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 - <p style="text-align: justify;">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.</p>