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Méthodes de décomposition de domaine : application à la résolution de problèmes de contrôle optimal

Abstract : This work deals with the use of domain decomposition methods to solve optimal control problems governed by partial differential equations. The problem on the whole domain decomposes into local subproblems with the cost function naturally resulting from the sum of the local cost functions provided that the perfect matching of the local solutions is preserved on both the sides of the interface between non-overlapping neighbour subdomains. In a first part, we especially study the elliptic case: we take simultaneously into account the minimization of the cost function and the links between subdomains. These links are satisfied by ensuring the flux continuity of the direct state given explicitly on the local system whereas the continuity constraint is considered with multipliers. The resulting optimization problem is treated by augmented Lagrangian techniques and is solved thanks to a variant of algorithms existing in the literature. In a second part, an extension to optimal control problems governed by parabolic system is developed considering only a space decomposition of the calculation domain. In the last part, we combine Schwarz algorithms and optimal control problems using an overlapping domain decomposition at each step of the minimization process. We design a parallel algorithm based on a multiplicative Schwarz method used as a solver. The adjoint state is naturally deduced from the transposition of the local direct. This mixed algorithm constitutes a robust but expensive method to solve the optimal control problem. For particularly large problems, another algorithm combining a Quasi-Newton method and a Krylov solver (\bic) preconditioned by an additive Schwarz method is proven to be more competitive since good parallel efficiencies are obtained. Results are presented to show the behaviour of the optimization solvers when Schwarz algorithms are used.
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Submitted on : Wednesday, February 18, 2004 - 11:13:55 AM
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Aïcha Bounaim. Méthodes de décomposition de domaine : application à la résolution de problèmes de contrôle optimal. Modélisation et simulation. Université Joseph-Fourier - Grenoble I, 1999. Français. ⟨tel-00004809⟩

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