Estimation of Regularization Parameters in Elliptic Optimal Control Problems by POD Model Reduction

  • Martin Kahlbacher
  • Stefan Volkwein
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 312)


In this article parameter estimation problems for a nonlinear elliptic problem are considered. Using Tikhonov regularization techniques the identification problems are formulated in terms of optimal control problems which are solved numerically by an augmented Lagrangian method combined with a globalized sequential quadratic programming algorithm. For the discretization of the partial differential equations a Galerkin scheme based on proper orthogonal decomposition (POD) is utilized, which leads to a fast optimization solver. This method is utilized in a bilevel optimization problem to determine the parameters for the Tikhonov regularization. Numerical examples illustrate the efficiency of the proposed approach.


  1. 1.
    Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus de’l Académie des Sciences Paris I(339), 667–672 (2004)MathSciNetMATHGoogle Scholar
  2. 2.
    Bertsekas, D.P.: Constrained Optimization and Lagrange Multipliers. Academic Press, New York (1982)MATHGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)MATHGoogle Scholar
  4. 4.
    Casas, E., Raymond, J.-P., Zidani, H.: Optimal control problem governed by semilinear elliptic equations with integral control constraints and pointwise state constraint. In: Desch, W., et al. (eds.) Control and estimation of distributed parameter systems. International conference in Vorau, Austria, July 14-20. Birkhauser, Basel (1996); ISNM, Int. Ser. Numer. Math. 126, 89–102 (1998)Google Scholar
  5. 5.
    Fukuda, K.: Introduction to Statistical Recognition. Academic Press, New York (1990)Google Scholar
  6. 6.
    Hintermüller, M.: A primal-dual active set algorithm for bilaterally control constrainted optimal control problems. Quarterly of Applied Mathematics 61, 131–160 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. In: Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge (1996)Google Scholar
  8. 8.
    Ito, K., Ravindran, S.S.: Reduced basis method for unsteady viscous flows. Int. J. of Comp. Fluid Dynamics 15, 97–113 (2001)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Lall, S., Marsden, J.E., Glavaski, S.: Empirical model reduction of controlled nonlinear systems. In: Proceedings of the IFAC Congress, vol. F, pp. 473–478 (1999)Google Scholar
  10. 10.
    Ly, H.V., Tran, H.T.: Modelling and control of physical processes using proper orthogonal decomposition. Mathematical and Computer Modeling 33, 223–236 (2001)CrossRefMATHGoogle Scholar
  11. 11.
    Kahlbacher, M., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems. Discussiones Mathematicae: Differential Inclusions, Control and Optimization 27, 95–117 (2007)MathSciNetMATHGoogle Scholar
  12. 12.
    Kahlbacher, M., Volkwein, S.: Model reduction by proper orthogonal decomposition for estimation of scalar parameters in elliptic PDEs. In: Wesseling, P., Onate, E., Periaux, J. (eds.) Proceedings of ECCOMAS CFD, Egmont aan Zee (2006)Google Scholar
  13. 13.
    Kahlbacher, M., Volkwein, S.: Estimation of diffusion coefficients in a scalar Ginzburg-Landau equation by using model reduction. Submitted (2007),
  14. 14.
    Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40, 492–515 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Sirovich, L.: Turbulence and the dynamics of coherent structures, parts I-III. Quarterly of Applied Mathematics XLV, 561–590 (1987)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Machiels, L., Maday, Y., Patera, A.T.: Output bounds for reduced-order approximations of elliptic partial differential equations. Computer Methods in Applied Mechanics and Engineering 190, 3413–3426 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Maday, Y., Patera, A.T., Turinici, G.: Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. Comptes Rendus de’l Académie des Sciences Paris I(335), 289–294 (2002)MathSciNetMATHGoogle Scholar
  18. 18.
    Maday, Y., Rønquist, E.M.: A reduced-basis element method. Journal of Scientific Computing 17, 1–4 (2002)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Rowley, C.W.: Model reduction for fluids, using balanced proper orthogonal decomposition. International Journal of Bifurcation and Chaos 15, 997–1013 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Vogel, C.R.: Computational Methods for Inverse Problems, Philadlphia. SIAM Frontiers in Applied Mathematics (2002)Google Scholar
  21. 21.
    Volkwein, S.: Model Reduction using Proper Orthogonal Decomposition. Lecture Notes, Institute of Mathematics and Scientific Computing, University of Graz,
  22. 22.
    Volkwein, S., Hepberger, A.: Impedance Identification by POD Model Reduction Techniques (2008) (submitted),
  23. 23.
    Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. American Institute of Aeronautics and Astronautics (AIAA) 40, 2323–2330 (2002)CrossRefGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2009

Authors and Affiliations

  • Martin Kahlbacher
    • 1
  • Stefan Volkwein
    • 1
  1. 1.Institute for Mathematics and Scientific ComputingUniversity of GrazGrazAustria

Personalised recommendations