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Smith-Type Methods for Balanced Truncation of Large Sparse Systems

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Dimension Reduction of Large-Scale Systems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 45))

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References

  1. A. C. Antoulas, D. C. Sorensen, and S. Gugercin. A survey of model reduction methods for large-scale systems. Contemporary Mathematics, AMS Publications, 280, 193–219 (2001).

    MathSciNet  Google Scholar 

  2. A.C. Antoulas, D.C. Sorensen, and Y.K. Zhou. On the decay rate of Hankel singular values and related issues. Systems and Control Letters, 46:5, 323–342 (2002).

    Article  MathSciNet  Google Scholar 

  3. A.C. Antoulas and D.C. Sorensen. The Sylvester equation and approximate balanced reduction. Linear Algebra and Its Applications, 351–352, 671–700 (2002).

    MathSciNet  Google Scholar 

  4. A.C. Antoulas. Lectures on the approximation of linear dynamical systems. Advances in Design and Control, SIAM, Philadelphia (2005).

    Google Scholar 

  5. R. H. Bartels and G. W. Stewart. Solution of the matrix equation AX + XA = C: Algorithm 432. Comm. ACM, 15, 820–826 (1972).

    Article  Google Scholar 

  6. P. Benner and E. S. Quintana-Ortí. Solving stable generalized Lyapunov equation with the matrix sign function. Numerical Algorithms, 20, 75–100 (1999).

    Article  MathSciNet  Google Scholar 

  7. P. Benner, E. S. Quintana-Ortí, and G. Quintana-Ortí. Efficient Numerical Algorithms for Balanced Stochastic Truncation. International Journal of Applied Mathematics and Computer Science, 11:5, 1123–1150 (2001).

    MathSciNet  Google Scholar 

  8. D. Calvetti and L, Reichel. Application of ADI iterative methods to the restoration of noisy images. SIAM J. Matrix Anal. Appl., 17, 165–186 (1996).

    Article  MathSciNet  Google Scholar 

  9. U.B. Desai and D. Pal. A transformation approach to stochastic model reduction. IEEE Trans. Automat. Contr., vol AC-29, 1097–1100 (1984).

    Article  MathSciNet  Google Scholar 

  10. N. Ellner and E. Wachspress. Alternating direction implicit iteration for systems with complex spectra. SIAM J. Numer. Anal., 28, 859–870 (1991).

    Article  MathSciNet  Google Scholar 

  11. D. Enns. Model reduction with balanced realizations: An error bound and a frequency weighted generalization. In Proc. 23rd IEEE Conf. Decision and Control (1984).

    Google Scholar 

  12. M. Green. A relative error bound for balanced stochastic truncation. IEEE Trans. Automat. Contr., AC-33:10, 961–965 (1988).

    Article  Google Scholar 

  13. M. Green. Balanced stochastic realizations. Journal of Linear Algebra and its Applications, 98, 211–247 (1988).

    Article  MATH  Google Scholar 

  14. W. Gawronski and J.-N. Juang. Model reduction in limited time and frequency intervals. Int. J. Systems Sci., 21:2, 349–376 (1990).

    MathSciNet  Google Scholar 

  15. K. Glover. All Optimal Hankel-norm Approximations of Linear Mutilvariable Systems and their L∞-error Bounds. Int. J. Control, 39, 1115–1193 (1984).

    MATH  MathSciNet  Google Scholar 

  16. G. Golub. and C. Van Loan. Matrix computations, 3rd Ed., Johns Hopkins University Press, Baltimore, MD (1996).

    Google Scholar 

  17. S. Gugercin, D.C. Sorensen, and A.C. Antoulas. A modified low-rank Smith method for large-scale Lyapunov equations. Numerical Algorithms, 32:1, 27–55 (2003).

    Article  MathSciNet  Google Scholar 

  18. S. Gugercin and A. C. Antoulas. Approximation of the International Space Station 1R and 12A models. In Proc. 40th CDC (2001).

    Google Scholar 

  19. S. Gugercin. Projection methods for model reduction of large-scale dynamical systems. Ph.D. Dissertation, ECE Dept., Rice University, Houston, TX, USA, May 2003.

    Google Scholar 

  20. S. Gugercin and A.C. Antoulas. A survey of model reduction by balaned truncation and some new results. Int. J. Control, 77:8, 748–766 (2004).

    Article  MathSciNet  Google Scholar 

  21. M.-P. Istace and J.-P. Thiran. On the third and fourth Zolotarev problems in the complex plane. SIAM J. Numer. Anal., 32:1, 249–259 (1995).

    Article  MathSciNet  Google Scholar 

  22. S. Hammarling. Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J. Numer. Anal., 2, 303–323 (1982).

    MATH  MathSciNet  Google Scholar 

  23. A. S. Hodel, K.P. Poola, and B. Tenison. Numerical solution of the Lyapunov equation by approximate power iteration. Linear Algebra Appl., 236, 205–230 (1996).

    Article  MathSciNet  Google Scholar 

  24. D. Y. Hu and L. Reichel. Krylov subspace methods for the Sylvester equation. Linear Algebra Appl., 172, 283–313, (1992).

    Article  MathSciNet  Google Scholar 

  25. I. M. Jaimoukha and E. M. Kasenally. Krylov subspace methods for solving large Lyapunov equations. SIAM J. Numerical Anal., 31, 227–251 (1994).

    Article  MathSciNet  Google Scholar 

  26. I.M. Jaimoukha, E.M. Kasenally. Implicitly restarted Krylov subspace methods for stable partial realizations. SIAM J. Matrix Anal. Appl., 18, 633–652 (1997).

    Article  MathSciNet  Google Scholar 

  27. M. Kamon and F. Wang and J. White. Recent improvements for fast inductance extracton and simulation [packaging]. Proceedings of the IEEE 7th Topical Meeting on Electrical Performance of Electronic Packaging, 281–284 (1998).

    Google Scholar 

  28. J.-R. Li and J. White. Low rank solution of Lyapunov equations. SIAM J. Matrix Anal. Appl., 24:1, 260–280 (2002).

    Article  MathSciNet  Google Scholar 

  29. J.-R. Li and J. White. Efficient model reduction of interconnect via approximate system Gramians. In Proc. IEEE/ACM Intl. Conf. CAD, 380–383, San Jose, CA (1999).

    Google Scholar 

  30. J.-R. Li and J. White. Reduction of large-circuit models via approximate system Gramians. Int. J. Appl. Math. Comp. Sci., 11, 1151–1171 (2001).

    MathSciNet  Google Scholar 

  31. C.-A Lin and T.-Y Chiu. Model reduction via frequency weighted balanced realization. Control Theory and Advanced Technol., 8, 341–351 (1992).

    MathSciNet  Google Scholar 

  32. A. Lu and E. Wachspress. Solution of Lyapunov equations by alternating direction implicit iteration. Comput. Math. Appl., 21:9, 43–58 (1991).

    Article  MathSciNet  Google Scholar 

  33. B. C. Moore. Principal Component Analysis in Linear System: Controllability, Observability and Model Reduction. IEEE Transactions on Automatic Control, AC-26, 17–32 (1981).

    Article  Google Scholar 

  34. C. T. Mullis and R. A. Roberts. Synthesis of minimum roundoff noise fixed point digital filters. IEEE Trans. on Circuits and Systems, CAS-23, 551–562, (1976).

    Article  MathSciNet  Google Scholar 

  35. P.C. Opdenacker and E.A. Jonckheere. A contraction mapping preserving balanced reduction scheme and its infinity norm error bounds. IEEE Trans. Circuits and Systems, (1988).

    Google Scholar 

  36. D. W. Peaceman and H. H. Rachford. The numerical solutions of parabolic and elliptic differential equations. J. SIAM, 3, 28–41 (1955).

    MathSciNet  Google Scholar 

  37. T. Penzl. Eigenvalue Decay Bounds for Solutions of Lyapunov Equations: The Symmetric Case. Systems and Control Letters, 40: 139–144 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  38. T. Penzl. A cyclic low-rank Smith method for large sparse Lyapunov equations. SIAM J. Sci. Comput., 21;4, 1401–1418 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  39. T. Penzl. Algorithms for model reduction of large dynamical systems. Technical Report SFB393/99-40, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz (1999). Avaliable from http://www.tu-chemnitz.de/sfb393/sfb99pr.html.

    Google Scholar 

  40. J. D. Roberts. Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. International Journal of Control, 32, 677–687 (1980).

    MATH  MathSciNet  Google Scholar 

  41. Y. Saad. Numerical solution of large Lyapunov equations. In Signal Processing, Scattering, Operator Theory and Numerical Methods, M. Kaashoek, J.V. Schuppen, and A. Ran, eds., Birkhäuser, Boston, MA, 503–511 (1990).

    Google Scholar 

  42. M. Silveira, M. Kamon, I. Elfadel and J. White. A coordinate-transformed Arnoldi algorithm for generating guaranteed stable reduced-order models of RLC circuits. In Proc. IEEE/ACM Intl. Conf. CAD, San Jose, CA, 288–294 (1996).

    Google Scholar 

  43. R. A. Smith. Matrix Equation, XA + BX = C. SIAM J. Appl. Math, 16, 198–201 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  44. V. Sreeram, B.D.O Anderson and A.G. Madievski. Frequency weighted balanced reduction technique: A generalization and an error bound. In Proc. 34th IEEE Conf. Decision and Control (1995).

    Google Scholar 

  45. G. Wang, V. Sreeram and W.Q. Liu. A new frequency weighted balanced truncation method and an error bound. IEEE Trans. Automat. Contr., 44;9, 1734–1737 (1999).

    Article  MathSciNet  Google Scholar 

  46. G. Starke. Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations. SIAM J. Numer. Anal., 28:5, 1431–1445 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  47. G. Starke. Fejer-Walsh points for rational functions and their use in the ADI iterative method. J. Comput. Appl. Math., 46, 129–141, (1993).

    Article  MATH  MathSciNet  Google Scholar 

  48. A. Varga and B.D.O Anderson. Accuracy enhancing methods for the frequency-weighted balancing related model reduction. In Proc. 40th IEEE Conf. Decision and Control (2001).

    Google Scholar 

  49. E. Wachspress. Optimum alternating-direction-implicit iteration parameters for a model problem. J. Soc. Indust. Appl. Math., 10, 339–350 (1962).

    Article  MATH  MathSciNet  Google Scholar 

  50. E. Wachspress. Iterative solution of the Lyapunov matrix equation. Appl. Math. Lett., 1, 87–90 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  51. E. Wachspress. The ADI minimax problem for complete spectra. Appl. Math. Lett., 1, 311–314 (1988).

    Article  MATH  MathSciNet  Google Scholar 

  52. E. Wachspress. The ADI minimax problem for complex spectra. In Iterative Methods for Large Linear Systems, D. Kincaid and L. Hayes, edss, Academic Press, San Diego, 251–271 (1990).

    Google Scholar 

  53. E. Wachspress. The ADI model problem. Self published, Windsor, CA (1995).

    Google Scholar 

  54. Y, Zhou. Numerical methods for large scale matrix equations with applications in LTI system model reduction. Ph. D. Thesis, CAAM Department, Rice University, Houston, TX, USA, May (2002).

    Google Scholar 

  55. K. Zhou. Frequency-weighted \(L\)∞ norm and optimal Hankel norm model reduction. IEEE Trans. Automat. Contr., 40:10, 1687–1699 (1995).

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. of Mathematics, Virginia Tech., Blacksburg, VA, 24061-0123, USA

    Serkan Gugercin

  2. Projet Ondes, Domaine de Voluceau - Rocquencourt, INRIA-Rocquencourt, B.P. 105, 78153, Le Chesnay Cedex, France

    Jing-Rebecca Li

Authors
  1. Serkan Gugercin
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  2. Jing-Rebecca Li
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Editor information

Editors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Peter Benner

  2. Department of Computational and Applied Mathematics, Rice University, Main Street 6100, 77005-1892, Houston, TX, USA

    Danny C. Sorensen

  3. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Volker Mehrmann

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© 2005 Springer-Verlag Berlin Heidelberg

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Gugercin, S., Li, JR. (2005). Smith-Type Methods for Balanced Truncation of Large Sparse Systems. In: Benner, P., Sorensen, D.C., Mehrmann, V. (eds) Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27909-1_2

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