Skip to main content
Springer Nature Link
Log in

An Adaptive Rational Block Lanczos-Type Algorithm for Model Reduction of Large Scale Dynamical Systems

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Multipoint moment matching based methods are considered as powerful methods for model-order reduction problems. They are related to rational Krylov subspaces (classical or block ones) and are based on the selection of some interpolation points which is the major problem for these methods. In this work, an adaptive rational block Lanczos-type algorithm is proposed and applied for model order reduction of dynamical multi-input and multi-output linear time independent dynamical systems. We give some algebraic properties of the proposed algorithm and derive an explicit formulation of the error between the original and the reduced transfer functions. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive rational block Lanczos-type process.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Anderson, B.O., Antoulas, A.C.: Rational interpolation and state-variable realizations. Linear Algebra Appl. 137–138, 479–509 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antoulas, A.C., Sorensen, D.C.: Approximation of large-scale dynamical systems: an overview. Int. J. Appl. Math. Comput. Sci. 11(5), 1093–1121 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Antoulas, A.C., Sorensen, D.C., Gugercin, S.: A survey of model reduction methods for large scale systems. Contemp. Math. 280, 193–219 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Z., Day, D., Ye, Q.: ABLE: An adaptive block Lanczos method for non-Hermitian eigenvalue problems. SIAM J. Matrix Anal. Appl. 20, 1060–1082 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bai, Z., Slone, R.D., Smith, W.T., Ye, Q.: Error bounds for reduced system model by Padé approximation via the Lanczos process. IEEE Trans. Comput. Aided Design. 18(2), 133–141 (1999)

    Article  Google Scholar 

  6. Baker Jr, G.A.: Essentials of Padé approximation. Academic Press, New York (1975)

    MATH  Google Scholar 

  7. Beattie, C., Gugercin, S.: Krylov-based minimization for optimal \({\fancyscript {H}}_{2}\) model reduction. In: Proceedings of the 46th IEEE Conference on Decision and Control., 4385–4390 (2007)

  8. Beattie, C., Gugercin, S.: Realization-independent \({\fancyscript {H}}_{2}\) approximation. In: 51st IEEE Conference on Decision and Control. (2012)

  9. Bodendiek, A., Bollhöfer, M.: Adaptive-order rational Arnoldi-type methods in computational electromagnetism, BIT Numer. Math., 1-24 (2013)

  10. Druskin, V., Simoncini, V.: Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst. Control Lett. 60(8), 546–560 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feldman, P., Freund, R.W.: Efficient linear circuit analysis by Padé approximation via a Lanczos method. IEEE Trans. Comput. Aided Des. 14, 639–649 (1995)

    Article  Google Scholar 

  12. Frangos, M., Jaimoukha, I.M.: Adaptive rational Krylov algorithms for model reduction, In Proceedings of the European Control Conference, 4179–4186 (2007)

  13. Frangos, M., Jaimoukha, I.M.: Adaptive rational interpolation: Arnoldi and Lanczos-like equations. Eur. J. Control 14(4), 342–354 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gallivan, K., Grimme, E., Van Dooren, P.: Padé Approximation of Large-Scale Dynamic Systems with Lanczos Methods. In Proceedings of the 33rd IEEE Conference on Decision and Control, 443–448 (1994)

  15. Gallivan, K., Grimme, E., Van Dooren, P.: A rational Lanczos algorithm for model reduction. Num. Algebra 12, 33–63 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gallivan, K., Vandendorpe, A., Van Dooren, P.: Sylvester equations and projection-based model reduction. J. Comp. Appl. Math. 162, 213–229 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Glover, K.: All optimal Hankel-norm approximation of linear multivariable systems and their \(L^{\infty }\)-error bounds. Int. J. Control 39(6), 1115–1193 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gragg, W.B., Lindquist, A.: On the partial realization problem. Linear Algebra Appl. 50, 277–319 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grimme, E.: Krylov Projection methods for model reduction, PhD thesis, ECE Department, University of Illinois, Urbana-Champaign (1997)

  20. Grimme, E., Gallivan, K., Van Dooren, P.: Rational Lanczos algorithm for model reduction II: interpolation point selection, Technical report, University of Illinois at Urbana Champaign (1998)

  21. Gugercin, S., Antoulas, A.C.: Model reduction of large scale systems by least squares. Linear Algebra Appl. 415(2–3), 290–321 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gugercin, S., Antoulas, A.C., Beattie, C.: Rational Krylov methods for optimal \({\fancyscript {H}}_{2}\) model reduction, submitted for publication (2006)

  23. Heyouni, M., Jbilou, K., Messaoudi, A., Tabaa, K.: Model reduction in large scale MIMO dynamical systems via the block Lanczos method. Comput. Appl. Math. 27(2), 211–236 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand. 45, 225–280 (1950)

    Article  MathSciNet  Google Scholar 

  25. Lee, H.J., Chu, C.C., Feng, W.S.: An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems. Linear Algebra Appl. 415, 235–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nguyen, T.V., Li, JR.: Multipoint Padé Approximation using a rational block Lanczos algorithm, In: Proceedings of IEEE International Conference on Computer Aided Design, 72–75 (1997)

  27. Panzer, H., Jaensch, S., Wolf, T., Lohmann, B.: A greedy rational Krylov method for \({\fancyscript {H}}_{2}\)-pseudo-optimal model order reduction with preservation of stability. In: American Control Conference (ACC), 5532–5537 (2013)

  28. Penzl, T.: LYAPACK-A MATLAB toolbox for large Lyapunov and Riccati equations, model reduction problems, and linear-quadratic optimal control problems, http://www.tu-chemintz.de/sfb393/lyapack

  29. Ruhe, A.: Rational Krylov algorithms for nonsymmetric eigenvalue problems II: matrix pairs. Linear Algebra Appl. 197, 283–295 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Soppa, A.: Krylov-Unterraum basierte Modellreduktion zur Simulation von Werkzeugmaschinen, PhD thesis, University Braunschweig (2011)

  31. Van Dooren, P.: The Lanczos algorithm and Padé approximations. In: Short Course, Benelux Meeting on Systems and Control (1995)

  32. Yousuff, A., Skelton, R.E., et al.: Covariance equivalent realizations with applications to model reduction of large-scale systems. In: Leondes, C.T. (ed.) Control and Dynamic Systems, vol. 22, pp. 273–348. Academic Press, Waltham (1985)

    Google Scholar 

Download references

Acknowledgments

We would like to thank the two referees for their helpful remarks and valuable suggestions. We also thank Serkan Gugercin for providing us with the matlab program of IRKA.

Author information

Authors and Affiliations

  1. Faculté des Sciences et Techniques-Gueliz, Laboratoire de Mathématiques Appliquées et Informatique, Marrakech, Morocco

    H. Barkouki & A. H. Bentbib

  2. Université du Littoral, Côte d’Opale, batiment H. Poincarré, 50 rue F. Buisson, 62280, Calais Cedex, France

    H. Barkouki & K. Jbilou

Authors
  1. H. Barkouki
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. A. H. Bentbib
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. K. Jbilou
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to K. Jbilou.

Ethics declarations

Conflicts of interest

The Authors declare that there is no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barkouki, H., Bentbib, A.H. & Jbilou, K. An Adaptive Rational Block Lanczos-Type Algorithm for Model Reduction of Large Scale Dynamical Systems. J Sci Comput 67, 221–236 (2016). https://doi.org/10.1007/s10915-015-0077-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-015-0077-5

Keywords

Mathematics Subject Classification