An DRCS preconditioning iterative method for a constrained fractional optimal control problem

The optimal control problem constrained by a fractional diffusion equation arises in a great deal of applications. Fast and efficient numerical methods for solving such kinds of problems have attracted much attention in recent years. In this paper, we consider an optimal control problem constrained by a fractional diffusion equation (FDE). After the state and costate equations are derived, the closed form of the optimal control variable is then given. The decoupled gradient projection method is applied to solve the coupled system of the state and costate equations to obtain the solution of the optimal control problem. The second-order Crank-Nicolson method as well as the weighted and shifted Grünwald difference (CN-WSGD) methods are utilized to discretize these two equations. We get the discretized state and costate equations as systems of linear equations with both coefficient matrices having the structure of the sum of a diagonal and a Toeplitz matrix. A diagonal and a R.Chan’s circulant splitting (DRCS) preconditioner is developed and combined in the Krylov subspace methods to solve the resulting discretized linear systems. Theoretical analysis of spectral distributions of the preconditioned matrix is also given. Numerical results exhibit that the proposed preconditioner can significantly improve the convergence of the Krylov subspace iteration methods.