  In this paper, a fast solver is studied for saddle point system arising from
a second-order Crank-Nicolson discretization of an initial-valued parabolic PDE
constrained optimal control problem, which is indefinite and ill-conditioned.
Different from the saddle point system arising from the first-order Euler
discretization, the saddle point system arising from Crank-Nicolson
discretization has a dense and non-symmetric Schur complement, which brings
challenges to fast solver designing. To remedy this, a novel symmetrization
technique is applied to the saddle point system so that the new Schur
complement is symmetric definite and the well-known matching-Schur-complement
(MSC) preconditioner is applicable to the new Schur complement. Nevertheless,
the new Schur complement is still a dense matrix and the inversion of the
corresponding MSC preconditioner is not parallel-in-time (PinT) and thus time
consuming. For this concern, a modified MSC preconditioner for the new Schur
complement system. Our new preconditioner can be implemented in a fast and PinT
way via a temporal diagonalization technique. Theoretically, the eigenvalues of
the preconditioned matrix by our new preconditioner are proven to be lower and
upper bounded by positive constants independent of matrix size and the
regularization parameter. With such spectrum, the preconditioned conjugate
gradient (PCG) solver for the Schur complement system is proven to have a
convergence rate independent of matrix size and regularization parameter. To
the best of my knowledge, it is the first time to have an iterative solver with
problem-independent convergence rate for the saddle point system arising from
Crank-Nicolson discretization of the optimal control problem.
  Numerical results are reported to show that the performance of the proposed
preconditioner.
