Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T16:52:11.007Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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