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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:xingyangye@163.com
Corresponding author.Email address:cjxu@xmu.edu.cn
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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

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

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