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.
Similar content being viewed by others
References
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)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)
Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B.: Large-scale PDE-constrained optimization: an introduction. Springer, Berlin (2003)
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)
Biros, G., Ghattas, O.: Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I. SIAM J. Sci. Comput. 27, 687–713 (2005)
Choi, S.C.T. : Minimal residual methods for complex symmetric, skew symmetric, and skew hermitian systems. arXiv preprint arXiv:1304.6782 (2013)
Choi, Y.: Simultaneous analysis and design in PDE-constrained optimization. Ph.D. thesis, Stanford University (2012)
Day, D., Heroux, M.A.: Solving complex-valued linear systems via equivalent real formulations. SIAM J. Sci. Comput. 23, 480–498 (2001)
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)
Dollar, H.S., Wathen, A.J.: Approximate factorization constraint preconditioners for saddle-point matrics. SIAM J. Sci. Comput. 27, 1555–1572 (2006)
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)
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)
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)
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)
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)
Forsgren, A., Gill, P.E., Griffin, J.D.: Iterative solution of augmented systems arising in interior methods. SIAM J. Optim. 18, 666–690 (2007)
Freund, R.W.: Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices. SIAM J. Sci. Stat. Comput. 13, 425–448 (1992)
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)
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)
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)
Gill, P.E., Murray, W.: Numerical Methods for Constrained Optimization, vol. 1. Academic Press, London (1974)
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)
Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21, 1300–1317 (2000)
Lahaye, D., De Gersem, H., Vandewalle, S., Hameyer, K.: Algebraic multigrid for complex symmetric systems. IEEE Trans. Magn. 36, 1535–1538 (2000)
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)
Mandel, J., Tezaur, R.: Convergence of a substructuring method with lagrange multipliers. Numer. Math. 73, 473–487 (1996)
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)
Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–624 (1975)
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)
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)
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)
Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32, 271–298 (2010)
Reitzinger, S., Schreiber, U., Van Rienen, U.: Algebraic multigrid for complex symmetric matrices and applications. J. Comput. Appl. Math. 155(2), 405–421 (2003)
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)
Simoncini, V.: Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems. Comput. Optim. Appl. 53(2), 591–617 (2012)
Stoll, M., Wathen, A.: Combination preconditioning and the Bramble–Pasciak\(^{+}\) preconditioner. SIAM J. Matrix Anal. Appl. 3(2), 582–608 (2011)
Thorne, H.S.: Properties of linear systems in PDE-constrained optimization. Part I: Distributed control. Technical report, Rutherford Appleton Laboratory (2009)
Thorne, H.S.: Properties of linear systems in PDE-constrained optimization. Part II: Neumann boundary control. Technical report, Rutherford Appleton Laboratory (2009)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-014-9976-0
Keywords
- PDE-constrained optimization
- Schur complement
- Poisson operator
- FETI
- Range-space method
- Full-space method
- Distributed optimal control