Two-Level Space–Time Domain Decomposition Methods for Flow Control Problems
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For time-dependent control problems, the class of sub-optimal algorithms is popular and the parallelization is usually applied in the spatial dimension only. In the paper, we develop a class of fully-optimal methods based on space–time domain decomposition methods for some boundary and distributed control of fluid flow and heat transfer problems. In the fully-optimal approach, we focus on the use of an inexact Newton solver for the necessary optimality condition arising from the implicit discretization of the optimization problem and the use of one-level and two-level space–time overlapping Schwarz preconditioners for the Jacobian system. We show that the numerical solution from the fully-optimal approach is generally better than the solution from the sub-optimal approach in terms of meeting the objective of the optimization problem. To demonstrate the robustness and parallel scalability and efficiency of the proposed algorithm, we present some numerical results obtained on a parallel computer with a few thousand processors.
KeywordsTime-dependent PDE-constrained optimization Boundary and distributed flow control Domain decomposition Space–time method Parallel computing
The authors would like to express their appreciations to the anonymous reviewers for the invaluable comments that have greatly improved the quality of the manuscript. This research was supported by the NSFC Grants 11571100 and 91330111. Haijian Yang was also supported in part by the NSFC Grant 11272352 and the Planned Science and Technology Project of Hunan Province under grant 2015JC3055.
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