A Fast Gradient Projection Method for a Constrained Fractional Optimal Control

Optimal control problems governed by a fractional diffusion equation tends to provide a better description than one by a classical second-order Fickian diffusion equation in the context of transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significantly increased computational complexity and storage requirement than the corresponding classical control problem, due to the nonlocal nature of fractional differential operators. We develop a fast gradient projection method for a pointwise constrained optimal control problem governed by a time-dependent space-fractional diffusion equation, which requires the computational cost from O(MN3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(M N^3)$$\end{document} of a conventional solver to O(MNlogN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(M N\log N)$$\end{document} and memory requirement from O(N2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N^2)$$\end{document} to O(N) for a problem of size N and of M time steps. Numerical experiments show the utility of the method.