PARAOPT: A parareal algorithm for optimality systems

Martin J. Gander (1) , Félix Kwok (2) , Julien Salomon (3, 4, 5)

1 Section de mathématiques [Genève]
2 HKBU - Department of Mathematics [Hong Kong Baptist University]
3 SU - Sorbonne Université
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
5 ANGE - Numerical Analysis, Geophysics and Ecology

The time parallel solution of optimality systems arising in PDE constrained opti- mization 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 nonlinear 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 for both linear and nonlinear optimality systems.

Mots clés: Preconditionning; Optimal control; Parareal algorithm

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]; Analyse numérique [math.NA]

Martin J. Gander, Félix Kwok, Julien Salomon. PARAOPT: A parareal algorithm for optimality systems. SIAM Journal on Scientific Computing, 2020, 42 (5), pp.A2773--A2802. ⟨10.1137/19M1292291⟩. ⟨hal-02346535v2⟩