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A Practical Factorization of a Schur Complement for PDE-Constrained Distributed Optimal Control

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Abstract

A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution of which is crucial. A new factorization of the Schur complement for such a system is proposed and its characteristics discussed. The factorization introduces two complex factors that are complex conjugate to each other. The proposed solution methodology involves the application of a parallel linear domain decomposition solver—FETI-DPH—for the solution of the subproblems with the complex factors. Numerical properties of FETI-DPH in this context are demonstrated, including numerical and parallel scalability and regularization dependence. The new factorization can be used to solve Schur complement systems arising in both range-space and full-space formulations. In both cases, numerical results indicate that the complex factorization is promising. Especially, in the full-space method with the new factorization, the number of iterations required for convergence is independent of regularization parameter values.

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Acknowledgments

The authors thank Philip Avery in the Farhat Research Group for his valuable comments and essential help with coding the physics-based C++ PDE solver Aero-S. The authors also thank anonymous reviewers for their valuable comments that improve the paper tremendously.

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Correspondence to Youngsoo Choi.

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Youngsoo Choi and Charbel Farhat acknowledge partial support by the Army Research Laboratory through the Army High Performance Computing Research Center under Cooperative Agreement W911NF-07-2-0027. Walter Murray and Michael Saunders acknowledge partial support by the ONR Grant N000141110067.

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Choi, Y., Farhat, C., Murray, W. et al. A Practical Factorization of a Schur Complement for PDE-Constrained Distributed Optimal Control. J Sci Comput 65, 576–597 (2015). https://doi.org/10.1007/s10915-014-9976-0

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