In this talk we discuss a parallel-in-time (PinT) and flexible preconditioned iterative method for the large scale all-at-once system arising from parabolic PDE-constrained optimization problems.  The construction of the new preconditioner lies in approximating the Schur complement of the all-at-once system by replacing the time stepping matrices (of Toeplitz-type) by the alpha-circulant matrices, where 0<alpha<1 is a free parameter. The new precondtioner has the advantage of parallel implementation across all the time steps, without needing additional complexity and memory storage. Similar to the classical matching Schur complement preconditiner, we show that the new preconditioner is flexible to handle quite a large rang of optimal control problems, including distributed control, local control, Neumann boundary control and boxing-type control. This is a series talk and consists of two parts.

Part-I: in this second talk, we introduce the PinT preconditioner for optimal control of wave  l equations. Since the wave equations have more complicated properties than the parabolic equations, the preconditioner needs more careful treatment.