  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. 
PARAOPT: A parareal algorithm for optimality systems