Abstract
In this paper, we present a method that enables to solve in parallel the Euler–Lagrange system associated with the optimal control of a parabolic equation. Our approach is based on an iterative update of a sequence of intermediate targets that gives rise to independent sub-problems that can be solved in parallel. This method can be coupled with the parareal in time algorithm. Numerical experiments show the efficiency of our method.
Part of this work was funded by the ANR-06-CIS6-007-01 project PITAC.
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References
G. Bal, Y. Maday, A parareal time discretization for non-linear PDEs with application to the pricing of an American put, in Recent Developments in Domain Decomposition Methods, Lect. Notes Comput. Sci. Eng. (Springer, Berlin, 2002), pp. 189–202
A. Bellen, M. Zennaro, Parallel algorithms for initial value problems for nonlinear vector difference and differential equations. J. Comput. Appl. Math. 25, 341–350 (1989)
K. Burrage, Parallel and Sequential Methods for Ordinary Differential Equations. Numerical Mathematics and Scientific Computation (Clarendon Press, Oxford, 1995)
J.-L. Lions, Virtual and effective control for distributed systems and decomposition of everything. J. Anal. Math. 80, 257–297 (2000)
J.-L. Lions, Y. Maday, G. Turinici, Résolution d’EDP par un shéma pararréel. C. R. Acad. Sci. Paris, I 332, 661–668 (2001)
Y. Maday, J. Salomon, G. Turinici, Parareal in time control for quantum systems. SIAM J. Numer. Anal. 45(6), 2468–2482 (2007)
Y. Maday, G. Turinici, A parareal in time procedure for the control of partial differential equations. C. R. Math. Acad. Sci. Paris 335(4), 387–392 (2002)
T.P. Mathew, M. Sarkis, C.E. Schaerer, Analysis of block parareal preconditioners for parabolic optimal control problems. SIAM J. Sci. Comput. 32(3), 1180–1200 (2010)
O. Pironneau, F. Hecht, K. Ohtsuka, FreeFem++-mpi, http://www.freefem.org
M.-K. Riahi, Conception et analyse d’algorithmes parallèles en temps pour l’accélération de simulations numériques d’équations d’évolution. Thèse de doctorat de l’université, Pierre et Marie Curie, Paris 6, July 2012
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Maday, Y., Riahi, MK., Salomon, J. (2013). Parareal in Time Intermediate Targets Methods for Optimal Control Problems. In: Bredies, K., Clason, C., Kunisch, K., von Winckel, G. (eds) Control and Optimization with PDE Constraints. International Series of Numerical Mathematics, vol 164. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0631-2_5
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DOI: https://doi.org/10.1007/978-3-0348-0631-2_5
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