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A Parareal Algorithm for Coupled Systems Arising from Optimal Control Problems

Banff International Research Station for Mathematical Innovation and Discovery

When solving optimal control problems over a long time horizon, one can introduce additional parallelism in time by subdividing the time horizon into smaller, non-overlapping time intervals and by solving these subproblems in parallel. If the intermediate state and adjoint between time intervals are known exactly, this procedure yields the exact solution. Thus, the problem reduces to solving a nonlinear system in these intermediate states, which are related via certain propagation operators. In this talk, we present a parareal approach for solving this nonlinear system: here, the global problem is approximated by a simpler one using coarse propagators, while the fine propagation is performed in parallel over different time intervals. One then iterates until the intermediate states are consistent across time intervals. Unlike parareal for initial value problems, the coarse problem still contains a forward-backward coupling, but it is much cheaper to solve than the global fine problem. We analyze the convergence of the new method for a model linear problem and illustrate its behaviour numerically for nonlinear problems in which the control enters as an additive source term.

Mathematics; Numerical analysis; Partial differential equations

video/mp4

Author affiliation: Hong Kong Baptist University

2017-05-15

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61655

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/