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Volume 6, Issue 3
Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

Xingyang Ye & Chuanju Xu

DOI: 10.4208/nmtma.2013.1207nm

Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 499-519.

Published online: 2013-06

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  • 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

Time fractional diffusion equation, inverse problem, spectral method, error estimate, conjugate gradient method.

  • AMS Subject Headings

65M12, 65M32, 65M70, 35S10, 49J20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-6-499, author = {Xingyang Ye and Chuanju Xu}, title = {Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2013}, volume = {6}, number = {3}, pages = {499--519}, 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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2013.1207nm}, url = {http://global-sci.org/intro/article_detail/nmtma/5915.html} }
TY - JOUR T1 - Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem AU - Xingyang Ye & Chuanju Xu JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 499 EP - 519 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1207nm UR - https://global-sci.org/intro/article_detail/nmtma/5915.html KW - Time fractional diffusion equation, inverse problem, spectral method, error estimate, conjugate gradient method. AB -

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.

Xingyang Ye and Chuanju Xu. (2013). Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem. Numerical Mathematics: Theory, Methods and Applications. 6 (3). 499-519. doi:10.4208/nmtma.2013.1207nm
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