  The time parallel solution of optimality systems arising in PDE constraint
optimization could be achieved by simply applying any time parallel algorithm,
such as Parareal, to solve the forward and backward evolution problems arising
in the optimization loop. We propose here a different strategy by devising
directly a new time parallel algorithm, which we call ParaOpt, for the coupled
forward and backward non-linear partial differential equations. ParaOpt is
inspired by the Parareal algorithm for evolution equations, and thus is
automatically a two-level method. We provide a detailed convergence analysis
for the case of linear parabolic PDE constraints. We illustrate the performance
of ParaOpt with numerical experiments both for linear and nonlinear optimality
systems.
