiu, S., Ruan, Z. and Zhang, W., A regularized optimization method for identifying the space-dependent source and the initial value simultaneously in a parabolic equation, Comput. Math. Appl., 67(7) (2014), pp. 1345–1357.Google Scholar
[36]Wei, T. and Zhang, Z., Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Eng. Anal. Boundary Elements, 37(1) (2013), pp. 23–31.CrossRefGoogle Scholar
[37]Xiong, X., Wang, J. and Li, M., An optimal method for fractional heat conduction problem backward in time, Appl. Anal., 91(4) (2012), pp. 823–840.Google Scholar
[38]Ye, X. and Xu, C., Spectral optimization methods for the time fractional diffusion inverse problem, Numer. Math. Theory Methods Appl., 6(3) (2013), pp. 499–519.Google Scholar
[39]Zhang, Y. and Xu, X., Inverse source problem for a fractional diffusion equation, Inverse Problems, 27(3) (2011), 035010.CrossRefGoogle Scholar
[40]Zheng, G. and Wei, T., A new regularization method for the time fractional inverse advection-dispersion problem, SIAM J. Numer. Anal., 49(5) (2011), pp. 1972–1990.Google Scholar
[41]Zou, H. and Hastie, T., Regularization and variable selection via the elastic net, J. Royal Statistical Society Series B (Statistical Methodology), 67(2) (2005), pp. 301–320.CrossRefGoogle Scholar