An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier–Stokes Equations

  • Peter Benner
  • Pawan Goyal


We discuss an interpolatory model reduction framework for quadratic-bilinear (QB) descriptor systems, arising especially from the semi-discretization of the Navier–Stokes equations. Several recent results indicate that directly applying interpolatory model reduction frameworks, developed for systems of ordinary differential equations, to descriptor systems, may lead to an unbounded error between the original and reduced-order systems, e.g., in the \(\mathscr {H}_2\)-norm, due to an inappropriate treatment of the polynomial part of the original system. Thus, the main goal of this article is to extend the recently studied interpolation-based optimal model reduction framework for QB ordinary differential equations (QBODEs) to aforementioned descriptor systems while ensuring bounded error. For this, we first aim at transforming the descriptor system into an equivalent ODE system by means of projectors for which standard model reduction techniques can be applied. Subsequently, we discuss how to construct optimal reduced systems corresponding to an equivalent ODE, without requiring explicit computation of the expensive projection used in the analysis. The efficiency of the proposed algorithm is illustrated by means of a numerical example, obtained via semi-discretization of the Navier–Stokes equations.



The authors would like to thank Dr. Jan Heiland for providing the lid-driven cavity model, used in the numerical experiment.


  1. 1.
    Ahmad, M.I., Benner, P., Goyal, P., Heiland, J.: Moment-matching based model reduction for Navier-Stokes type quadratic-bilinear descriptor systems. Z. Angew. Math. Mech (2017)Google Scholar
  2. 2.
    Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia, PA (2005)CrossRefGoogle Scholar
  3. 3.
    Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Autom. Control 53(10), 2237–2251 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baur, U., Benner, P., Feng, L.: Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Arch. Comput. Methods Eng. 21(4), 331–358 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Behr, M., Benner, P., Heiland, J.: Example setups of Navier-Stokes equations with control and observation: Spatial discretization and representation via linear-quadratic matrix coefficients (2017). arXiv:1707.08711
  6. 6.
    Benner, P., Breiten, T.: Interpolation-based \(\cal{H}_2\)-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl. 33(3), 859–885 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Benner, P., Breiten, T.: Krylov-subspace based model reduction of nonlinear circuit models using bilinear and quadratic-linear approximations. In: Günther, M., Bartel, A., Brunk, M., Schöps, S., Striebel, M. (eds.) Progress in Industrial Mathematics at ECMI 2010. Mathematics in Industry, vol. 17, pp. 153–159. Springer-Verlag, Berlin (2012)CrossRefGoogle Scholar
  8. 8.
    Benner, P., Breiten, T.: Two-sided moment matching methods for nonlinear model reduction. Preprint MPIMD/12-12, MPI Magdeburg (2012).
  9. 9.
    Benner, P., Breiten, T.: Two-sided projection methods for nonlinear model reduction. SIAM J. Sci. Comput. 37(2), B239–B260 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Benner, P., Damm, T.: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control Optim. 49(2), 686–711 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Benner, P., Goyal, P.: Multipoint interpolation of Volterra series and \(\cal{H}_2\)-model reduction for a family of bilinear descriptor systems. Syst. Control Lett. 97, 1–11 (2016)CrossRefGoogle Scholar
  12. 12.
    Benner, P., Goyal, P.: Balanced truncation model order reduction for quadratic-bilinear control systems (2017). arXiv:1705.00160
  13. 13.
    Benner, P., Goyal, P., Gugercin, S.: \(\cal{H}_2 \)-quasi-optimal model order reduction for quadratic-bilinear control systems (2016). arXiv:1610.03279
  14. 14.
    Benner, P., Mehrmann, V., Sorensen, D.C.: Dimension Reduction of Large-Scale Systems, vol. 45. Lecture Notes in Computational Science and Engineering. Springer, Heidelberg (2005)Google Scholar
  15. 15.
    Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Flagg, G., Gugercin, S.: Multipoint Volterra series interpolation and \(\cal{H}_2\) optimal model reduction of bilinear systems. SIAM J. Matrix Anal. Appl. 36(2), 549–579 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Goyal, P., Benner, P.: An iterative model order reduction scheme for a special class of bilinear descriptor systems appearing in constraint circuit simulation. In: ECCOMAS Congress 2016, VII European Congress on Computational Methods in Applied Sciences and Engineering, vol. 2, pp. 4196–4212 (2016)Google Scholar
  18. 18.
    Gu, C.: QLMOR: a projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 30(9), 1307–1320 (2011)CrossRefGoogle Scholar
  19. 19.
    Gugercin, S., Stykel, T., Wyatt, S.: Model reduction of descriptor systems by interpolatory projection methods. SIAM J. Sci. Comput. 35(5), B1010–B1033 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II-Stiff and Differential-Algebraic Problems, 2nd edn. Springer Series in Computational Mathematics. Springer (2002)Google Scholar
  21. 21.
    Heinkenschloss, M., Sorensen, D.C., Sun, K.: Balanced truncation model reduction for a class of descriptor systems with applications to the Oseen equations. SIAM J. Sci. Comput. 30(2), 1038–1063 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control. In: [14], pp. 261–306Google Scholar
  23. 23.
    Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40(2), 492–515 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kunisch, K., Volkwein, S.: Proper orthogonal decomposition for optimality systems. ESAIM Math. Model. Numer. Anal. 42(1), 1–23 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schilders, W.H.A., van der Vorst, H.A., Rommes, J.: Model Order Reduction: Theory, Research Aspects and Applications. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  27. 27.
    Zhang, L., Lam, J.: On \(H_2\) model reduction of bilinear systems. Automatica 38(2), 205–216 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

Personalised recommendations