Skip to main content
Springer Nature Link
Account
Menu
Find a journal Publish with us Track your research
Search
Cart
  1. Home
  2. Advances in Computational Mathematics
  3. Article

Improved a posteriori error bounds for reduced port-Hamiltonian systems

  • Open access
  • Published: 11 September 2024
  • Volume 50, article number 100, (2024)
  • Cite this article
Download PDF

You have full access to this open access article

Advances in Computational Mathematics Aims and scope Submit manuscript
Improved a posteriori error bounds for reduced port-Hamiltonian systems
Download PDF
  • Johannes Rettberg1,
  • Dominik Wittwar2,
  • Patrick Buchfink2,
  • Robin Herkert2,
  • Jörg Fehr  ORCID: orcid.org/0000-0003-2850-14401 &
  • …
  • Bernard Haasdonk2 

  • 163 Accesses

  • Explore all metrics

Abstract

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem, to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid–structure interaction.

Article PDF

Download to read the full article text

Similar content being viewed by others

Theory and Implementation of Coupled Port-Hamiltonian Continuum and Lumped Parameter Models

Article Open access 13 July 2021

Model Reduction for a Port-Hamiltonian Formulation of the Euler Equations

Chapter © 2022

Structure-Preserving Interpolatory Model Reduction for Port-Hamiltonian Differential-Algebraic Systems

Chapter © 2022
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Availability of data and materials

The port-Hamiltonian system matrices are available at [57].

References

  1. Mehrmann, V., Unger, B.: Control of port-Hamiltonian differential-algebraic systems and applications. Acta Numer 32, 395–515 (2023). https://doi.org/10.1017/S0962492922000083

    Article  MathSciNet  Google Scholar 

  2. Duindam, V., Macchelli, A., Stramigioli, S., Bruyninckx, H.: Modeling and control of complex physical systems - the port-Hamiltonian approach., 1st edn. Springer, Berlin Heidelberg (2009). https://doi.org/10.1007/978-3-642-03196-0

  3. Rettberg, J., Wittwar, D., Buchfink, P., Brauchler, A., Ziegler, P., Fehr, J., Haasdonk, B.: Port-Hamiltonian fluid-structure interaction modeling and structure-preserving model order reduction of a classical guitar. Math. Comput. Model. Dyn. Syst. (2023). https://doi.org/10.1080/13873954.2023.2173238

    Article  Google Scholar 

  4. Haasdonk, B., Ohlberger, M.: Space-adaptive reduced basis simulation for time-dependent problems. In: MATHMOD, 6th Vienna International Conference on Mathematical Modelling, Vienna, Austria, pp. 718–723 (2009). https://www.argesim.org/fileadmin/user_upload_argesim/ARGESIM_Publications_OA/MATHMOD_Publications_OA/MATHMOD_2009_AR34_35/full_papers/184.pdf

  5. Haasdonk, B., Kleikamp, H., Ohlberger, M., Schindler, F., Wenzel, T.: A new certified hierarchical and adaptive RB-ML-ROM surrogate model for parametrized PDEs. SIAM J. Sci. Comput. 45(3), 1039–1065 (2023). https://doi.org/10.1137/22M1493318

    Article  MathSciNet  Google Scholar 

  6. Haasdonk, B., Ohlberger, M.: Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition. MCMDS 17(2), 145–161 (2011). https://doi.org/10.1080/13873954.2010.514703

    Article  MathSciNet  Google Scholar 

  7. Hain, S., Ohlberger, M., Radic, M., Urban, K.: A hierarchical a posteriori error estimator for the reduced basis method. Adv. Comput. Math. 45(5), 2191–2214 (2019). https://doi.org/10.1007/s10444-019-09675-z

    Article  MathSciNet  Google Scholar 

  8. Schmidt, A., Wittwar, D., Haasdonk, B.: Rigorous and effective a-posteriori error bounds for nonlinear problems-application to RB methods. Adv. Comput. Math. 46(2), 32 (2020). https://doi.org/10.1007/s10444-020-09741-x

    Article  MathSciNet  Google Scholar 

  9. Mehl, C., Mehrmann, V., Wojtylak, M.: Linear algebra properties of dissipative Hamiltonian descriptor systems. SIAM J. Matrix Anal. Appl. 39(3), 1489–1519 (2018). https://doi.org/10.1137/18M1164275

    Article  MathSciNet  Google Scholar 

  10. Willems, J.C.: Dissipative dynamical systems part i: General theory. Arch. Ration. Mech. Anal. 45(5), 321–351 (1972). https://doi.org/10.1007/BF00276493

    Article  Google Scholar 

  11. Cherifi, K., Gernandt, H., Hinsen, D.: The difference between port-Hamiltonian, passive and positive real descriptor systems. Mathematics of Control, Signals, and Systems (2023). https://doi.org/10.1007/s00498-023-00373-2

  12. Beattie, C.A., Mehrmann, V., Van Dooren, P.: Robust port-Hamiltonian representations of passive systems. Automatica 100, 182–186 (2019). https://doi.org/10.1016/j.automatica.2018.11.013

    Article  MathSciNet  Google Scholar 

  13. Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory I. Springer, Berlin Heidelberg (2005). https://doi.org/10.1007/b137541

  14. van der Schaft, A., Jeltsema, D.: Port-Hamiltonian systems theory: an introductory overview. Fndn Trends in Syst Contr 1(2–3), 173–378 (2014). https://doi.org/10.1561/2600000002

    Article  Google Scholar 

  15. Gillis, N., Sharma, P.: Finding the nearest positive-real system. SIAM J. Numer. Anal. 56(2), 1022–1047 (2018). https://doi.org/10.1137/17M1137176

    Article  MathSciNet  Google Scholar 

  16. Rashad, R., Califano, F., van der Schaft, A., Stramigioli, S.: Twenty years of distributed port-Hamiltonian systems: a literature review. IMA J. Math. Control. Inf. 37(4), 1400–1422 (2020). https://doi.org/10.1093/imamci/dnaa018

    Article  MathSciNet  Google Scholar 

  17. Goyal, P., Pontes Duff, I., Benner, P.: Guaranteed stable quadratic models and their applications in SINDy and operator inference. arXiv Preprint (2023). https://doi.org/10.48550/arXiv.2308.13819

  18. Gillis, N., Sharma, P.: On computing the distance to stability for matrices using linear dissipative Hamiltonian systems. Automatica 85, 113–121 (2017). https://doi.org/10.1016/j.automatica.2017.07.047

    Article  MathSciNet  Google Scholar 

  19. Gillis, N., Mehrmann, V., Sharma, P.: Computing the nearest stable matrix pairs. Numerical Linear Algebra with Applications 25(5), 2153 (2018). https://doi.org/10.1002/nla.2153.e2153nla.2153

  20. Mehl, C., Mehrmann, V., Sharma, P.: Stability radii for linear Hamiltonian systems with dissipation under structure-preserving perturbations. SIAM J. Matrix Anal. Appl. 37(4), 1625–1654 (2016). https://doi.org/10.1137/16M1067330

    Article  MathSciNet  Google Scholar 

  21. Mehl, C., Mehrmann, V., Sharma, P.: Stability radii for real linear Hamiltonian systems with perturbed dissipation. BIT Numer. Math. 57(3), 811–843 (2017). https://doi.org/10.1007/s10543-017-0654-0

    Article  MathSciNet  Google Scholar 

  22. Liljegren-Sailer, B.: On port-Hamiltonian modeling and structure-preserving model reduction. Doctoral thesis, Universität Trier (2020). https://nbn-resolving.org/urn:nbn:de:hbz:385-1-14498

  23. Beattie, C., Gugercin, S., Mehrmann, V.: Structured Dynamical Systems. In: Beattie, C., Benner, P., Embree, M., Gugercin, S., Lefteriu, S. (eds.) Structure-preserving interpolatory model reduction for port-Hamiltonian differential-algebraic systems, pp. 235–254. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-95157-3_13

  24. Gugercin, S., Polyuga, R.V., Beattie, C., van der Schaft, A.: Structure-preserving tangential interpolation for model reduction of port-Hamiltonian systems. Automatica 48(9), 1963–1974 (2012). https://doi.org/10.1016/j.automatica.2012.05.052

    Article  MathSciNet  Google Scholar 

  25. Breiten, T., Unger, B.: Passivity preserving model reduction via spectral factorization. Automatica 142, 110368 (2022). https://doi.org/10.1016/j.automatica.2022.110368

    Article  MathSciNet  Google Scholar 

  26. Polyuga, R.V., van der Schaft, A.: Structure preserving model reduction of port-Hamiltonian systems by moment matching at infinity. Automatica 46(4), 665–672 (2010). https://doi.org/10.1016/j.automatica.2010.01.018

    Article  MathSciNet  Google Scholar 

  27. Ionescu, T.C., Astolfi, A.: Families of moment matching based, structure preserving approximations for linear port Hamiltonian systems. Automatica 49(8), 2424–2434 (2013). https://doi.org/10.1016/j.automatica.2013.05.006

    Article  MathSciNet  Google Scholar 

  28. Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N., Mehrmann, V.: On structure-preserving model reduction for damped wave propagation in transport networks. SIAM J. Sci. Comput. 40(1), 331–365 (2018). https://doi.org/10.1137/17M1125303

    Article  MathSciNet  Google Scholar 

  29. Moser, T., Lohmann, B.: A new Riemannian framework for efficient H2-optimal model reduction of port-Hamiltonian systems. In: 2020 59th IEEE Conference on Decision and Control (CDC), pp. 5043–5049 (2020). https://doi.org/10.1109/CDC42340.2020.9304134

  30. Schwerdtner, P., Voigt, M.: SOBMOR: structured optimization-based model order reduction. SIAM J. Sci. Comput. 45(2), 502–529 (2023). https://doi.org/10.1137/20M1380235

    Article  MathSciNet  Google Scholar 

  31. Polyuga, R.V., van der Schaft, A.J.: Effort- and flow-constraint reduction methods for structure preserving model reduction of port-Hamiltonian systems. Syst. Contr. Lett. 61(3), 412–421 (2012). https://doi.org/10.1016/j.sysconle.2011.12.008

    Article  MathSciNet  Google Scholar 

  32. Hauschild, S.-A., Marheineke, N., Mehrmann, V.: Model reduction techniques for linear constant coefficient port-Hamiltonian differential-algebraic systems. Control and Cybernetics 48, 125–152 (2019). https://doi.org/10.48550/arXiv.1901.10242

  33. Guiver, C., Opmeer, M.R.: Error bounds in the gap metric for dissipative balanced approximations. Linear Algebra Appl. 439(12), 3659–3698 (2013). https://doi.org/10.1016/j.laa.2013.09.032

    Article  MathSciNet  Google Scholar 

  34. Breiten, T., Morandin, R., Schulze, P.: Error bounds for port-Hamiltonian model and controller reduction based on system balancing. Comp. Math. Appl. 116, 100–115 (2022). https://doi.org/10.1016/j.camwa.2021.07.022

    Article  MathSciNet  Google Scholar 

  35. Borja, P., Scherpen, J.M.A., Fujimoto, K.: Extended balancing of continuous LTI systems: a structure-preserving approach. IEEE Trans. Autom. Control 68(1), 257–271 (2023). https://doi.org/10.1109/TAC.2021.3138645

    Article  MathSciNet  Google Scholar 

  36. Schulze, P., Unger, B.: Model reduction for linear systems with low-rank switching. SIAM J. Control. Optim. 56(6), 4365–4384 (2018). https://doi.org/10.1137/18M1167887

    Article  MathSciNet  Google Scholar 

  37. Schulze, P.: Energy-based model reduction of transport-dominated phenomena. Doctoral thesis, Technische Universität Berlin (2023). https://doi.org/10.14279/depositonce-17843

  38. Beattie, C., Gugercin, S.: Structure-preserving model reduction for nonlinear port-Hamiltonian systems. Proceed. IEEE Conf. Decision. Contr. 6564–6569 (2011). https://doi.org/10.1109/CDC.2011.6161504

  39. Chaturantabut, S., Beattie, C., Gugercin, S.: Structure-preserving model reduction for nonlinear port-Hamiltonian systems. SIAM J. Sci. Comput. 38(5), 837–865 (2016). https://doi.org/10.1137/15M1055085

    Article  MathSciNet  Google Scholar 

  40. Wolf, T., Lohmann, B., Eid, R., Kotyczka, P.: Passivity and structure preserving order reduction of linear port-Hamiltonian systems using Krylov subspaces. Eur. J. Control. 16(4), 401–406 (2010). https://doi.org/10.3166/EJC.16.401-406

    Article  MathSciNet  Google Scholar 

  41. Nguyen, N.-C., Veroy, K., Patera, A.T.: Certified real-time solution of parametrized partial differential equations. In: Yip, S. (ed.) Handbook of Materials Modeling, pp. 1529–1564. Springer, Dordrecht (2005). https://doi.org/10.1007/978-1-4020-3286-8_76

  42. Veroy, K., Patera, A.T.: Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47(8–9), 773–788 (2005). https://doi.org/10.1002/fld.867

    Article  MathSciNet  Google Scholar 

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

  44. Knezevic, D.J., Nguyen, N.-C., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for the parametrized unsteady Boussinesq equations. Math. Models Methods Appl. Sci. 21(07), 1415–1442 (2011). https://doi.org/10.1142/S0218202511005441

    Article  MathSciNet  Google Scholar 

  45. Grepl, M.A., Maday, Y., Nguyen, N.-C., Patera, A.T.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41(3), 575–605 (2007). https://doi.org/10.1051/m2an:2007031

  46. Grunert, D., Fehr, J., Haasdonk, B.: Well-scaled, a-posteriori error estimation for model order reduction of large second-order mechanical systems. ZAMM - J. Appl. Math. Mech. 100(8), 201900186 (2020). https://doi.org/10.1002/zamm.201900186

    Article  MathSciNet  Google Scholar 

  47. Glas, S., Patera, A.T., Urban, K.: A reduced basis method for the wave equation. Int. J. Comput. Fluid Dyn. 34(2), 139–146 (2020). https://doi.org/10.1080/10618562.2019.1686486

    Article  MathSciNet  Google Scholar 

  48. Stahl, N., Liljegren-Sailer, B., Marheineke, N.: Certified reduced basis method for the damped wave equations on networks. IFAC-PapersOnLine 55(20), 289–294 (2022). https://doi.org/10.1016/j.ifacol.2022.09.110

    Article  Google Scholar 

  49. Antoulas, A.C., Benner, P., Feng, L.: Model reduction by iterative error system approximation. MCMDS 24(2), 103–118 (2018). https://doi.org/10.1080/13873954.2018.1427116

    Article  MathSciNet  Google Scholar 

  50. Feng, L., Lombardi, L., Antonini, G., Benner, P.: Multi-fidelity error estimation accelerates greedy model reduction of complex dynamical systems. Int. J. Numer. Meth. Eng. 124(23), 5312–5333 (2023). https://doi.org/10.1002/nme.7348

    Article  MathSciNet  Google Scholar 

  51. Güttel, S., Nakatsukasa, Y.: Scaled and squared subdiagonal Padé approximation for the matrix exponential. SIAM J. Matrix Anal. Appl. 37(1), 145–170 (2016). https://doi.org/10.1137/15M1027553

    Article  MathSciNet  Google Scholar 

  52. Söderlind, G.: The logarithmic norm. history and modern theory. BIT Numerical Mathematics 46(3), 631–652 (2006). https://doi.org/10.1007/s10543-006-0069-9

  53. Desoer, C.A., Vidyasagar, M.: Feedback Systems. Society for Industrial and Applied Mathematics, Philadelphia (2009). https://doi.org/10.1137/1.9780898719055

  54. Wittwar, D.: Approximation with matrix-valued kernels and highly effective error estimators for reduced basis approximations. Dissertation. University library Stuttgart, Stuttgart (2022). https://doi.org/10.18419/opus-12526

  55. Hackbusch, W.: Hierarchical matrices: algorithms and analysis. Springer, Berlin Heidelberg (2015). https://doi.org/10.1007/978-3-662-47324-5

  56. Gugercin, S., Antoulas, A.C.: A survey of model reduction by balanced truncation and some new results. Int. J. Control 77(8), 748–766 (2004). https://doi.org/10.1080/00207170410001713448

    Article  MathSciNet  Google Scholar 

  57. Rettberg, J., Wittwar, D., Buchfink, P., Brauchler, A., Ziegler, P., Fehr, J., Haasdonk, B.: Replication data for: port-Hamiltonian fluid-structure interaction modeling and structure-preserving model order reduction of a classical guitar. DaRUS (2023). https://doi.org/10.18419/darus-3248

  58. Rettberg, J., Wittwar, D., Herkert, R.: Softwarepackage CCMOR2. DaRUS (2023). https://doi.org/10.18419/darus-3839

  59. Peng, L., Mohseni, K.: Symplectic model reduction of Hamiltonian systems. SIAM J. Sci. Comput. 38(1), 1–27 (2016). https://doi.org/10.1137/140978922

    Article  MathSciNet  Google Scholar 

  60. Buchfink, P., Bhatt, A., Haasdonk, B.: Symplectic model order reduction with non-orthonormal bases. Math. Comput. Appl. 24(2) (2019). https://doi.org/10.3390/mca24020043

  61. Chellappa, S., Feng, L., de la Rubia, V., Benner, P.: Inf-sup-constant-free state error estimator for model order reduction of parametric systems in electromagnetics. IEEE Trans. Microw. Theory Tech. 71(11), 4762–4777 (2023). https://doi.org/10.1109/TMTT.2023.3288642

    Article  Google Scholar 

  62. Buchfink, P., Haasdonk, B., Rave, S.: PSD-Greedy basis generation for structure-preserving model order reduction of Hamiltonian systems. In: Frolkovič, P., Mikula, K., Ševčovič, D. (eds.) Proceedings of the Conference Algoritmy 2020, pp. 151–160. Vydavateľstvo SPEKTRUM, Vysoke Tatry, Podbanske (2020). http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1577/829

Download references

Acknowledgements

The authors thank the anonymous reviewers for their insightful comments that helped to improve the paper.

Funding

Open Access funding enabled and organized by Projekt DEAL. This work is supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project No. 314733389, and under Germany’s Excellence Strategy - EXC 2075 - 390740016. We acknowledge the support by the Stuttgart Center for Simulation Science (SimTech).

Author information

Authors and Affiliations

  1. Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569, Stuttgart, Baden-Württemberg, Germany

    Johannes Rettberg & Jörg Fehr

  2. Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Baden-Württemberg, Germany

    Dominik Wittwar, Patrick Buchfink, Robin Herkert & Bernard Haasdonk

Authors
  1. Johannes Rettberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dominik Wittwar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Patrick Buchfink
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Robin Herkert
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jörg Fehr
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Bernard Haasdonk
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Conceptualization: J. Rettberg, D. Wittwar, P. Buchfink, J. Fehr, B. Haasdonk. Methodology: J. Rettberg, D. Wittwar, P. Buchfink, J. Fehr, B. Haasdonk. Formal analysis and investigation: J. Rettberg, D. Wittwar, P. Buchfink, R. Herkert, J. Fehr, B. Haasdonk. Software: J. Rettberg, D. Wittwar. Writing—original draft preparation: J. Rettberg, D. Wittwar. Writing—review and editing: J. Rettberg, D. Wittwar, P. Buchfink, R. Herkert, J. Fehr, B. Haasdonk. Funding acquisition: J. Fehr, B. Haasdonk, P. Buchfink.

Corresponding author

Correspondence to Jörg Fehr.

Ethics declarations

Ethics approval

The research does not involve human participants or animals.

Conflict of interest

The authors declare no competing interests.

Additional information

Communicated by: Tobias Breiten

Publisher's Note

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

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rettberg, J., Wittwar, D., Buchfink, P. et al. Improved a posteriori error bounds for reduced port-Hamiltonian systems. Adv Comput Math 50, 100 (2024). https://doi.org/10.1007/s10444-024-10195-8

Download citation

  • Received: 31 March 2023

  • Accepted: 26 August 2024

  • Published: 11 September 2024

  • DOI: https://doi.org/10.1007/s10444-024-10195-8

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Structure-preserving model order reduction
  • A posteriori error control
  • Port-Hamiltonian system
  • Fluid–structure interaction

Mathematics Subject Classification (2010)

  • 65L70
  • 34C20
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us
  • Track your research

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Journal finder
  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support
  • Cancel contracts here

Not affiliated

Springer Nature

© 2024 Springer Nature