For optimal control problems constrained by a initial-valued parabolic PDE,
we have to solve a large scale saddle point algebraic system consisting of
considering the discrete space and time points all together. A popular strategy
to handle such a system is the Krylov subspace method, for which an efficient
preconditioner plays a crucial role. The matching-Schur-complement
preconditioner has been extensively studied in literature and the
implementation of this preconditioner lies in solving the underlying PDEs
twice, sequentially in time. In this paper, we propose a new preconditioner for
the Schur complement, which can be used parallel-in-time (PinT) via the so
called diagonalization technique. We show that the eigenvalues of the
preconditioned matrix are low and upper bounded by positive constants
independent of matrix size and the regularization parameter. The uniform
boundedness of the eigenvalues leads to an optimal linear convergence rate of
conjugate gradient solver for the preconditioned Schur complement system. To
the best of our knowledge, it is the first time to have an optimal convergence
analysis for a PinT preconditioning technique of the optimal control problem.
Numerical results are reported to show that the performance of the proposed
preconditioner is robust with respect to the discretization step-sizes and the
regularization parameter