Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T05:37:35.303Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

Published online by Cambridge University Press:  28 May 2015

Xingyang Ye  and
Chuanju Xu
Xingyang Ye*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, China School of Science, Jimei University, Xiamen 361021, Fujian, China
Chuanju Xu*
Affiliation:
School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, China
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Get access

Abstract

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework, the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established, and a time-space spectral method is proposed to numerically solve the resulting minimization problem. The contribution of the paper is threefold: 1) a priori error estimate for the spectral approximation is derived; 2) a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem; 3) some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition, and that the convergence rate of the method is exponential if the optimal initial condition is smooth.

Keywords

65M1265M3265M7035S1049J20Time fractional diffusion equationinverse problemspectral methoderror estimateconjugate gradient method
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 3 , August 2013 , pp. 499 - 519
Copyright
Copyright © Global Science Press Limited 2013

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] Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
[2] Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A., Spectral Methods Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.CrossRefGoogle Scholar
[3] 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), pp. 115002.CrossRefGoogle Scholar
[4] V Ervin, J. and Roop, J. P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations, 22(3) (2006), pp. 558–576.CrossRefGoogle Scholar
[5] V Ervin, J. and Roop, J. P., Variational solution of fractional advection dispersion equations on bounded domains in R d , Numer. Methods Partial Differential Equations, 23(2) (2007), pp. 256–281.CrossRefGoogle Scholar
[6] Isakov, V., Inverse Problems for Partial Differential Equations, Springer Science, New York, 2006.Google Scholar
[7] Langlands, T. A.M. and Henry, B. I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205(2) (2005), pp. 719–736.CrossRefGoogle Scholar
[8] Li, Xianjuan and Xu, Chuanju, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47(3) (2009), pp. 2108–2131.CrossRefGoogle Scholar
[9] Lin, Yumin and Xu, Chuanju, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225(2) (2007), pp. 1533–1552.CrossRefGoogle Scholar
[10] Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, volume 170 of Grundlehren Math. Wiss., Springer-Verlag, Berlin, 1971.Google Scholar
[11] Liu, F., Shen, S., Anh, V and Turner, I., Analysis of a discrete non-markovian random walk approximation for the time fractional diffusion equation, ANZIAM J, 46(4) (2005), pp. 488–504.CrossRefGoogle Scholar
[12] Luchko, Y., Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl, 351(1) (2009), pp. 218–223.CrossRefGoogle Scholar
[13] Luchko, Y., Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl, 59(5) (2010), pp. 1766–1772.CrossRefGoogle Scholar
[14] Podlubny, I., Fractional Differential Equations, Acad. Press, New York, 1999.Google Scholar
[15] Roop, J. P., Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in Ш2 , J. Comput. Appl. Math., 193(1) (2006), pp. 243–268.CrossRefGoogle Scholar
[16] 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.CrossRefGoogle Scholar
[17] Samarskii, A. A. and Vabishchevich, P. N., Numerical Methods for Solving Inverse Problems of Mathematical Physics, De Gruyter, Berlin, 2007.CrossRefGoogle Scholar
[18] Schneider, W.R. and Wyss, W., Fractional diffusion and wave equations, J. Math. Phys, 30(1) (1989), pp. 134–144.CrossRefGoogle Scholar
[19] Sun, Z. and Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56(2) (2006), pp. 193–209.CrossRefGoogle Scholar
[20] Wyss, W., The fractional diffusion equation, J. Math. Phys., 27(11) (1986), pp. 2782–2785.CrossRefGoogle Scholar
[21] Zhang, Y. and Xu, X., Inverse source problem for a fractional diffusion equation, Inverse Problems, 27(3) (2011), pp. 035010.CrossRefGoogle Scholar