STRUCTURE-PRESERVING MODEL REDUCTION USING A KRYLOV SUBSPACE PROJECTION FORMULATION*

Communications in Mathematical Sciences

Volumn 3, Issue 2, 2005, Pages 179-199

  Li, Ren Cang ^a   Bai, Zhaojun ^b  

^a UNIVERSITY OF KENTUCKY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Arnoldi process; Krylov subspace; Lanczos process; Model reduction; Moment matching; Structure preserving

Indexed keywords

EID: 34547792690     PISSN: 15396746     EISSN: 19450796     Source Type: Journal    
DOI: 10.4310/CMS.2005.v3.n2.a6     Document Type: Article

References (55)

1. [1] A. C. Antoulas, D. C. Sorensen and S. Gugercin, A survey of model reduction methods for large-scale systems, Structured Matrices in Mathematics, Computer Science, and Engineering I: Proceedings of an AMS-IMS-SIAM joint summer research conference, University of Colorado, Boulder, June 27—July 1, 1999, V. Olshevsky, ed., of Contemporary Mathematics, American Mathematical Society, Providence, Rhode Island, 280, 193-219, 2001.
2. [2] W. E. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quarterly of Applied Math., 9, 17-29, 1951.
3. [3] Z. Bai, Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem, Mathematics of Computation, 62, 209-226, 1994.
4. [4] Z. Bai, Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems,Applied Numerical Mathematics, 43, 9-44, 2002.
5. [5] Z. Bai, D. Bindel, J. Clark, J. Demmel, K. S. J. Pister and N. Zhou, New numerical techniques and tools in SUGAR for 3D MEMS simulation, in Technical Proceedings of the Fourth International Conference on Modeling and Simulation of Microsystems, 31—34, 2000.
6. [6] Z. Bai and R. Freund, A partial Padé-via-Lanczos method for reduced-order modeling, Linear Algebra and its Applications, 332—334, 139—164, 2001.
7. [7] Z. Bai and Y. Su, SOAR: A second-order Arnoldi method for the solution of the quadratic eigen-value problem, Computer Science Technical Report CSE-2003-21, University of California, Davis, California, 2003, Submitted to SIMAX.
8. [8] Z. Bai and Y. Su, Dimension reduction of second-order dynamical systems via a second-order Arnoldi method, Computer Science Technical Report CSE-2004-1, University of California, Davis, California, 2004.
9. [9] Z. Bai and Q. Ye, Error estimation of the Pade approximation of transfer functions via the Lanczos process, Electronic Transactions on Numerical Analysis, 7, 1—17, 1998.
10. [10] M. J. Balas, Trends in large space structure control theory: fondest theory, wildest dreams,IEEE Trans. Automat. Control, AC-27, 522—535, 1982.
11. [11] A. Bjorck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996.
12. [12] J. V. Clark, N. Zhou, D. Bindel, L. Schenato, W. Wu, J. Demmel and K. S. J. Pister, 3D MEMS simulation using modified nodal analysis, in Proceedings of Microscale Systems: Mechanics and Measurements Symposium, 68—75, 2000.
13. [13] Jr. R. R. Craig, Structural Dynamics: An Introduction to Computer Methods, John Wiley & Sons, 1981.
14. [14] J. Cullum, A. Ruehli and T. Zhang, A method for reduced-order modeling and simulation of large interconnect circuits and its application to PEEC models with retardation, IEEE Trans. Circuits and Systems — II, 47, 261-273, 2000.
15. [15] J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
16. [16] P. Feldman and R. W. Freund, Efficient linear circuit analysis by Pade approximation via the Lanczos process, IEEE Trans. Computer-Aided Design, 14, 639-649, 1995.
17. [17] P. Feldmann and R. W. Freund, Circuit noise evaluation by Padé approximation based model-reduction techniques, in Technical Digest of the 1997 IEEE/ACM International Conference on Computer-Aided Design, IEEE Computer Society Press, 132—138, 1997.
18. [18] R. W. Freund, Pade-type reduced-order modeling of higher-order systems, Presentation at Ober-wolfach Mini-Workshop on Dimensional Reduction of Large-Scale Systems, October, 2003.
19. [19] R. W. Freund, Krylov-subspace methods for reduced-order modeling in circuit simulation, Jour-nal of Computational and Applied Mathematics, 123, 395-421, 2000.
20. [20] R. W. Freund, Model reduction methods based on Krylov subspaces, Acta Numerica, 12, 267-319,2003.
21. [21] R. W. Freund and P. Feldmann, The SyMPVL algorithm and its applications to interconnec simulation, in Proc. 1997 International Conference on Simulation of Semiconductor Processes and Devices, Piscataway, New Jersey, IEEE, 113-116, 1997.
22. [22] R. W. Freund, M. H. Gutknecht and N. M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Comput., 14, 137-158, 1993.
23. [23] K. Gallivan, E. Grimme and P. Van Dooren, Asymptotic waveform evaluation via a Lanczos method, Appl. Math. Lett., 7, 75-80, 1994.
24. [24] S. D. Garvey, Z. Chen, M. I. Friswell and U. Prells, Model reduction using structure-preserving transformations, in Proceedings of the International Modal Analysis Conference IMAC XXI, Kissimmee, Florida, Feb., 361-377, 2003.
25. [25] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York,1982.
26. [26] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland, 3rd ed., 1996.
27. [27] W. B. Gragg, Matrix interpretations and applications of the continued fraction algorithm,Rocky Mountain J. Math., 4, 213-225, 1974.
28. [28] E. J. Grimme, Krylov Projection Methods For Model Reduction, PhD thesis, University of Illinois at Urbana-Champaign, Urbana, Illinois, 1997.
29. [29] L. Hoffnung, Subspace Projection Methods for the Quadratic Eigenvalue Problem, PhD thesis,University of Kentucky, Lexington, KY, August 2004.
30. [30] L. Hoffnung, R. C. Li and Q. Ye, Krylov type subspace methods for matrix polynomials, Linear Algebra and its Applications, 2003, to appear.
31. [31] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 2nd ed., 1970.
32. [32] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, Journal of Research of the National Bureau of Standards, 45, 255— 282, 1950.
33. [33] R. C. Li, Structural preserving model reductions, Technical Report 2004-02, Department of Mathematics, University of Kentucky, January 2004. Avaliable at http://www.ms.uky.edu /〜math/MAreport/.
34. [34] R. C. Li and Z. Bai, Structure-preserving model reductions using a Krylov subspace projection formulation, Tech. Report CSE-2004-24, Department of Computer Science, University of California, Davis, March 2004.
35. [35] R. C. Li and Z. Bai, Structure-preserving model reduction, in PARA,04 State-of-the-Art in Scientific Computing, June, 20-23, 2004. J. Dongarra, K. Madsen, and J. Wasniewski, eds., Lecture Notes in Computer Science, Springer, 2005, to appear.
36. [36] R. C. Li and Q. Ye, A Krylov subspace method for quadratic matrix polynomials with application to constrained least squares problems, SIAM J. Matrix Anal. Appl., 25, 405—428, 2003.
37. [37] D. Meyer and S. Srinivasan, Balancing and model reduction for second-order form linear sys-tems, IEEE Transactions on Automatic Control, 41, 1632—1644, 1996.
38. [38] A. Odabasioglu, M. Celik and L. T. Pileggi, PRIMA: passive reduced-order interconnect macro-modeling algorithm, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 17, 645—654, 1998.
39. [39] B. N. Parlett, Reduction to tridiagonal form and minimal realization, SIAM Journal on Matrix Analysis and Applications, 13, 567—593, 1992.
40. [40] B. N. Parlett, D. R. Taylor and Z. A. Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Mathematics of Computation, 44, 105—124, 1985.
41. [41] L. T. Pillage and R. A. Rohrer, Asymptotic waveform evaluation for timing analysis, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 9, 352-366, 1990.
42. [42] D. Ramaswamy and J. White, Automatic generation of small-signal dynamic macromodels from 3-D simulation, in Technical Proceedings of the Fourth International Conference on Modeling and Simulation of Microsystems, 27-30, 2000.
43. [43] A. Ruhe, Rational Krylov sequence methods for eigenvalue computation, Linear Algebra and its Applications, 58, 391-405, 1984.
44. [44] A. Ruhe, Rational Krylov algorithms for nonsymmetric eigenvalue problems. II. matrix pairs, Linear Algebra and its Applications, 197-198, 283-295, 1994.
45. [45] B. Salimbahrami and B. Lohmann, Structure preserving order reduction of large scale second order systems, in Proceeding of 10th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems: Theory and Applications, Osaka, Japan, July, 245-250, 2004.
46. [46] R. D. Slone, A computationally efficient method for solving electromagnetic interconnect prob-lems: the Padé approximation via the Lanczos process with an error bound, master's thesis, University of Kentucky, Lexington, KY, 1997.
47. [47] R. D. Slone, Fast frequency sweep model order reduction of polynomial matrix equations result-ing from finite element discretization, PhD thesis, Ohio State University, Columbus, OH, 2002.
48. [48] R. D. Slone, R. Lee and J. F. Lee, Broadband model order reduction of polynomial matrix equations using single point well-conditioned asymptotic waveform evaluation: derivation and theory, Inter. J. Numer. Methods in Eng., 2002, submitted.
49. [49] T. J. Su and R. R. Craig, Model reduction and control offlexible structures using Krylov vectors,J. Guidance, Control, and Dynamics, 14, 260-267, 1991.
50. [50] A. Vandendorpe and P. Van Dooren, Krylov techniques for model reduction of second-order systems, Unpublished Note, March 2, 2004.
51. [51] C. E. Villemagne and R. E. Skelton, Model reduction using a projection formulation, Int. J.Control, 46, 2141-2169, 1987.
52. [52] T. Wittig, I. Munteanu, R. Schuhmann and T. Weiland, Two-step Lanczos algorithm for model order reduction, IEEE Trans. Magn., 38, 673-676, 2002.
53. [53] Q. Ye, A convergence analysis for nonsymmetric Lanczos algorithms, Mathematics of Compu-tation, 56, 677-691, 1991.
54. [54] Q. Ye, A breakdown-free variation of the nonsymmetric Lanczos algorithms, Mathematics of Computation, 62, 179-207, 1994.
55. [55] H. Zheng and L. Pileggi, Robust and passive model order reduction for circuits constaining sus-ceptance elements, in Tech. Dig. 2002 IEEE/ACM International Conference on Computer-Aided Design, IEEE, 2002.