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

Abstract

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 $\rho \to 1$ as $\Delta x\to 0$ for the parareal algorithm, where $\Delta x$ 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 $\rho \approx 1/5$ , which is independent of $\Delta x$ . Numerical results are provided to show the efficiency of the proposed algorithm.