A fast second-order parareal solver for fractional optimal control problems

2017 ◽  
Vol 24 (15) ◽  
pp. 3418-3433 ◽  
Author(s):  
Shu-Lin Wu ◽  
Ting-Zhu Huang

The gradient projection technique has recently been used to solve the optimal control problems governed by a fractional diffusion equation. It lies in repeatedly solving the state and co-state equations derived from the optimality conditions, and the Crank–Nicolson (CN) scheme, which gives a second-order numerical solution, is a widely used method to solve these two equations. The goal of this paper is to implement the CN scheme in a parallel-in-time manner in the framework of the parareal algorithm. Because of the stiffness of the approximation matrix of the fractional operator, direct use of the CN scheme results in a convergence factor ρ satisfying [Formula: see text] as [Formula: see text] for the parareal algorithm, where [Formula: see text] denotes the space step-size. Here, we provide a new idea to let the parareal algorithm use the CN scheme as the basic component possessing a constant convergence factor [Formula: see text], which is independent of [Formula: see text]. Numerical results are provided to show the efficiency of the proposed algorithm.

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