Kolmogorov n-widths for linear dynamical systems


Kolmogorov n-widths and Hankel singular values are two commonly used concepts in model reduction. Here, we show that for the special case of linear time-invariant (LTI) dynamical systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov n-width of an LTI system equals its (n + 1)st Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov n-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov n-width.

This is a preview of subscription content, log in to check access.


  1. 1.

    Adamjan, V., Arov, D., Krein, M.: Analytic properties of Schmidt pairs for a Hankel operator and the generalized schur-Takagi problem. Math. USSR-Sbornik 15(1), 31–73 (1971). https://doi.org/10.1070/SM1971v015n01ABEH001531

    Article  MATH  Google Scholar 

  2. 2.

    Antoulas, A.C.: Approximation of large-scale dynamical systems. Advances in Design and Control. Society for Industrial and Applied Mathematics, Philadelphia. https://doi.org/10.1137/1.9780898718713 (2005)

  3. 3.

    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). https://doi.org/10.1007/s11831-014-9111-2

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Benner, P., Cohen, A., Ohlberger, M., Willcox, K.: Model Reduction and Approximation. SIAM, Philadelphia (2017). https://doi.org/10.1137/1.9781611974829

    Google Scholar 

  5. 5.

    Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Review 57(4), 483–531 (2015). https://doi.org/10.1137/130932715

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G., Wojtaszczyk, P.: Convergence rates for greedy algorithms in the reduced basis method. SIAM J. Math. Anal. 43, 1457–1472 (2011). https://doi.org/10.1137/100795772

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Bui-Thanh, T., Willcox, K., Ghattas, O.: Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput. 30 (6), 3270–3288 (2008). https://doi.org/10.1137/070694855

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Cagniart, N., Maday, Y., Stamm, B.: Model Order Reduction for Problems with Large Convection Effects, pp. 131–150. Springer International Publishing, Cham (2019). https://doi.org/10.1007/978-3-319-78325-3

    Google Scholar 

  9. 9.

    Chinesta, F., Huerta, A., Rozza, G., Willcox, K.: Model order reduction: a survey. Wiley Encyclopedia of Computational Mechanics (2016)

  10. 10.

    Constantine, P.G.: Active subspaces: Emerging ideas for dimension reduction in parameter studies. SIAM spotlights society for industrial and applied mathematics. https://doi.org/10.1137/1.9781611973860 (2015)

  11. 11.

    Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics. Springer , New York (1995). https://doi.org/10.1007/978-1-4612-4224-6

    Google Scholar 

  12. 12.

    Djouadi, S.M.: On the optimality of the proper orthogonal decomposition and balanced truncation. Proc. IEEE Conf. Decis. Control (3): 4221–4226 (2008). https://doi.org/10.1109/CDC.2008.4739458

  13. 13.

    Djouadi, S.M.: On the connection between balanced proper orthogonal decomposition, balanced truncation, and metric complexity theory for infinite dimensional systems. In: Am. Control Conf, pp. 4911–4916 (2010). https://doi.org/10.1109/ACC.2010.5530920

  14. 14.

    Francis, B.A.: A Course in H Control Theory. Lecture Notes in Control and Information Sciences. Springer, Berlin (1987). https://doi.org/10.1007/BFb0007371

    Google Scholar 

  15. 15.

    Fujimoto, K., Scherpen, J.M.A.: Balanced realization and model order reduction for nonlinear systems based on singular value analysis. SIAM J. Control Optim. 48 (7), 4591–4623 (2010). https://doi.org/10.1137/070695332

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L -error bounds. Int. J. Control 39(6), 1115–1193 (1984). https://doi.org/10.1080/00207178408933239

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Grepl, M., Patera, A.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM-Mathematical Modelling and Numerical Analysis (M2AN) 39(1), 157–181 (2005). https://doi.org/10.1051/m2an:2005006

    Article  MATH  Google Scholar 

  18. 18.

    Haasdonk, B.: Convergence rates of the POD–greedy method. ESAIM: Mathematical Modelling and Numerical Analysis 47(3), 859–873 (2013). https://doi.org/10.1051/m2an/2012045

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Mathematical Modelling and Numerical Analysis 42(2), 277–302 (2008). https://doi.org/10.1051/m2an:2008001

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Hesthaven, J.S., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics. Springer, Switzerland (2016). https://doi.org/10.1186/2190-5983-1-3

    Google Scholar 

  21. 21.

    Kolmogorov, A.: ÜBer die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. Math. 37(1), 107–110 (1936). https://doi.org/10.2307/1968691

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Lassila, T., Manzoni, A., Quarteroni, A., Rozza, G. Quarteroni, A., Rozza, G. (eds.): Model Order Reduction in Fluid Dynamics: Challenges and Perspectives. Springer International Publishing, Cham (2014)

  23. 23.

    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 Mathematique 335(3), 289–294 (2002). https://doi.org/10.1016/S1631-073X(02)02466-4

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Maday, Y., Patera, A.T., Turinici, G.: A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. 17(1), 437–446 (2002). https://doi.org/10.1023/A:1015145924517

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Pinkus, A.: N-Widths in Approximation Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1985). https://doi.org/10.1007/978-3-642-69894-1

    Google Scholar 

  26. 26.

    Prud’homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A., Turinici, G.: Reliable real-time solution of parameterized partial differential equations: Reduced-basis output bound methods. J. Fluids Eng. 124, 70–80 (2002). https://doi.org/10.1115/1.1448332

    Article  Google Scholar 

  27. 27.

    Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations: an Introduction. UNITEXT. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-15431-2

    Google Scholar 

  28. 28.

    Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1(1), 3 (2011). https://doi.org/10.1186/2190-5983-1-3

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Reed, M., Simon, B.: Analysis of Operators. Methods of Modern Mathematical Physics. Academic Press, Cambridge (1978)

    Google Scholar 

  30. 30.

    Wittmuess, P., Tarin, C., Keck, A., Arnold, E., Sawodny, O.: Parametric model order reduction via balanced truncation with taylor series representation. IEEE Trans. Automat. Contr. 61(11), 3438–3451 (2016). https://doi.org/10.1109/TAC.2016.2521361

    MathSciNet  Article  MATH  Google Scholar 

Download references


We thank Profs. Christopher Beattie, Volker Mehrmann, and Karen Willcox for their valuable comments.


The work of the first author was funded by the DFG Collaborative Research Center 910 Control of self-organizing nonlinear systems: theoretical methods and concepts of application. The work of the second author was financially supported in parts by NSF through Grant DMS-1720257 and by the Alexander von Humboldt Foundation.

Author information



Corresponding author

Correspondence to Benjamin Unger.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by: Anthony Nouy

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Unger, B., Gugercin, S. Kolmogorov n-widths for linear dynamical systems. Adv Comput Math 45, 2273–2286 (2019). https://doi.org/10.1007/s10444-019-09701-0

Download citation


  • Model reduction
  • Hankel singular values
  • Kolmogorov n-width
  • Hankel operator
  • Reduced basis method
  • Active subspaces

Mathematics Subject Classification (2010)

  • 37M99
  • 47B35
  • 65P99
  • 34A45
  • 35A35
  • 93C05