A fast and stable preconditioned iterative method for optimal control problem of wave equations

In this paper, we develop a new central finite difference scheme in terms of both time and space for solving the first-order necessary optimality systems that characterize the optimal control of wave equations. The obtained new scheme is proved to be unconditionally convergent with a second-order accuracy, without the requirement of the Courant-Friedrichs-Lewy condition on the corresponding grid ratio. An efficient preconditioned iterative method is further developed for solving the discretized sparse linear system based on the relationship between the resultant matrix structure and the coupled PDE optimality system. Numerical examples are presented to verify the theoretical analysis and to demonstrate the high efficiency of the proposed preconditioned iterative solver.