Skip to main content
SpringerLink
Log in

Krylov projection methods for linear Hamiltonian systems

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We study geometric properties of Krylov projection methods for large and sparse linear Hamiltonian systems. We consider in particular energy-preservation. We discuss the connection to structure preserving model reduction. We illustrate the performance of the methods by applying them to Hamiltonian PDEs.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Van der Schaft, A., Jeltsema, D.: Port-Hamiltonian systems theory: an introductory overview. Foundations and Trends in Systems and Control 1(2–3), 173 (2014)

    Article  Google Scholar 

  2. Feng, K., Qin, M.Z.: .. In: Numerical Methods for Partial Differential Equations, pp. 1–37. Springer (1987)

  3. McLachlan, R.: Symplectic integration of Hamiltonian wave equations. Numer. Math. 66(1), 465 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Marsden, J.E., Weinstein, A.: The Hamiltonian structure of the Maxwell-Vlasov equations. Physica D: Nonlinear Phenomena 4(3), 394 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sun, Y., Tse, P.: Symplectic and multisymplectic methods for Maxwell’s equations. J. Comput. Phys. 230(5), 2076 (2010). https://doi.org/10.1016/j.jcp.2010.12.006

    Article  MathSciNet  MATH  Google Scholar 

  6. Taylor Michael, E.: Partial Differential Equations, I. Basic Theory, vol. 115. Springer, New York (2011)

    Book  Google Scholar 

  7. Richtmyer, R.D., Morton, K.W.: Difference methods for initial-value problems. Interscience Publishers John Wiley & Sons, Inc., Academia Publishing House of the Czechoslovak Acad (1967)

  8. LaBudde, R.A., Greenspan, D.: Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion. Numer. Math. 25(4), 323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  9. McLachlan, R.I., Quispel, G., Robidoux, N.: Geometric integration using discrete gradients. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 357(1754), 1021 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brugnano, L., Iavernaro, F., Trigiante, D.: Hamiltonian boundary value methods (energy preserving discrete line integral methods). Journal of Numerical Analysis Industrial Applied Mathematics 5(1), 17 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D., Owren, B., O’neale, D., Quispel, G.: Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method. J. Comput. Phys. 231(20), 6770 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ge, Z., Marsden, J.: Lie-Poisson Hamilton-Jacobi theory and Lie-Poisson integrators. Phys. Lett. A 133, 134 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Botchev, M.A., Verwer, J.G.: Numerical integration of damped Maxwell equations. SIAM J. Sci. Comput. 31(2), 1322 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eirola, T., Koskela, A.: Krylov integrators for Hamiltonian systems. BIT Numer. Math. 1–20. https://doi.org/10.1007/s10543-018-0732-y (2018)

  15. Lopez, L., Simoncini, V.: Preserving geometric properties of the exponential matrix by block Krylov subspace methods. BIT Numer. Math. 46(4), 813 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Archid, A., Bentbib, A.H.: Approximation of the matrix exponential operator by a structure-preserving block Arnoldi-type method. Appl. Numer. Math. 75, 37 (2014). https://doi.org/10.1016/j.apnum.2012.11.008

    Article  MathSciNet  MATH  Google Scholar 

  17. Benner, P., Mehrmann, V., Xu, H.: A numerically stable structure-preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils. Numer. Math. 78(3), 329 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Arnol’d, V.I., Dubrovin, B., Kirillov, A., Krichever, I.: Dynamical Systems IV: Symplectic Geometry and Its Applications, vol. 4. Springer Science & Business Media, Berlin (2001)

    Book  Google Scholar 

  19. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structurepreserving Algorithms for Ordinary Differential Equations, vol. 31. Springer Science & Business Media, Berlin (2006)

    MATH  Google Scholar 

  20. Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9(1), 17 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  21. Benner, P., Faßbender, H., Stoll, M.: A Hamiltonian Krylov–Schur-type method based on the symplectic Lanczos process. Linear Algebra Appl. 435(3), 578 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lall, S., Krysl, P., Marsden, J.E.: Structure-preserving model reduction for mechanical systems. Physica D: Nonlinear Phenomena 184(1), 304 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Celledoni, E., Li, L.: .. In: Proceedings of the ECMI Conference, pp. 663–559 (2016)

  24. Benner, P., Faßbender, H.: An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem. Linear Algebra Appl. 263, 75 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Watkins, D.S.: On Hamiltonian and symplectic Lanczos processes. Linear Algebra Appl. 385, 23 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Faßbender, H.: A detailed derivation of the parameterized SR algorithm and the symplectic Lanczos method for Hamiltonian matrices. Preprint (2006)

  27. Agoujil, S., Bentbib, A., Kanber, A.: A structure preserving approximation method for hamiltonian exponential matrices. Appl. Numer. Math. 62(9), 1126 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Addison Wesley (2001)

  29. Benner, P., Mehrmann, V., Sorensen, D.C.: Dimension Reduction of Large-Scale Systems, vol. 35. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  30. Ward, R.C.: Numerical computation of the matrix exponential with accuracy estimate. SIAM J. Num. Anal. 14, 600 (1977)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgment

The second author would like to thank Dr. Long Pei for the helpful discussions and suggestions on previous versions of this paper. We are grateful to the anonymous referees for many useful comments.

Funding

This work was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie, grant agreement No. 691070.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, NTNU, 7491, Trondheim, Norway

    Lu Li & Elena Celledoni

Authors
  1. Lu Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elena Celledoni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lu Li.

Additional information

Publisher’s note

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

Appendix

Appendix

figure a
figure b

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, L., Celledoni, E. Krylov projection methods for linear Hamiltonian systems. Numer Algor 81, 1361–1378 (2019). https://doi.org/10.1007/s11075-018-00649-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-018-00649-8

Keywords

Search

Navigation