Advertisement

Archives of Computational Methods in Engineering

, Volume 21, Issue 4, pp 331–358 | Cite as

Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective

  • Ulrike Baur
  • Peter Benner
  • Lihong Feng
Article

Abstract

In the past decades, Model Order Reduction (MOR) has demonstrated its robustness and wide applicability for simulating large-scale mathematical models in engineering and the sciences. Recently, MOR has been intensively further developed for increasingly complex dynamical systems. Wide applications of MOR have been found not only in simulation, but also in optimization and control. In this survey paper, we review some popular MOR methods for linear and nonlinear large-scale dynamical systems, mainly used in electrical and control engineering, in computational electromagnetics, as well as in micro- and nano-electro-mechanical systems design. This complements recent surveys on generating reduced-order models for parameter-dependent problems (Benner et al. in 2013; Boyaval et al. in Arch Comput Methods Eng 17(4):435–454, 2010; Rozza et al. Arch Comput Methods Eng 15(3):229–275, 2008) which we do not consider here. Besides reviewing existing methods and the computational techniques needed to implement them, open issues are discussed, and some new results are proposed.

Notes

Acknowledgments

We would like to thank Tobias Breiten, Heiko Panzer and Thomas Wolf for reading a draft version of this manuscript and providing various suggestions for improvement.

References

  1. 1.
    Achar R, Nakhla MS (2001) Simulation of high-speed interconnects. Proc IEEE 89(5):693–728Google Scholar
  2. 2.
    Al-Baiyat SA, Bettayeb M (1993) A new model reduction scheme for k-power bilinear systems. In: Proceedings of the 32nd IEEE conference on decision and control 1:22–27. doi: 10.1109/CDC.1993.325196.
  3. 3.
    Aldhaheri RW (1991) Model order reduction via real Schur-form decomposition. Int J Control 53(3):709–716zbMATHMathSciNetGoogle Scholar
  4. 4.
    Anderson BDO, Antoulas AC (1990) Rational interpolation and state-variable realizations. Linear Algebra Appl 137(138):479–509MathSciNetGoogle Scholar
  5. 5.
    Antoulas AC (2005) Approximation of large-scale dynamical systems. SIAM Publications, PhiladelphiazbMATHGoogle Scholar
  6. 6.
    Antoulas AC, Sorensen DC, Gugercin S (2001) A survey of model reduction methods for large-scale systems. Contemp Math 280:193–219MathSciNetGoogle Scholar
  7. 7.
    Antoulas AC, Sorensen DC, Zhou Y (2002) On the decay rate of Hankel singular values and related issues. Syst Control Lett 46(5):323–342zbMATHMathSciNetGoogle Scholar
  8. 8.
    Astrid P, Weiland S, Willcox K, Backx T (2008) Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans Autom Control 53(10):2237–2251MathSciNetGoogle Scholar
  9. 9.
    Badía J, Benner P, Mayo R, Quintana-Ortí ES (2002) Solving large sparse Lyapunov equations on parallel computers. In: Monien B, Feldmann R (eds) Euro-Par 2002–parallel processing, no. 2400 in lecture notes in computer science, pp 687–690. Springer, Berlin.Google Scholar
  10. 10.
    Badía JM, Benner P, Mayo R, Quintana-Ortí ES, Quintana-Ortí G, Remón A (2006) Balanced truncation model reduction of large and sparse generalized linear systems. Tech. Rep. Chemnitz Scientific Computing Preprints 06–04, Fakultät für Mathematik, TU Chemnitz.Google Scholar
  11. 11.
    Bai Z (2002) Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl Numer Math 43(1–2):9–44zbMATHMathSciNetGoogle Scholar
  12. 12.
    Bai Z, Dewilde PM, Freund RW (2005) Reduced-order modeling. In: Handbook of numerical analysis. Vol. XIII, Handb. Numer. Anal., XIII, pp 825–891. North-Holland, Amsterdam.Google Scholar
  13. 13.
    Bai Z, Feldmann P, Freund RW (1998) How to make theoretically passive reduced-order models passive in practice. In: Proceedings of the IEEE 1998 custom integrated circuits conference, pp 207–210. IEEE.Google Scholar
  14. 14.
    Bai Z, Freund RW (2001) A partial Padé-via-Lanczos method for reduced-order modeling. In: Proceedings of the eighth conference of the international linear algebra society (Barcelona, 1999), vol. 332/334, pp 139–164.Google Scholar
  15. 15.
    Bai Z, Freund RW (2001) A symmetric band Lanczos process based on coupled recurrences and some applications. SIAM J Sci Comput 23(2):542–562 (electronic).Google Scholar
  16. 16.
    Bai Z, Skoogh D (2006) A projection method for model reduction of bilinear dynamical systems. Linear Algebra Appl 415(2–3):406–425zbMATHMathSciNetGoogle Scholar
  17. 17.
    Bai Z, Slone RD, Smith WT, Ye Q (1999) Error bound for reduced system model by Padé approximation via the Lanczos process. IEEE Trans Comput-Aided Des Integr Circuits Syst 18(2):133–141Google Scholar
  18. 18.
    Bai Z, Su YF (2005) Dimension reduction of large-scale second-order dynamical systems via a second-order Arnoldi method. SIAM J Sci Comput 26(5):1692–1709zbMATHMathSciNetGoogle Scholar
  19. 19.
    Baker G (1975) Essentials of Padé approximation. Academic Press, LondonGoogle Scholar
  20. 20.
    Barrachina S, Benner P, Quintana-Ortí ES, Quintana-Ortí G (2005) Parallel algorithms for balanced truncation of large-scale unstable systems. In: Proceedings of 44th IEEE conference decision control ECC 2005, pp 2248–2253.Google Scholar
  21. 21.
    Barrault M, Maday Y, Nguyen NC, Patera AT (2004) An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C R Acad Sci Paris Ser I 339:667–672zbMATHMathSciNetGoogle Scholar
  22. 22.
    Baur U, Benner P (2006) Factorized solution of Lyapunov equations based on hierarchical matrix arithmetic. Computing 78(3):211–234zbMATHMathSciNetGoogle Scholar
  23. 23.
    Baur U, Benner P (2008) Cross-gramian based model reduction for data-sparse systems. Electron Trans Numer Anal 31:256– 270Google Scholar
  24. 24.
    Baur U, Benner P (2008) Gramian-based model reduction for data-sparse systems. SIAM J Sci Comput 31(1):776–798zbMATHMathSciNetGoogle Scholar
  25. 25.
    Bechtold T, Rudnyi EB, Korvink JG (2005) Error indicators for fully automatic extraction of heat-transfer macromodels for MEMS. J Micromech Microeng 15(3):430–440Google Scholar
  26. 26.
    Benner P (2004) Solving large-scale control problems. IEEE Control Syst Mag 14(1):44–59MathSciNetGoogle Scholar
  27. 27.
    Benner P (2010) Advances in balancing-related model reduction for circuit simulation. In: Roos J, Costa L (eds) Scientific computing in electrical engineering SCEE 2008, mathematics in industry, vol 14. Springer, Berlin, pp 469–482Google Scholar
  28. 28.
    Benner P, Breiten T (2012) Interpolation-based \(h_2\)-model reduction of bilinear control systems. SIAM J Matrix Anal Appl 33(3):859–885zbMATHMathSciNetGoogle Scholar
  29. 29.
    Benner P, Breiten T (2012) Krylov-subspace based model reduction of nonlinear circuit models using bilinear and quadratic-linear approximations. In: Günther M, Bartel A, Brunk M, Schöps S, Striebel M (eds) Progress in industrial mathematics at ECMI 2010, mathematics in industry, vol 17. Springer, Berlin, pp 153–159Google Scholar
  30. 30.
    Benner P, Breiten T (2012) Two-sided moment matching methods for nonlinear model reduction. Preprint MPIMD/12-12, Max Planck Institute Magdeburg Preprints. Available from http://www.mpi-magdeburg.mpg.de/preprints/
  31. 31.
    Benner P, Breiten T (2013) Low rank methods for a class of generalized Lyapunov equations and related issues. Numer Math 124(3):441–470zbMATHMathSciNetGoogle Scholar
  32. 32.
    Benner P, Breiten T (2013) On optimality of approximate low rank solutions of large-scale matrix equations. Syst Control Lett 67(1):55–64MathSciNetGoogle Scholar
  33. 33.
    Benner P, Castillo M, Quintana-Ortí ES, Quintana-Ortí G (2004) Parallel model reduction of large-scale unstable systems. In: Joubert G, Nagel W, Peters F, Walter W (eds) Parallel computing: software technology, algorithms, architectures & applications. Proceedings of International Conference ParCo2003, Dresden, Germany, Advances in Parallel Computing, vol. 13, pp 251–258. Elsevier BV (North-Holland).Google Scholar
  34. 34.
    Benner P, Claver JM, Quintana-Ortí ES (1998) Efficient solution of coupled Lyapunov equations via matrix sign function iteration. In: Dourado A et al. (ed) Proceedings of 3rd Portuguese conference on automatic control CONTROLO’98, Coimbra, pp 205–210.Google Scholar
  35. 35.
    Benner P, Damm T (2011) Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J Control Optim 49(2):686–711zbMATHMathSciNetGoogle Scholar
  36. 36.
    Benner P, Ezzatti P, Kressner D, Quintana-Ortí ES, Remón A: A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU-GPU platforms. Parallel Comput 37(8):439–450 (2011). doi: 10.1016/j.parco.2010.12.002. http://www.sciencedirect.com/science/article/pii/S0167819110001560
  37. 37.
    Benner P, Gugercin S, Willcox K (2013) A survey of model reduction methods for parametric systems. Preprint MPIMD/13-14, Max Planck Institute Magdeburg Preprints. Available from http://www.mpi-magdeburg.mpg.de/preprints/
  38. 38.
    Benner P, Kürschner P, Saak J (2012) A goal-oriented dual LRCF-ADI for balanced truncation. In: Troch I, Breitenecker F (eds) 7th Vienna international conference on mathematical modelling, IFAC-PapersOnlines, mathematical modelling, vol. 7, pp. 752–757. Vienna Univ. of Technology.Google Scholar
  39. 39.
    Benner P, Kürschner P, Saak J (2013) An improved numerical method for balanced truncation for symmetric second order systems. Math Comput Model Dyn Syst 19(6):593–615. doi: 10.1080/13873954.2013.794363 MathSciNetGoogle Scholar
  40. 40.
    Benner P, Li JR, Penzl T (2008) Numerical solution of large Lyapunov equations, Riccati equations, and linear-quadratic control problems. Numer Algorithms 15(9):755–777zbMATHMathSciNetGoogle Scholar
  41. 41.
    Benner P, Mehrmann V, Sorensen DC (eds) (2005) Dimension reduction oflarge-scale systems, vol 45., lecture notes in computational science and engineeringSpringer, BerlinGoogle Scholar
  42. 42.
    Benner P, Quintana-Ortí ES: Model reduction based on spectral projection methods. Chapter 1 (pp 5–48) of [41].Google Scholar
  43. 43.
    Benner P, Quintana-Ortí ES (1999) Solving stable generalized Lyapunov equations with the matrix sign function. Numer Algorithms 20(1):75–100Google Scholar
  44. 44.
    Benner P, Quintana-Ortí ES, Quintana-Ortí G (1999) A portable subroutine library for solving linear control problems on distributed memory computers. In: Cooperman G, Jessen E, Michler G (eds) Workshop on Wide Area Networks and high performance computing, Essen (Germany), September 1998. Lecture notes in control and information Springer, Berlin, pp 61–88Google Scholar
  45. 45.
    Benner P, Quintana-Ortí ES, Quintana-Ortí G (2000) Balanced truncation model reduction of large-scale dense systems on parallel computers. Math Comput Model Dyn Syst 6(4):383– 405Google Scholar
  46. 46.
    Benner P, Quintana-Ortí ES, Quintana-Ortí G (2000) Singular perturbation approximation of large, dense linear systems. Proceedings of 2000 IEEE international symposium CACSD, Anchorage, Alaska, USA, September 25–27, 2000. IEEE Press, Piscataway, pp 255–260Google Scholar
  47. 47.
    Benner P, Quintana-Ortí ES, Quintana-Ortí G (2001) Efficient numerical algorithms for balanced stochastic truncation. Int J Appl Math Comput Sci 11(5):1123–1150zbMATHMathSciNetGoogle Scholar
  48. 48.
    Benner P, Quintana-Ortí ES, Quintana-Ortí G (2003) State-space truncation methods for parallel model reduction of large-scale systems. Parallel Comput 29:1701–1722MathSciNetGoogle Scholar
  49. 49.
    Benner P, Quintana-Ortí ES, Quintana-Ortí G (2004) Computing optimal Hankel norm approximations of large-scale systems. Proceedings of 43rd IEEE conference decision Control. Omnipress, Madison, WI, pp 3078–3083Google Scholar
  50. 50.
    Benner P, Schneider A (2012) Model reduction for linear descriptor systems with many ports. In: Günther M, Bartel A, Brunk M, Schöps S, Striebel M (eds) Progress in industrial mathematics at ECMI 2010, mathematics in industry, vol 17. Springer, Berlin, pp 137–143Google Scholar
  51. 51.
    Bollhöfer M, Bodendiek A (2012) Adaptive-order rational Arnoldi method for Maxwell’s equations. In: Scientific computing in electrical engineering (Abstracts), pp 77–78.Google Scholar
  52. 52.
    Bond B, Daniel L (2005) Parameterized model order reduction of nonlinear dynamical systems. In: Proceedings of international conference on computer-aided design, pp 487–494.Google Scholar
  53. 53.
    Bond B, Daniel L (2009) Stable reduced models for nonlinear descriptor systems through piecewise-linear approximation and projection. IEEE Trans Comput-Aided Des Integr Circuits Syst 28(10):1467–1480Google Scholar
  54. 54.
    Boyaval S, Le Bris C, Lelièvre T, Maday Y, Nguyen NC, Patera AT (2010) Reduced basis techniques for stochastic problems. Arch Comput Methods Eng 17(4):435–454. doi: 10.1007/s11831-010-9056-z zbMATHMathSciNetGoogle Scholar
  55. 55.
    Breiten T, Damm T (2010) Krylov subspace methods for model order reduction of bilinear control systems. Syst Control Lett 59(10):443–450zbMATHMathSciNetGoogle Scholar
  56. 56.
    Bunse-Gerstner A, Kubalinska D, Vossen G, Wilczek D (2010) \(h_2\)-norm optimal model reduction for large scale discrete dynamical MIMO systems. J Comput Appl Math 233(5):1202–1216. doi:  10.1016/j.cam.2008.12.029 zbMATHMathSciNetGoogle Scholar
  57. 57.
    Chaturantabut S, Sorensen DC (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737–2764zbMATHMathSciNetGoogle Scholar
  58. 58.
    Chen Y (1999) Model order reduction for nonlinear systems. Master’s thesis, Massachusetts Institute of Technology.Google Scholar
  59. 59.
    Chiprout E, Nakhla M (1995) Analysis of interconnect networks using complex frequency hopping (CFH). IEEE Trans Comput-Aided Des Integr Circuits Syst 14(2):186–200Google Scholar
  60. 60.
    Chu CC, Lai MH, Feng WS (2006) MIMO interconnects order reductions by using the multiple point adaptive-order rational global Arnoldi algorithm. IEICE Trans Electron E89-C(6):792–802.Google Scholar
  61. 61.
    Condon M, Ivanov R (2007) Krylov subspaces from bilinear representations of nonlinear systems. COMPEL Math Electr Electron Eng 26(2):399–406zbMATHMathSciNetGoogle Scholar
  62. 62.
    Craig RR, Bampton MCC (1968) Coupling of substructures for dynamic analysis. AIAA J 6:1313–1319zbMATHGoogle Scholar
  63. 63.
    Davison EJ (1966) A method for simplifying linear dynamic systems. IEEE Trans Autom Control AC-11:93–101.Google Scholar
  64. 64.
    Druskin V, Knizhnerman L, Simoncini V (2011) Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation. SIAM J Numer Anal 49(5):1875–1898zbMATHMathSciNetGoogle Scholar
  65. 65.
    Dong N, Roychowdhury J (2003) Piecewise polynomial nonlinear model reduction. In: Proceedings of design automation conference, pp 484–489.Google Scholar
  66. 66.
    Dong N, Roychowdhury J (2004) Automated extraction of broadly applicable nonlinear analog macromodels from SPICE-level descriptions. In: Custom integrated circuits conference, 2004. Proceedings of the IEEE 2004, pp 117–120.Google Scholar
  67. 67.
    Druskin V, Simoncini V (2011) Adaptive rational Krylov subspaces for large-scale dynamical systems. Syst Control Lett 60:546–560zbMATHMathSciNetGoogle Scholar
  68. 68.
    Enns D (1984) Model reduction with balanced realization: an error bound and a frequency weighted generalization. Proceedings of 23rd IEEE conference decision control. Las Vegas, NV, pp 127–132Google Scholar
  69. 69.
    Eppler A, Bollhöfer M (2010) An alternative way of solving large Lyapunov equations. Proc Appl Math Mech 10(1):547–548Google Scholar
  70. 70.
    Eppler A, Bollhöfer M (2012) Structure-preserving GMRES methods for solving large Lyapunov equations. In: Günther M, Bartel A, Brunk M, Schöps S, Striebel M (eds) Progress in industrial mathematics at ECMI 2010, mathematics in industry, vol 17. Springer, Berlin, pp 131–136Google Scholar
  71. 71.
    Faßbender H, Soppa A (2011) Machine tool simulation based on reduced order FE models. Math Comput Simul 82(3):404–413zbMATHGoogle Scholar
  72. 72.
    Feldmann P, Freund RW (1994) Efficient linear circuit analysis by Padé approximation via the Lanczos process. In: Proceedings of EURO-DAC ’94 with EURO-VHDL ’94, Grenoble, France, pp 170–175. IEEE Computer Society Press.Google Scholar
  73. 73.
    Feldmann P, Freund RW (1995) Efficient linear circuit analysis by Padé approximation via the Lanczos process. IEEE Trans Comput-Aided Des Integr Circuits Syst 14:639–649Google Scholar
  74. 74.
    Feldmann P, Liu F (2004) Sparse and efficient reduced order modeling of linear subcircuits with large number of terminals. In: Proceedings of international conference on computer-aided design, pp 88–92.Google Scholar
  75. 75.
    Feng L, Benner P (2007) A note on projection techniques for model order reduction of bilinear systems. In: Numerical analysis and applied mathematics: international conference of numerical analysis and applied mathematics, pp 208–211.Google Scholar
  76. 76.
    Feng L, Benner P (2012) Automatic model order reduction by moment-matching according to an efficient output error bound. In: Scientific computing in electrical engineering (Abstracts), pp 71–72.Google Scholar
  77. 77.
    Feng L, Benner P, Korvink JG (2013) System-level modeling of MEMS by means of model order reduction (mathematical approximation)-mathematical background. In: Bechtold T, Schrag G, Feng L (eds) System-level modeling of MEMS, advanced micro & nanosystems. Wiley-VCHGoogle Scholar
  78. 78.
    Feng L, Korvink JG, Benner P (2012) A fully adaptive scheme for model order reduction based on moment-matching. Preprint MPIMD/12-14, Max Planck Institute Magdeburg Preprints. Available from http://www.mpi-magdeburg.mpg.de/preprints/
  79. 79.
    Feng L, Zeng X, Chiang C, Zhou D, Fang Q (2004) Direct nonlinear order reduction with variational analysis. In: Proceedings of design automation and test in Europe, pp 1316–1321.Google Scholar
  80. 80.
    Fernando KV, Hammarling SJ (1988) A product induced singular value decmoposition for two matrices and balanced realization. In: Datta B et al (eds) Linear algebra in signals, systems and control. SIAM, Philadelphia, pp 128–140Google Scholar
  81. 81.
    Fernando KV, Nicholson H (1983) On the structure of balanced and other principal representations of SISO systems. IEEE Trans Autom Control 28(2):228–231zbMATHMathSciNetGoogle Scholar
  82. 82.
    Fernando KV, Nicholson H (1984) On a fundamental property of the cross-Gramian matrix. IEEE Trans Circuits Syst CAS-31(5):504–505.Google Scholar
  83. 83.
    Flagg GM (2010) H2-optimal interpolation: new properties and applications. In: Talk given at the, (2010) SIAM annual meeting. Pittsburgh, PAGoogle Scholar
  84. 84.
    Flagg GM (2012) Interpolation methods for the model reduction of bilinear systems. Ph.D. thesis, Virginia Polytechnic Institute and State University.Google Scholar
  85. 85.
    Flagg GM, Beattie CA, Gugercin S (2013) Interpolatory \({\cal H}_\infty \) model reduction. Syst Control Lett 62(7):567–574zbMATHMathSciNetGoogle Scholar
  86. 86.
    Flagg GM, Gugercin S (2013) On the ADI method for the Sylvester equation and the optimal-\({\cal H}_2\) points. Appl Numer Math 64:50–58Google Scholar
  87. 87.
    Fortuna L, Nummari G, Gallo A (1992) Model order reduction techniques with applications in electrical engineering. Springer, BerlinGoogle Scholar
  88. 88.
    Freund RW (2000) Krylov subspace methods for reduced-order modeling in circuit cimulation. J Comput Appl Math 123:395–421zbMATHMathSciNetGoogle Scholar
  89. 89.
    Freund RW (2003) Model reduction methods based on Krylov subspaces. Acta Numer 12:267–319zbMATHMathSciNetGoogle Scholar
  90. 90.
    Freund RW (2004) SPRIM: structure-preserving reduced-order interconnect macromodeling. In.Technical digest of the 2004 IEEE/ACM international conference on computer-aided design, pp 80–87. IEEE Computer Society Press.Google Scholar
  91. 91.
    Freund RW, Feldmann P (1996) Reduced-order modeling of large passive linear circuits by means of the SyPVL algorithm. In: Technical digest of the 1996 IEEE/ACM international conference on computer-aided design, pp 280–287. IEEE Computer Society Press.Google Scholar
  92. 92.
    Freund RW, Feldmann P (1997) The SyMPVL algorithm and its applications to interconnect simulation. In: Proceedings of the 1997 international conference on simulation of semiconductor processes and devices, pp. 113–116. IEEE.Google Scholar
  93. 93.
    Freund RW, Feldmann P (1998) Reduced-order modeling of large linear passive multi-terminal circuits using matrix-Padé approximation. In: Proceedings of the design, automation and test in Europe conference 1998, pp 530–537. IEEE Computer Society Press.Google Scholar
  94. 94.
    Gallivan K, Grimme E, Van Dooren P (1994) Asymptotic waveform evaluation via a Lanczos method. Appl Math Lett 7(5):75–80zbMATHMathSciNetGoogle Scholar
  95. 95.
    Gawronski W, Juang JN (1990) Model reduction in limited time and frequency intervals. Int J Syst Sci 21(2):349–376zbMATHMathSciNetGoogle Scholar
  96. 96.
    Glover K (1984) All optimal Hankel-norm approximations of linear multivariable systems and their L\(^{\infty }\) norms. Int J Control 39:1115–1193zbMATHMathSciNetGoogle Scholar
  97. 97.
    Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  98. 98.
    Gragg WB, Lindquist A (1983) On the partial realization problem. Linear Algebra Appl 50:277–319zbMATHMathSciNetGoogle Scholar
  99. 99.
    Grasedyck L (2004) Existence of a low rank or \({\cal H}\)-matrix approximant to the solution of a Sylvester equation. Numer Linear Algebra Appl 11(4):371–389zbMATHMathSciNetGoogle Scholar
  100. 100.
    Grasedyck L, Hackbusch W (2007) A multigrid method to solve large scale Sylvester equations. SIAM J Matrix Anal Appl 29(3):870–894zbMATHMathSciNetGoogle Scholar
  101. 101.
    Grasedyck L, Hackbusch W, Khoromskij BN (2003) Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices. Computing 70(2):121–165zbMATHMathSciNetGoogle Scholar
  102. 102.
    Grepl MA, Maday Y, Nguyen NC, Patera AT (2007) Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Math Model Numer 41(03):575–605.Google Scholar
  103. 103.
    Grimme EJ (1997) Krylov projection methods for model reduction. Ph.D. thesis, Univ. Illinois, Urbana-Champaign.Google Scholar
  104. 104.
    Grimme EJ, Sorensen DC, Van Dooren P (1996) Model reduction of state space systems via an implicitly restarted Lanczos method. Numer Algorithms 12:1–31zbMATHMathSciNetGoogle Scholar
  105. 105.
    Gu C (2009) QLMOR: A new projection-based approach for nonlinear model order reduction. In: Proceedings of international conference on computer-aided design, pp 389–396.Google Scholar
  106. 106.
    Gu C (2011) QLMOR: a projection-based nonlinear model order reduction approach using quadratic-linear representation of nonlinear systems. IEEE Trans Comput-Aided Des 30(9):1307–1320Google Scholar
  107. 107.
    Gugercin S, Antoulas AC (2004) A survey of model reduction by balanced truncation and some new results. Int J Control 77(8):748–766zbMATHMathSciNetGoogle Scholar
  108. 108.
    Gugercin S, Antoulas AC, Beattie CA (2008) \({\cal H}_2\) model reduction for large-scale linear dynamical systems. SIAM J Matrix Anal Appl 30(2):609–638zbMATHMathSciNetGoogle Scholar
  109. 109.
    Gugercin S, Li JR, Smith-type methods for balanced truncation of large systems. Chapter 2 (pp 49–82) of [41].Google Scholar
  110. 110.
    Gugercin S, Sorensen DC, Antoulas AC (2003) A modified low-rank Smith method for large-scale Lyapunov equations. Numer Algorithms 32(1):27–55zbMATHMathSciNetGoogle Scholar
  111. 111.
    Heath MT, Laub AJ, Paige CC, Ward RC (1987) Computing the SVD of a product of two matrices. SIAM J Sci Stat Comput 7:1147–1159MathSciNetGoogle Scholar
  112. 112.
    Heinkenschloss M, Reis T, Antoulas AC (2011) Balanced truncation model reduction for systems with inhomogeneous initial conditions. Automatica 47:559–564zbMATHMathSciNetGoogle Scholar
  113. 113.
    Hochbruck M, Starke G (1995) Preconditioned Krylov subspace methods for Lyapunov matrix equations. SIAM J Matrix Anal Appl 16(1):156–171zbMATHMathSciNetGoogle Scholar
  114. 114.
    Hochman A, Vasilyev DM, Rewieński MJ, White JK (2013) Projection-based nonlinear model order reduction. In: Bechtold T, Schrag G, Feng L (eds) System-level modeling of MEMS, advanced micro & nanosystems. Wiley-VCHGoogle Scholar
  115. 115.
    Hodel AS, Tenison B, Poolla KR (1996) Numerical solution of the Lyapunov equation by approximate power iteration. Linear Algebra Appl 236:205–230zbMATHMathSciNetGoogle Scholar
  116. 116.
    Hu DY, Reichel L (1992) Krylov-subspace methods for the Sylvester equation. Linear Algebra Appl 172:283–313zbMATHMathSciNetGoogle Scholar
  117. 117.
    Ito K, Kunisch K (2006) Reduced order control based on approximate inertial manifolds. Linear Algebra Appl 415(2–3):531–541zbMATHMathSciNetGoogle Scholar
  118. 118.
    Jaimoukha IM, Kasenally EM (1994) Krylov subspace methods for solving large Lyapunov equations. SIAM J Numer Anal 31(1):227–251zbMATHMathSciNetGoogle Scholar
  119. 119.
    Jaimoukha IM, Kasenally EM (1997) Implicitly restarted Krylov subspace methods for stable partial realizations. SIAM J Matrix Anal Appl 18(3):633–652zbMATHMathSciNetGoogle Scholar
  120. 120.
    Jbilou K, Riquet AJ (2006) Projection methods for large Lyapunov matrix equations. Linear Algebra Appl 415:344–358zbMATHMathSciNetGoogle Scholar
  121. 121.
    Kamon M, Wang F, White J (2000) Generating nearly optimally compact models from krylov-subspace based reduced-order models. IEEE Trans Circuits Syst 47(4):239–248Google Scholar
  122. 122.
    Kerns KJ, Wemple IL, Yang AT (1995) Stable and efficient reduction of substrate model networks using congruence transforms. ICCAD ’95: proceedings of the 1995 IEEE/ACM international conference on Computer-aided design. DC, USA, Washington, pp 207–214Google Scholar
  123. 123.
    Konkel Y, Farle O, Dyczij-Edlinger R (2008) Ein Fehlerschätzer für die Krylov-Unterraum basierte Ordnungsreduktion zeitharmonischer Anregungsprobleme (in German). In: Lohmann B, Kugi A (eds) Tagungsband GMA-FA 1.30, ’Modellbildung, Identifikation und Simulation in der Automatisierungstechnik’, pp 139–149.Google Scholar
  124. 124.
    Konkel Y, Farle O, Sommer A, Burgard S, Dyczij-Edlinger R (2014) A posteriori error bounds for Krylov-based fast frequency sweeps of finite-element systems. IEEE Trans Magn 50(2):441–444Google Scholar
  125. 125.
    Kressner D (2005) Numerical methods for general and structured eigenvalue problems, vol 46., lecture notes in computational science and engineeringSpringer, BerlinGoogle Scholar
  126. 126.
    Kressner D, Tobler C (2010) Krylov subspace methods for linear systems with tensor product structure. SIAM J Matrix Anal Appl 31(4):1688–1714Google Scholar
  127. 127.
    Lancaster P (1970) Explicit solutions of linear matrix equations. SIAM Rev 12:544–566zbMATHMathSciNetGoogle Scholar
  128. 128.
    Laub AJ, Heath MT, Paige CC, Ward RC (1987) Computation of system balancing transformations and other application of simultaneous diagonalization algorithms. IEEE Trans Autom Control 34:115–122Google Scholar
  129. 129.
    Laub AJ, Silverman LM, Verma M (1983) A note on cross-Grammians for symmetric realizations. Proc IEEE Trans Circuits Syst 71(7):904–905Google Scholar
  130. 130.
    Lee H, Chu C, Feng W (2006) An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems. Linear Algebra Appl 415(23):235–261zbMATHMathSciNetGoogle Scholar
  131. 131.
    Li JR, Wang F, White J (1999) An efficient Lyapunov equation-based approach for generating reduced-order models of interconnect. In. Proceedings of design automation conference, pp 1–6.Google Scholar
  132. 132.
    Li JR, White J (2001) Reduction of large circuit models via low rank approximate gramians. Int J Appl Math Comput Sci 11(5):1151–1171zbMATHMathSciNetGoogle Scholar
  133. 133.
    Li JR, White J (2002) Low rank solution of Lyapunov equations. SIAM J Matrix Anal Appl 24(1):260–280MathSciNetGoogle Scholar
  134. 134.
    Li P, Pileggi LT (2003) NORM: compact model order reduction of weakly nonlinear systems. In: Proceedings of design automation conference, pp 472–477.Google Scholar
  135. 135.
    Li P, Shi W (2006) Model order reduction of linear networks with massive ports via frequency-dependent port packing. In: Proceedings of international conference on computer-aided design, pp 267–272.Google Scholar
  136. 136.
    Li R, Bai Z (2005) Structure-preserving model reduction using a Krylov subspace projection formulation. Commun Math Sci 3(2):179–199zbMATHMathSciNetGoogle Scholar
  137. 137.
    Lin Y, Bao L, Wei Y (2007) A model-order reduction method based on Krylov subspace for MIMO bilinear dynamical systems. J Appl Math Comput 25(1–2):293–304MathSciNetGoogle Scholar
  138. 138.
    Lin Y, Simoncini V (2013) Minimal residual methods for large scale Lyapunov equations. Appl Numer Math 72:52–71zbMATHMathSciNetGoogle Scholar
  139. 139.
    Liu P, Tan XD, Li H, Qi Z, Kong J, McGaughy B, He L (2005) An efficient method for termical reduction of interconnect circuits considering delay variations. In: Proceedings of international conference on computer-aided design, pp 820–825.Google Scholar
  140. 140.
    Liu WQ, Sreeram V (2000) Model reduction of singular systems. In: Proceedings of 39th IEEE conference on decision and control 2000, pp 2373–2378.Google Scholar
  141. 141.
    Liu Y, Anderson BDO (1986) Controller reduction via stable factorization and balancing. Int J Control 44:507–531zbMATHGoogle Scholar
  142. 142.
    Marschall SA (1966) An approximate method for reducing the order of a linear system. Control Eng. 10:642–648Google Scholar
  143. 143.
    Mehrmann V, Stykel T. Balanced truncation model reduction for large-scale systems in descriptor form. Chapter 3 (pp 83–115) of [41].Google Scholar
  144. 144.
    Moore BC (1981) Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans Autom Control AC-26:17–32.Google Scholar
  145. 145.
    Mullis C, Roberts RA (1976) Synthesis of minimum roundoff noise fixed point digital filters. IEEE Trans Circuits Syst CAS-23(9):551–562.Google Scholar
  146. 146.
    Nakhla N, Nakhla MS, Achar R (2007) Sparse and passive reduction of massively coupled large multiport interconnects. In: Proceedings of international conference on computer-aided design, pp 622–626.Google Scholar
  147. 147.
    Nguyen N, Patera AT, Peraire J (2008) A best points interpolation method for efficient approximation of parametrized functions. Int J Numer Methods Eng 73(4):521–543zbMATHMathSciNetGoogle Scholar
  148. 148.
    Nouy A (2009) Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch Comput Methods Eng 16(3):251–285. doi: 10.1007/s11831-009-9034-5 MathSciNetGoogle Scholar
  149. 149.
    Odabasioglu A, Celik M, Pileggi LT (1998) PRIMA: passive reduced-order interconnect macromodeling algorithm. IEEE Trans Comput-Aided Des Integr Circuits Syst 17(8):645–654Google Scholar
  150. 150.
    Opmeer MR (2012) Model order reduction by balanced proper orthogonal decomposition and by rational interpolation. IEEE Trans Autom Control 57(2):472–477MathSciNetGoogle Scholar
  151. 151.
    Panzer H, Jaensch S, Wolf T, Lohmann B (2013) A greedy rational Krylov method for h2-pseudooptimal model order reduction with preservation of stability. In: Proceedings of the American control conference, pp 5512–5517.Google Scholar
  152. 152.
    Panzer H, Wolf T, Lohmann B (2013) \(H_2\) and \(H_{\infty }\) error bounds for model order reduction of second order systems by Krylov subspace methods. In: Proceedings of the European control conference, pp 4484–4489.Google Scholar
  153. 153.
    Penzl T (1997) A multi-grid method for generalized Lyapunov equations. Tech. Rep. SFB393/97-24, Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, FRG. Available from http://www.tu-chemnitz.de/sfb393/sfb97pr.html
  154. 154.
    Penzl T (1999) /00) A cyclic low-rank Smith method for large sparse Lyapunov equations. SIAM, J Sci Comput 21(4):1401–1418MathSciNetGoogle Scholar
  155. 155.
    Penzl T (2000) Eigenvalue decay bounds for solutions of Lyapunov equations: the symmetric case. Syst Control Lett 40:139–144zbMATHMathSciNetGoogle Scholar
  156. 156.
    Penzl T (2006) Algorithms for model reduction of large dynamical systems. Linear Algebra Appl 415(2–3):322–343 Reprint of Technical Report SFB393/99-40, TU Chemnitz, 1999.Google Scholar
  157. 157.
    Perev K, Shafai B (1994) Balanced realization and model reduction of singular systems. Int J Syst Sci 25(6):1039–1052zbMATHMathSciNetGoogle Scholar
  158. 158.
    Pernebo L, Silverman LM (1982) Model reduction via balanced state space representations. IEEE Trans Autom Control 27(2):382–387zbMATHMathSciNetGoogle Scholar
  159. 159.
    Phillips JR (2000) Projection frameworks for model reduction of weakly nonlinear systems. In: Proceedings of design automation conference, pp 184–189.Google Scholar
  160. 160.
    Phillips JR (2003) Projection-based approaches for model reduction of weakly nonlinear time-varying systems. IEEE Trans Comput-Aided Des Integr Circuits Syst 22(2):171–187Google Scholar
  161. 161.
    Phillips JR, Daniel L, Silveira LM (2003) Guaranteed passive balancing transformations for model order reduction. IEEE Trans Comput-Aided Des Integr Circuits Syst 22(8):1027–1041Google Scholar
  162. 162.
    Phillips JR, Silveira LM (2005) Poor man’s TBR: a simple model reduction scheme. IEEE Trans Comput-Aided Des Integr Circuits Syst 24(1):43–55Google Scholar
  163. 163.
    Pillage LT, Rohrer RA (1990) Asymptotic waveform evaluation for timing analysis. IEEE Trans Comput-Aided Des 9:325–366Google Scholar
  164. 164.
    Rabiei P, Pedram M (1999) Model order reduction of large circuits using balanced truncation. In: Proceedings of Asia and South Pacific design automation conference pp 237–240.Google Scholar
  165. 165.
    Reis T, Stykel T (2010) Positive real and bounded real balancing for model reduction of descriptor systems. Int J Control 83(1):74–88zbMATHMathSciNetGoogle Scholar
  166. 166.
    Rewieński M, White J (2003) A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Trans Comput-Aided Des Integr Circuits Syst 22(2):155–170Google Scholar
  167. 167.
    Rewieński M, White J (2006) Model order reduction for nonlinear dynamical systems based on trajectory piecewise-linear approximations. Linear Algebra Appl 415(2–3):426–454zbMATHMathSciNetGoogle Scholar
  168. 168.
    Rosen IG, Wang C (1995) A multi-level technique for the approximate solution of operator Lyapunov and algebraic Riccati equations. SIAM J Numer Anal 32(2):514–541zbMATHMathSciNetGoogle Scholar
  169. 169.
    Roychowdhury J (1999) Reduced-order modeling of time-varying system. IEEE Trans Circuits Syst II 46(10):1273–1288Google Scholar
  170. 170.
    Rozza G, Huynh DBP, Patera AT (2008) Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch Comput Methods Eng 15(3):229–275. doi: 10.1007/s11831-008-9019-9 zbMATHMathSciNetGoogle Scholar
  171. 171.
    Rugh WJ (1981) Nonlinear system theory. The Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  172. 172.
    Saad Y (1990) Numerical solution of large Lyapunov equations. In: Kaashoek MA, van Schuppen JH, Ran ACM (eds) Signal processing, scattering, operator theory and numerical methods. Birkhäuser, pp 503–511.Google Scholar
  173. 173.
    Safonov MG, Chiang RY (1989) A Schur method for balanced truncation model reduction. IEEE Trans Autom Control 34(7):729–733zbMATHMathSciNetGoogle Scholar
  174. 174.
    Salimbahrami B, Lohmann B (2006) Order reduction of large scale second-order systems using Krylov subspace methods. Linear Algebra Appl 415(2):385–405zbMATHMathSciNetGoogle Scholar
  175. 175.
    Saraswat D, Achar R, Nakhla MS (2005) Projection based fast passive compact macromodeling of high-speed VLSI circuits and interconnects. In: Proceedings of 18th international conference VLSI design 2005, pp 629–633.Google Scholar
  176. 176.
    Sastry S (1999) Nonlinear systems: analysis, stability, and control, interdisciplinary applied mathematics, vol 10. Springer, New YorkGoogle Scholar
  177. 177.
    Siebert WM (1986) Circuits, signals, and systems. MIT Press, CambridgeGoogle Scholar
  178. 178.
    Silveira LM, Kamon M, Elfadel I, White J (1996) A coordinate-transformed Arnoldi algorithm for generating guaranteed stable reduced order models of arbitrary RLC circuits. In: Proceedings of international conference on computer-aided design, pp 288–294.Google Scholar
  179. 179.
    Silveira LM, Phillips JR (2004) Exploiting input information in a model reduction algorithm for massively coupled parasitic networks. In: Proceedings of design automation conference, pp 385–388.Google Scholar
  180. 180.
    Simoncini V (2007) A new iterative method for solving large-scale Lyapunov matrix equations. SIAM J Sci Comput 29(3):1268–1288zbMATHMathSciNetGoogle Scholar
  181. 181.
    Simoncini V, Druskin V (2009) Convergence analysis of projection methods for the numerical solution of large Lyapunov equations. SIAM J Numer Anal 47(2):828–843zbMATHMathSciNetGoogle Scholar
  182. 182.
    Sorensen DC, Antoulas AC (2002) The Sylvester equation and approximate balanced reduction. Linear Algebra Appl 351(352):671–700MathSciNetGoogle Scholar
  183. 183.
    Sorensen DC, Zhou Y (2002) Bounds on eigenvalue decay rates and sensitivity of solutions to Lyapunov equations. Tech. Rep. TR02-07, Dept. of Comp. Appl. Math., Rice University, Houston, TX. Available online from http://www.caam.rice.edu/caam/trs/tr02.html#TR02-07
  184. 184.
    Stykel T (2004) Gramian-based model reduction for descriptor systems. Math Control Signals Syst 16(4):297–319zbMATHMathSciNetGoogle Scholar
  185. 185.
    Stykel T (2006) Balanced truncation model reduction for semidiscretized Stokes equation. Linear Algebra Appl 415(2–3):262–289zbMATHMathSciNetGoogle Scholar
  186. 186.
    Su YF, Wang J, Zeng X, Bai Z, Chiang C, Zhou D (2004) SAPOR: second-order Arnoldi method for passive order reduction of RCS circuits. In: Proceedings of international conference on computer-aided design, pp 74–79.Google Scholar
  187. 187.
    Tiwary SK, Rutenbar RA (2005) Scalable trajectory methods for on-demand analog macromodel extraction. In: Proceedings of design automation conference, pp 403–408.Google Scholar
  188. 188.
    Tombs MS, Postlethwaite I (1987) Truncated balanced realization of a stable non-minimal state-space system. Int J Control 46(4):1319–1330zbMATHMathSciNetGoogle Scholar
  189. 189.
    The MathWorks Inc, http://www.matlab.com
  190. 190.
    Van Dooren P (2000) Gramian based model reduction of large-scale dynamical systems. In: Griffiths D, Watson G (eds) Numerical analysis 1999. Proceedings of 18th Dundee Biennial conference on numerical analysis, pp 231–247. London.Google Scholar
  191. 191.
    Van Dooren P, Gallivan KA, Absil PA (2008) \({{\cal H}}_2\)-optimal model reduction of MIMO systems. Appl Math Lett 21(12):1267–1273zbMATHMathSciNetGoogle Scholar
  192. 192.
    Vandereycken B, Vandewalle S (2010) A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations related databases. SIAM J Matrix Anal Appl 31(5):2553–2579zbMATHMathSciNetGoogle Scholar
  193. 193.
    Varga A (1991) Efficient minimal realization procedure based on balancing. Preparation of the IMACS symposium on modelling and control of technological systems 2:42–47Google Scholar
  194. 194.
    Vasilyev D, Rewieński M, White J (2003) A TBR-based trajectory piecewise-linear algorithm for generating accurate low-order models for nonlinear analog circuits and MEMS. In: Proceedings of design automation conference, pp 490–495.Google Scholar
  195. 195.
    Villena JF, Silveira LM (2011) Multi-dimensional automatic sampling schemes for multi-point modeling methodologies. IEEE Trans Comput-Aided Des Integr Circuits Syst 30(8):1141–1151Google Scholar
  196. 196.
    Willcox K, Peraire J (2002) Balanced model reduction via the proper orthogonal decomposition. AIAA J 40(11):2323–2330Google Scholar
  197. 197.
    Wittig T, Munteanu I, Schuhmann R, Weiland T (2002) Two-step Lanczos algorithm for model order reduction. IEEE Trans Magn 38:673–676Google Scholar
  198. 198.
    Wolf T, Panzer H, Lohmann B (2011) Gramian-based error bound in model reduction by Krylov subspace methods. In: Proceedings of IFAC World Congress, pp 3587–3591.Google Scholar
  199. 199.
    Wolf T, Panzer H, Lohmann B (2012) ADI-Lösung großer Ljapunow-Gleichungen mittels Krylov-Methoden und neue Formulierung des Residuums (in German). In: Sawodny O, Adamy J (eds) Tagungsband GMA-FA 1.30, ’Modellbildung, Identifikation und Simulation in der Automatisierungstechnik’, pp 291–303.Google Scholar
  200. 200.
    Wolf T, Panzer H, Lohmann B (2012) Sylvester equations and the factorization of the error system in Krylov subspace methods. In: Proceedings of the 7th Vienna conference on mathematical modelling (MATHMOD). Vienna, Austria. Google Scholar
  201. 201.
    Wolf T, Panzer H, Lohmann B (2013) Model reduction by approximate balanced truncation: a unifying framework. at-Automatisierungstechnik 61(8):545–556.Google Scholar
  202. 202.
    Yan B, Tan XD, Liu P, McGaughy B (2007) Passive interconnect macromodeling via balanced truncation of linear systems in descriptor form. In: Proceedings of design automation conference, pp 355–360.Google Scholar
  203. 203.
    Yan B, Tan XD, Liu P, McGaughy B (2007) SBPOR: second-order balanced truncation for passive order reduction of RLC circuits. In: Proceedings of design automation conference, pp 158–161.Google Scholar
  204. 204.
    Zhang L, Lam J (2002) On \({\cal H}_2\) model reduction of bilinear systems. Automatica 38(2):205–216zbMATHMathSciNetGoogle Scholar
  205. 205.
    Zhang Y, Liu H, Wang Q, Fong N, Wong N (2012) Fast nonlinear model order reduction via associated transforms of high-order Volterra transfer functions. In: Proceedings of design automation conference, pp 289–294.Google Scholar
  206. 206.
    Zhou K, Salomon G, Wu E (1999) Balanced realization and model reduction for unstable systems. Int J Robust Nonlinear Control 9(3):183–198zbMATHMathSciNetGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2014

Authors and Affiliations

  1. 1.Max Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany

Personalised recommendations