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The time parallel solution of optimality systems arising in PDE constraintoptimization could be achieved by simply applying any time parallel algorithm,such as Parareal, to solve the forward and backward evolution problems arisingin the optimization loop. We propose here a different strategy by devisingdirectly a new time parallel algorithm, which we call ParaOpt, for the coupledforward and backward non-linear partial differential equations. ParaOpt isinspired by the Parareal algorithm for evolution equations, and thus isautomatically a two-level method. We provide a detailed convergence analysisfor the case of linear parabolic PDE constraints. We illustrate the performanceof ParaOpt with numerical experiments both for linear and nonlinear optimalitysystems.

PARAOPT: A parareal algorithm for optimality systems