A solution for fractional PDE constrained optimization problems using reduced basis method

Abstract

In this paper, we employ a reduced basis method for solving PDE constrained optimization problem governed by a fractional parabolic equation with the fractional derivative in time. The fractional derivative used in this paper is Caputo fractional derivative whose order $$ \alpha \in (0,1) $$. A new approach is proposed based on optimize-then-discretize to solve the problem. Firstly, the optimality conditions for the problem is extracted. Then using finite difference method for time variable and reduced basis method, a numerical technique is obtained. Afterwards, we derive efficiently computable and rigorous a posteriori error bounds for various quantities of interest which is a tool to generate the reduced space. Extensive numerical results are presented to demonstrate the convergence property of the method.