Journal of Scientific Computing

, Volume 65, Issue 2, pp 576–597 | Cite as

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

  • Youngsoo ChoiEmail author
  • Charbel Farhat
  • Walter Murray
  • Michael Saunders


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.


PDE-constrained optimization Schur complement Poisson operator FETI Range-space method Full-space method Distributed optimal control 

Mathematics Subject Classification

65N22 65N55 65F10 65F50 



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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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Youngsoo Choi
    • 1
    Email author
  • Charbel Farhat
    • 2
  • Walter Murray
    • 3
  • Michael Saunders
    • 3
  1. 1.Aeronautics and AstronauticsStanford UniversityStanfordUSA
  2. 2.Aeronautics and Astronautics, Mechanical EngineeringStanford UniversityStanfordUSA
  3. 3.Management Science and EngineeringStanford UniversityStanfordUSA

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