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 Choi
  • 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.


  1. 1.
    Avery, P., Farhat, C.: The FETI family of domain decomposition methods for inequality-constrained quadratic programming: Application to contact problems with conforming and nonconforming interfaces. Comput. Methods Appl. Mech. Eng. 198, 1673–1683 (2009)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B.: Large-scale PDE-constrained optimization: an introduction. Springer, Berlin (2003)CrossRefGoogle Scholar
  4. 4.
    Biegler, L.T., Wächter, A.: SQP SAND strategies that link to existing modeling systems. In: Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B. (eds.) Large-Scale PDE-Constrained Optimization, vol. 30, pp. 199–217. Springer Verlag, Berlin (2003)Google Scholar
  5. 5.
    Biros, G., Ghattas, O.: Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I. SIAM J. Sci. Comput. 27, 687–713 (2005)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Choi, S.C.T. : Minimal residual methods for complex symmetric, skew symmetric, and skew hermitian systems. arXiv preprint arXiv:1304.6782 (2013)
  7. 7.
    Choi, Y.: Simultaneous analysis and design in PDE-constrained optimization. Ph.D. thesis, Stanford University (2012)Google Scholar
  8. 8.
    Day, D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23, 480–498 (2001)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dollar, H.S., Gould, N.I.M., Stoll, M., Wathen, A.J.: Preconditioning saddle-point systems with applications in optimization. SIAM J. Sci. Comput. 32, 249–270 (2010)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dollar, H.S., Wathen, A.J.: Approximate factorization constraint preconditioners for saddle-point matrics. SIAM J. Sci. Comput. 27, 1555–1572 (2006)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Draganescu, A., Soane, A.M.: Multigrid solution of a distributed optimal control problem constrained by the Stokes equations. Appl. Math. Comput. 219(10), 5622–5634 (2013)Google Scholar
  12. 12.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford Science Publications, Oxford (2005)Google Scholar
  13. 13.
    Farhat, C., Avery, P., Tezaur, R., Li, J.: FETI-DPH: a dual-primal domain decomposition method for acoustic scattering. J. Comput. Acoust. 13, 499–524 (2005)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Farhat, C., Lesoinne, M., LeTallec, P., Pierson, K., Rixen, D.: FETI-DP: a dual-primal unified FETI method. I. A faster alternative to the two-level FETI method. Int. J. Numer. Methods Eng. 50, 1523–1544 (2001)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Farhat, C., Roux, F.X.: A method of finite element tearing and interconnecting and its parallel solution algorithm. Int. J. Numer. Methods Eng. 32, 1205–1227 (1991)MATHCrossRefGoogle Scholar
  16. 16.
    Forsgren, A., Gill, P.E., Griffin, J.D.: Iterative solution of augmented systems arising in interior methods. SIAM J. Optim. 18, 666–690 (2007)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Freund, R.W.: Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J. Sci. Stat. Comput. 13, 425–448 (1992)MATHCrossRefGoogle Scholar
  18. 18.
    Freund, R.W., Nachtigal, N.M.: A new Krylov-subspace method for symmetric indefinite linear systems. In Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, pp. 1253–1256 (1994)Google Scholar
  19. 19.
    Gill, P.E., Gould, N., Murray, W., Saunders, M.A., Wright, M.H.: Range-space methods for convex quadratic programming. Technical report, Systems Optimization Laboratory, Stanford University, Stanford, CA (1982)Google Scholar
  20. 20.
    Gill, P.E., Gould, N.I.M., Murray, W., Saunders, M.A., Wright, M.H.: A weighted Gram–Schmidt method for convex quadratic programming. Math. Program. 30(2), 176–195 (1984)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Gill, P.E., Murray, W.: Numerical Methods for Constrained Optimization, vol. 1. Academic Press, London (1974)Google Scholar
  22. 22.
    Gill, P.E., Murray, W., Ponceleón, D.B., Saunders, M.A.: Preconditioners for indefinite systems arising in optimization. SIAM J. Matrix Anal. Appl. 13(1), 292–311 (1992)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21, 1300–1317 (2000)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Lahaye, D., De Gersem, H., Vandewalle, S., Hameyer, K.: Algebraic multigrid for complex symmetric systems. IEEE Trans. Magn. 36, 1535–1538 (2000)CrossRefGoogle Scholar
  25. 25.
    Lesoinne, M.: 19. a feti-dp corner selection algorithm for three-dimensional problems. In Domain Decomposition Methods in Science and Engineering, Cocoyoc, Mexico, Conference Presentation (2003)Google Scholar
  26. 26.
    Mandel, J., Tezaur, R.: Convergence of a substructuring method with lagrange multipliers. Numer. Math. 73, 473–487 (1996)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21(6), 1969–1972 (2000)MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–624 (1975)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Pearson, J.W., Stoll, M., Wathen, A.J.: Regularization-robust preconditioners for time-dependent PDE-constrained optimization problems. SIAM J. Matrix Anal. Appl. 33(4), 1126–1152 (2012)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Pearson, J.W., Wathen, A.J.: A new approximation of the Schur complement in preconditioners for PDE-constrained optimization. Numer. Linear Algebra Appl. 19, 816–829 (2012)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Prudencio, E., Byrd, R., Cai, X.C.: Parallel full space SQP Lagrange–Newton–Krylov–Schwarz algorithms for PDE-constrained optimization problems. SIAM J. Sci. Comput. 27, 1305–1328 (2006)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32, 271–298 (2010)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Reitzinger, S., Schreiber, U., Van Rienen, U.: Algebraic multigrid for complex symmetric matrices and applications. J. Comput. Appl. Math. 155(2), 405–421 (2003)MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)Google Scholar
  35. 35.
    Simoncini, V.: Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems. Comput. Optim. Appl. 53(2), 591–617 (2012)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Stoll, M., Wathen, A.: Combination preconditioning and the Bramble–Pasciak\(^{+}\) preconditioner. SIAM J. Matrix Anal. Appl. 3(2), 582–608 (2011)Google Scholar
  37. 37.
    Thorne, H.S.: Properties of linear systems in PDE-constrained optimization. Part I: Distributed control. Technical report, Rutherford Appleton Laboratory (2009)Google Scholar
  38. 38.
    Thorne, H.S.: Properties of linear systems in PDE-constrained optimization. Part II: Neumann boundary control. Technical report, Rutherford Appleton Laboratory (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Youngsoo Choi
    • 1
  • 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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