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Parameterized Model Order Reduction

  • Chapter
Coupled Multiscale Simulation and Optimization in Nanoelectronics

Part of the book series: Mathematics in Industry ((TECMI,volume 21))

Abstract

This Chapter introduces parameterized, or parametric, Model Order Reduction (pMOR). The Sections are offered in a prefered order for reading, but can be read independently. Section 5.1, written by Jorge Fernández Villena, L. Miguel Silveira, Wil H.A. Schilders, Gabriela Ciuprina, Daniel Ioan and Sebastian Kula, overviews the basic principles for pMOR. Due to higher integration and increasing frequency-based effects, large, full Electromagnetic Models (EM) are needed for accurate prediction of the real behavior of integrated passives and interconnects. Furthermore, these structures are subject to parametric effects due to small variations of the geometric and physical properties of the inherent materials and manufacturing process. Accuracy requirements lead to huge models, which are expensive to simulate and this cost is increased when parameters and their effects are taken into account. This Section introduces the framework of pMOR, which aims at generating reduced models for systems depending on a set of parameters.

Section 5.2, written by Gabriela Ciuprina, Alexandra Ştefănescu, Sebastian Kula and Daniel Ioan, provides robust procedures for pMOR. This Section proposes a robust specialized technique to extract reduced parametric compact models, described as parametric SPICE-like netlists, for long interconnects modeled as transmission lines with several field effects such as skin effect and substrate losses. The technique uses an EM formulation based on partial differential equations (PDE), which is discretized to obtain a finite state space model. Next, a variability analysis of the geometrical data is carried out. Finally, a method to extract an equivalent parametric circuit is proposed.

Section 5.3, written by Michael Striebel, Roland Pulch, E. Jan W. ter Maten, Zoran Ilievski, and Wil H.A. Schilders, covers ways to efficiently determine sensitivity of output with respect to parameters. First direct and adjoint techniques are considered with forward and backward time integration, respectively. Here also the use of MOR via POD (Proper Orthogonal Decomposition) is discussed. Next, techniques in Uncertainty Quantification are described. Here pMOR techniques can be used efficiently.

Section 5.4, written by Kasra Mohaghegh, Roland Pulch and E. Jan W. ter Maten, provides a novel way in extending MOR to Differential-Algebraic Systems. Here new MOR techniques for reducing semi-explicit system of DAEs are introduced. These techniques are extendable to all linear DAEs. Especially pMOR techniques are exploited for singularly perturbed systems.

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Notes

  1. 1.

    Section 5.1 has been writen by: Jorge Fernández Villena, L. Miguel Silveira, Wil H.A. Schilders, Gabriela Ciuprina, Daniel Ioan and Sebastian Kula. For additional topics and applications see also the Ph.D.-Thesis of the last author [20].

  2. 2.

    Section 5.2 has been written by: Gabriela Ciuprina, Alexandra Ştefănescu, Sebastian Kula and Daniel Ioan. For additional topics see also the Ph.D.-Theses of the second author [59] and of the third author [56].

  3. 3.

    Section 5.3 has been written by: Michael Striebel, Roland Pulch, E. Jan W. ter Maten, Zoran Ilievski, and Wil H.A. Schilders. Of parts of this Section extensions can be found in the Ph.D.-Thesis of the fourth author [95].

  4. 4.

    Section 5.4 has been written by: Kasra Mohaghegh, Roland Pulch and E. Jan W. ter Maten. For an extended version we refer to the Ph.D.-Thesis [135] of the first author and to the papers [136, 137].

References

References for Section 5.1

  1. Achar, R., Nakhla, M.S.: Simulation of high-speed interconnects. Proc. IEEE 89(5), 693–728 (2001)

    Article  Google Scholar 

  2. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  3. Benner, P., Mehrmann, V., Sorensen, D.C. (eds.): Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45. Springer, Berlin/Heidelberg (2005)

    Google Scholar 

  4. Bi, Y.: Effects of paramater variations on integrated circuits. MSc.-Thesis, Delft University of Technology/Technical University Eindhoven, The Netherlands (2007)

    Google Scholar 

  5. Bi, Y., van der Kolk, K.-J., Ioan, D., van der Meijs, N.P.: Sensitivity computation of interconnect capacitances with respect to geometric parameters. In: Proceedings of the IEEE International Conference on Electrical Performance of Electronic Packaging (EPEP), San Jose, CA, USA, pp. 209–212 (2008)

    Google Scholar 

  6. Ciuprina, G., Ioan, D., Niculae, D., Fernández Villena, J., Silveira, L.M.: Parametric models based on sensitivity analysis for passive components. In: Intelligent Computer Techniques in Applied Electromagnetics. Studies in Computational Intelligence Series, vol. 119, pp. 231–239. Springer, Berlin/Heidelberg (2008)

    Google Scholar 

  7. Ciuprina, G., Loan, D., Mihalache, D., Seebacher, E.: Domain partitioning based parametric models for passive on-chip components. In: Roos, J., Costa, L. (eds.) Scientific Computing in Electrical Engineering 2008. Mathematics in Industry, vol. 14, pp. 37–44. Springer, Berlin/Heidelberg (2010)

    Chapter  Google Scholar 

  8. Daniel, L., Siong, O.C., Low, S.C., Lee, K.H., White, J.K.: A multiparameter moment-matching model-reduction approach for generating geometrically parametrized interconnect performance models. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 23, 678–693 (2004)

    Article  Google Scholar 

  9. Davis, T.A.: Direct Methods for Sparse Linear Systems. The Fundamentals of Algorithms. SIAM, Philadelphia (2006)

    Book  MATH  Google Scholar 

  10. Davis, T.A., Palamadai Natarajan, E.: Algorithm 907: KLU, a direct sparse solver for circuit simulation problems. ACM Trans. Math. Softw. 37(3), Article 36 (2010)

    Google Scholar 

  11. Elfadel, I.M., Ling, D.L.: A block rational Arnoldi algorithm for multipoint passive model-order reduction of multiport RLC networks. In: Proceedings of the International Conference on Computer Aided-Design (ICCAD), San Jose, pp. 66–71 (1997)

    Google Scholar 

  12. El-Moselhy, T.A., Elfadel, I.M., Daniel, L.: A capacitance solver for incremental variation-aware extraction. In: Proceedings of the IEEE/ACM International Conference on Computer Aided-Design (ICCAD), San Jose, pp. 662–669 (2008)

    Google Scholar 

  13. Feldmann, P., Freund, R.W.: Efficient linear circuit analysis by Padé approximation via the Lanczos process. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 14(5), 639–649 (1995)

    Article  Google Scholar 

  14. Fernández Villena, J., Schilders, W.H.A., Silveira, L.M.: Parametric structure-preserving model order reduction. In: IFIP International Conference on Very Large Scale Integration, VLSI – SoC 2007, Atlanta, pp. 31–36 (2007)

    Google Scholar 

  15. Freund, R.W.: Sprim: structure-preserving reduced-order interconnect macro-modeling. In: Proceedings of the International Conference on Computer Aided-Design (ICCAD) 2004, San Jose, pp. 80–87 (2004)

    Google Scholar 

  16. Gunupudi, P.K., Khazaka, R., Nakhla, M.S., Smy, T., Celo, D.: Passive parameterized time-domain macromodels for high-speed transmission-line networks. IEEE Trans. Microw. Theory Tech. 51(12), 2347–2354 (2003)

    Article  Google Scholar 

  17. Heydari, P., Pedram, M.: Model reduction of variable-geometry interconnects using variational spectrally-weighted balanced truncation. In: Proceedings of the International Conference on Computer Aided-Design (ICCAD), San Jose, pp. 586–591 (2001)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Kamon, M., Wang, F., White, J.: Generating nearly optimally compact models from Krylov-subspace based reduced-order models. IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process. 47(4), 239–248 (2000)

    Article  Google Scholar 

  20. Kula, S.: Reduced order models of interconnects in high frequency integrated circuits. Ph.D.-Thesis, Politehnica University of Bucharest (2009)

    Google Scholar 

  21. Li, J.-R., Wang, F., White, J.: Efficient model reduction of interconnect via approximate system Grammians. In: Proceedings of the International Conference on Computer Aided-Design (ICCAD), San Jose, pp. 380–383 (1999)

    Google Scholar 

  22. Li, P., Liu, F., Li, X., Pileggi, L.T., Nassif, S.R.: Modeling interconnect variability using efficient parametric model order reduction. In: Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE), Munich, pp. 958–963 (2005)

    Google Scholar 

  23. Li, X., Li, P., Pileggi, L.: Parameterized interconnect order reduction with explicit-and-implicit multi-parameter moment matching for inter/intra-die variations. In: Proceedings of the International Conference on Computer Aided-Design (ICCAD), San Jose, pp. 806–812 (2005)

    Google Scholar 

  24. Li, Y., Bai, Z., Su, Y., Zeng, X.: Model order reduction of parameterized interconnect networks via a two-directional Arnoldi process. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 27(9), 1571–1582 (2008)

    Article  Google Scholar 

  25. Liu, Y., Pileggi, L.T., Strojwas, A.J.: Model order reduction of RC(L) interconnect including variational analysis. In: Proceedings of the 36th ACM/IEEE Design Automation Conference (DAC), New Orleans, pp. 201–206 (1999)

    Google Scholar 

  26. Moore, B.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control AC-26(1), 17–32 (1981)

    Article  Google Scholar 

  27. Odabasioglu, A., Celik, M., Pileggi, L.T.: PRIMA: passive reduced-order interconnect macromodeling algorithm. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 17(8), 645–654 (1998)

    Article  Google Scholar 

  28. Phillips, J.: Variational interconnect analysis via PMTBR. In: Proceedings of the International Conference on Computer Aided-Design (ICCAD), San Jose, pp. 872–879 (2004)

    Google Scholar 

  29. Phillips, J.R., Silveira, L.M.: Poor Man’s TBR: a simple model reduction scheme. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 24(1), 43–55 (2005)

    Article  Google Scholar 

  30. Phillips, J., Daniel, L., Silveira, L.M.: Guaranteed passive balancing transformations for model order reduction. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 22(8), 1027–1041 (2003)

    Article  Google Scholar 

  31. Pillage, L.T., Rohrer, R.A.: Asymptotic waveform evaluation for timing analysis. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 9(4), 352–366 (1990)

    Article  Google Scholar 

  32. Saad, Y.: Iterative Methods for Sparse Linear Systems. Pws Publishing Co., Boston (1996)

    MATH  Google Scholar 

  33. Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.): Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13. Springer, Berlin (2008)

    Google Scholar 

  34. Silveira, L.M., Kamon, M., Elfadel, I., White, J.K.: A coordinate-transformed Arnoldi algorithm for generating guaranteed stable reduced-order models of RLC circuits. In: Proceedings of the International Conference on Computer Aided-Design (ICCAD), San Jose, pp. 288–294 (1996)

    Google Scholar 

  35. Yang, F., Zeng, X., Su, Y., Zhou, D.: RLCSYN: RLC equivalent circuit synthesis for structure-preserved reduced-order model of interconnect. In: Proceedings of the International Symposium on Circuits and Systems, New Orleans, pp. 2710–2713 (2007)

    Google Scholar 

  36. Yu, H., He, L., Tan, S.X.D.: Block structure preserving model order reduction. In: BMAS – IEEE Behavioral Modeling and Simulation Workshop, San Jose, pp. 1–6 (2005)

    Google Scholar 

  37. Zhu, Z., Phillips, J.: Random sampling of moment graph: a stochastic Krylov-reduction algorithm. In: Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE), Nice, pp. 1502–1507 (2007)

    Google Scholar 

References for Section 5.2

  1. Bi, Y., van der Meijs, N., Ioan, D.: Capacitance sensitivity calculation for interconnects by adjoint field technique. In: Proceedings of the 12th IEEE Workshop on Signal Propagation on Interconnects (SPI-2008), pp. 1–4 (2008)

    Google Scholar 

  2. Bossavit, A.: Most general non-local boundary conditions for the Maxwell equations in a bounded region. COMPEL: Int. J. Comput. Math. Electr. Electron. Eng. 19(2), pp. 239–245 (2000)

    MATH  MathSciNet  Google Scholar 

  3. CHAMELEON-RF website. http://www.chameleon-rf.org

  4. Ciuprina, G., Ioan, D., Niculae, D., Fernandez Villena, J., Silveira, L.: Parametric models based on sensitivity analysis for passive components. In: Wiak, S., Krawczyk, A., Dolezel, I. (eds.) Intelligent Computer Techniques in Applied Electromangetics. Studies in Computational Intelligence, vol. 119, pp. 231–239. Springer, Berlin (2008)

    Google Scholar 

  5. Clemens, M., Weiland, T.: Discrete electromagnetism with the finite integration technique. Prog. Electromagn. Res. 32, 65–87 (2001)

    Article  Google Scholar 

  6. Deschrijver, D., Mrozowski, M., Dhaene, T., De Zutter, D.: Macromodeling of multiport systems using a fast implementation of the vector fitting method. IEEE Microw. Wirel. Compon. Lett. 18(6), 383–385 (2008)

    Article  Google Scholar 

  7. Ferranti, F., Antonini, G., Dhaene, T., Knockaert, L.: Passivity-preserving parameterized model order reduction for PEEC based full wave analysis. In: Proceedings of the 14th IEEE Workshop on Signal Propagation on Interconnects, pp. 65–68 (2010)

    Google Scholar 

  8. Goel, A.K.: High Speed VLSI Interconnections. Wiley Series in Microwave and Optical Engineering. Wiley-IEEE Press, John Wiley & Sons, Hoboken, NJ, USA (2007)

    Book  Google Scholar 

  9. Gustavsen, B.: Improving the pole relocating properties of vector fitting. IEEE Trans. Power Deliv. 21(3), 1587–1592 (2006)

    Article  Google Scholar 

  10. Gustavsen, B., Semlyen, A.: Rational approximation of frequency domain responses by vector fitting. IEEE Trans. Power Deliv. 14(3), 1052–1061 (1999)

    Article  Google Scholar 

  11. Ioan, D., Ciuprina, G.: Reduced order models of on-chip passive components and interconnects, workbench and test structures. In: Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.) Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13, pp. 447–467. Springer, Berlin (2008)

    Chapter  Google Scholar 

  12. Ioan, D., Ciuprina, G., Radulescu, M.: Algebraic sparsefied partial equivalent electric circuit – ASPEEC. In: Anile, A.M., Alì, G., Mascali, G. (eds.) Scientific Computing in Electrical Engineering. Series Mathematics in Industry vol. 9, pp. 45–50. Springer, Berlin (2006)

    Chapter  Google Scholar 

  13. Ioan, D., Ciuprina, G., Radulescu, M.: Absorbing boundary conditions for compact modeling of on-chip passive structures. COMPEL: Int. J. Comput. Math. Electr. Electron. Eng. 25(3), 652–659 (2006)

    Article  MATH  Google Scholar 

  14. Ioan, D., Ciuprina, G., Radulescu, M., Seebacher, E.: Compact modeling and fast simulation of on-chip interconnect lines. IEEE Trans. Magn. 42(4), 547–550 (2006)

    Article  Google Scholar 

  15. Ioan, D., Ciuprina, G., Kula, S.: Reduced order models for HF interconnect over lossy semiconductor substrate. In: Proceedings of the 11th IEEE Workshop on SPI, pp. 233–236 (2007)

    Google Scholar 

  16. Ioan, D., Ciuprina, G., Schilders, W.: Parametric models based on the adjoint field technique for RF passive integrated components. IEEE Trans. Magn. 44(6), 1658–1661 (2008)

    Article  Google Scholar 

  17. Ioan, D., Schilders, W., Ciuprina, G., van der Meijs, N., Schoenmaker, W.: Models for integrated components coupled with their environment. COMPEL: Int. J. Comput. Math. Electr. Electron. Eng. 27(4), 820–828 (2008)

    Article  MATH  Google Scholar 

  18. Kinzelbach, H.: Statistical variations of interconnect parasitics: extraction and circuit simulation. In: Proceedings of the 10th IEEE Workshop on Signal Propagation on Interconnects (SPI), pp. 33–36 (2006)

    Google Scholar 

  19. Kula, S.: Reduced order models of interconnects in high frequency integrated circuits. Ph.D.-Thesis, Politehnica University of Bucharest (2009)

    Google Scholar 

  20. Palenius, T., Roos, J.: Comparison of reduced-order interconnect macromodels for time-domain simulation. IEEE Trans. Microw. Theory Tech. 52(9), 2240-2250 (2004)

    Article  Google Scholar 

  21. Răduleţ, R., Timotin, A., Ţugulea, A.: The propagation equations with transient parameters for long lines with losses. Rev. Roum. Sci. Tech. 15(4), 585–599 (1979)

    Google Scholar 

  22. Ştefănescu, A.: Parametric models for interconnections from analogue high frequency integrated circuits. Ph.D.-Thesis, Politehnica University of Bucharest (2009)

    Google Scholar 

  23. Stefanescu, A., Ioan, D., Ciuprina, G.: Parametric models of transmission lines based on first order sensitivities. In: Roos, J., Costa, L.R.J. (eds.) Scientific Computing in Electrical Engineering 2008. Mathematics in Industry, vol. 14, pp. 29–36. Springer, Berlin/Heidelberg (2010)

    Chapter  Google Scholar 

  24. The Vector Fitting Web Site. http://www.energy.sintef.no/produkt/VECTFIT/index.asp

  25. van der Meijs, N., Fokkema, J.: VLSI circuit reconstruction from mask topology. VLSI J. Integr. 2(3), 85–119 (1984)

    Article  Google Scholar 

References for Section 5.3

  1. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM Publications, Philadelphia (2005)

    Book  MATH  Google Scholar 

  2. Armbruster, H., Feldmann, U., Frerichs, M.: Analysis based reduction using sensitivity analysis. In: Proceedings of the 10th IEEE Workshop on Signal Propagation on Interconnects (SPI), pp. 29–32 (2006)

    Google Scholar 

  3. Augustin, F., Gilg, A., Paffrath, M., Rentrop, P., Wever, U.: Polynomial chaos for the approximation of uncertainties: chances and limits. Eur. J. Appl. Math. 19, 149–190 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baur, U., Beattie, C., Benner, P., Gugercin, S.: Interpolatory projection methods for parameterized model reduction. SIAM J. Comput. 33, 2489–2518 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Benner, P.: Advances in balancing-related model reduction for circuit simulation. In: Roos, J., Costa, L.R.J. (eds.) Scientific Computing in Electrical Engineering SCEE 2008. Mathematics in Industry, vol. 14, pp. 469–482. Springer, Berlin/Heidelberg (2010)

    Chapter  Google Scholar 

  6. Benner, P., Schneider, A.: Model reduction for linear descriptor dystems 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, pp. 137–143. Springer, Berlin/New York (2012)

    Chapter  Google Scholar 

  7. Benner, P., Mehrmann, V., Sorensen, D. (eds.): Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering, vol. 45. Springer, Berlin (2005)

    Google Scholar 

  8. Benner, P., Hinze, M., ter Maten, E.J.W. (eds.): Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74. Springer, Berlin (2011)

    Google Scholar 

  9. Bi, Y., van der Kolk, K.-J., Fernández Villena, J., Silveira, L.M., van der Meijs, N.: Fast statistical analysis of RC nets subject to manufacturing variabilities. In: Proceedings of the DATE 2011, Grenoble, 14–18 Mar 2011

    Google Scholar 

  10. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods – Fundamentals in Single Domains. Springer, Berlin (2010)

    Google Scholar 

  11. Cao, Y., Petzold, L.R.: A posteriori error estimation and global error control for ordinary differential equations by the adjoint method. SIAM J. Sci. Comput. 26(2), 359–374 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cao, Y., Li, S., Petzold, L.: Adjoint sensitivity analysis for differential-algebraic equations: algorithms and software. SIAM J. Sci. Comput. 149, 171–191 (2002)

    MATH  MathSciNet  Google Scholar 

  13. Cao, Y., Li, S., Petzold, L., Serban, R.: Adjoint sensitivity for differential-algebraic equations: the adjoint DAE system and its numerical solution. SIAM J. Sci. Comput. 24(3), 1076–1089 (2002)

    Article  MathSciNet  Google Scholar 

  14. Conn, A.R., Haring, R.A., Visweswariah, C., Wu, C.W.: Circuit optimization via adjoint Lagrangians. In: Proceedings of the ICCAD, San Jose, pp. 281–288 Nov 1997

    Google Scholar 

  15. Conn, A.R., Coulman, P.K., Haring, R.A., Morrill, G.L., Visweswariah, C., Wu, C.W.: JiffyTune: circuit optimization using time-domain sensitivities. IEEE Trans. CAD ICs Syst. 17(12), 1292–1309 (1998)

    Article  Google Scholar 

  16. Daldoss, L., Gubian, P., Quarantelli, M.: Multiparameter time-domain sensitivity computation. IEEE Trans. Circuits Syst. I: Fund. Theory Appl. 48(11), 1296–1307 (2001)

    Article  Google Scholar 

  17. Echeverría Ciaurri, D.: Multi-level optimization: space mapping and manifold mapping. Ph.D.-Thesis, University of Amsterdam (2007). http://dare.uva.nl/document/45897

  18. Echeverría, D., Lahaye, D., Hemker, P.W.: Space mapping and defect correction. In: Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.) Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13, pp. 157–176. Springer, Berlin/Heidelberg (2008)

    Chapter  Google Scholar 

  19. Ernst, O.G., Mugler, A., Starkloff, H.-J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM: Math. Model. Numer. Anal. 46, 317–339 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Errico, R.M.: What is an adjoint model?. Bull. Am. Meteorol. Soc. 78, 2577–2591 (1997)

    Article  Google Scholar 

  21. Feng, L.: Parameter independent model order reduction. Math. Comput. Simul. 68(3), 221–234 (2005)

    Article  MATH  Google Scholar 

  22. Feng, L., Benner, P.: A robust algorithm for parametric model order reduction. PAMM Proc. Appl. Math. Mech. 7, 1021501–1021502 (2007)

    Article  Google Scholar 

  23. Fernández Villena, J., Silveira, L.M.: Multi-dimensional automatic sampling schemes for multi-point modeling methodologies. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 30(8), 1141–1151 (2011)

    Article  Google Scholar 

  24. Fijnvandraat, J.G., Houben, S.H.M.J., ter Maten, E.J.W., Peters, J.M.F.: Time domain analog circuit simulation. J. Comput. Appl. Math. 185, 441–459 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gautschi, W.: OPQ: a Matlab suite of programs for generating orthogonal polynomials and related quadrature rules (2002). http://www.cs.purdue.edu/archives/2002/wxg/codes

  26. Gautschi, W.: Orthogonal polynomials (in Matlab). J. Comput. Appl. Math. 178, 215–234 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  28. Günther, M., Feldmann, U., ter Maten, J.: Modelling an discretization of circuit problems. In: Schilders, W.H.A., ter Maten, E.J.W. (eds.) Handbook of Numerical Analysis, Volume XIII. Special Volume: Numerical Methods in Electromagnetics, pp. 523–659, Chapter 6. Elsevier Science, Amsterdam/Boston (2005)

    Google Scholar 

  29. Häusler, R., Kinzelbach, H.: Sensitivity-based stochastic analysis method for power variations. In: Proceedings of the Analog ’06. VDE Verlag (2006)

    Google Scholar 

  30. Hemker, P.W., Echeverría, D.: Manifold mapping for multilevel optimization. In: Ciuprina, G., Ioan, D. (eds.) Scientific Computing in Electrical Engineering SCEE 2006. Series Mathematics in Industry, vol. 11, pp. 325–330. Springer, Berlin/New York (2007)

    Google Scholar 

  31. Hocevar, D.E., Yang, P., Trick, T.N., Epler, B.D.: Transient sensitivity computation for MOSFET circuits. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 4(4), 609–620 (1985)

    Article  Google Scholar 

  32. Homescu, C., Petzold, L.R., Serban, R.: Error estimation for reduced-order models of dynamical systems. SIAM Rev. 49(2), 277–299 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Ilievski, Z.: Model order reduction and sensitivity analysis. Ph.D.-Thesis, TU Eindhoven (2010). http://alexandria.tue.nl/extra2/201010770.pdf

  34. Ilievski, Z., Xu, H., Verhoeven, A., ter Maten, E.J.W., Schilders, W.H.A., Mattheij, R.M.M.: Adjoint transient sensitivity analysis in circuit simulation. In: Ciuprina, G., Ioan, D. (eds.) Scientific Computing in Electrical Engineering SCEE 2006. Series Mathematics in Industry, vol. 11, pp. 183–189. Springer, Berlin/New York (2007)

    Google Scholar 

  35. Ionutiu, R.: Model order reduction for multi-terminal Systems – with applications to circuit simulation. Ph.D.-Thesis, TU Eindhoven (2011). http://alexandria.tue.nl/extra2/716352.pdf

  36. Knyazev, A.V., Argentati, M.E.: Principle angles between subspaces in an A-based scalar product: algorithms and perturbation estimates. SIAM J. Sci. Comput. 23(6), 2009-2041 (2002). [Algorithm available in Matlab, The Mathworks, http://www.mathworks.com/]

  37. Lang, J., Verwer, J.G.: On global error estimation and control for initial value problems. SIAM J. Sci. Comput. 29(4), 1460-1475 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  38. Le Maître, O.P., Knio, O.M.: Spectral Methods for Uncertainty Quantification, with Applications to Computational Fluid Dynamics. Springer, Dordrecht (2010)

    Book  MATH  Google Scholar 

  39. Lutowska, A.: Model order reduction for coupled systems using low-rank approximations. Ph.D.-Thesis, TU Eindhoven (2012). http://alexandria.tue.nl/extra2/729804.pdf

  40. Parks, M.L., de Sturler, E., Mackey, G., Johnson, D.D., Maiti, S.: Recycling Krylov subspaces for sequences of linear systems. SIAM J. Sci. Comput. 28(5), 1651–1674 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  41. Pinnau, R.: Model reduction via proper orthogonal decomposition. In: Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.) Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13, pp. 95–109. Springer, Berlin/Heidelberg (2008)

    Chapter  Google Scholar 

  42. Pulch, R., ter Maten, E.J.W.: Stochastic Galerkin methods and model order reduction for linear dynamical systems. Provisionally accepted for International Journal for Uncertainty Quantification (2015)

    Google Scholar 

  43. Pulch, R., ter Maten, E.J.W., Augustin, F.: Sensitivity analysis of linear dynamical systems in uncertainty quantification. PAMM - Proceedings in Applied Mathematics and Mechanics, Vol. 13, Issue 1, pp. 507–508 (2013) DOI:10.1002/pamm.201310246

    Article  Google Scholar 

  44. Pulch, R., ter Maten, E.J.W., Augustin, F.: Sensitivity analysis and model order reduction for random linear dynamical systems. Mathematics and Computers in Simulation 111, pp. 80–95 (2015) DOI: http://dx.doi.org/10.1016/j.matcom.2015.01.003

    Google Scholar 

  45. Rathinam, M., Petzold, L.R.: A new look at proper orthogonal decomposition. SIAM J. Numer. Anal. 41(5), 1893–1925 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  46. Schilders, W.H.A.: Introduction to model order reduction. In: Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.) Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13, pp. 3–32. Springer, Berlin/Heidelberg (2008)

    Chapter  Google Scholar 

  47. Schilders, W.H.A.: The need for novel model order reduction techniques in the electronics industry. In: Benner, P., Hinze, M., ter Maten, E.J.W. (eds.): Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol. 74, pp. 3–23. Springer, Berlin (2011)

    Google Scholar 

  48. Schilders, W.H.A., van der Vorst, H.A., Rommes, J. (eds.) Model Order Reduction: Theory, Research Aspects and Applications. Mathematics in Industry, vol. 13. Springer, Berlin/Heidelberg (2008)

    Google Scholar 

  49. Stavrakakis, K.K.: Model order reduction methods for parameterized systems in electromagnetic field simulations. Ph.D.-Thesis, TU-Darmstadt (2012)

    Google Scholar 

  50. Stavrakakis, K., Wittig, T., Ackermann, W., Weiland, T.: Linearization of parametric FIT-discretized systems for model order reduction. IEEE Trans. Magn. 45(3), 1380–1383 (2009)

    Article  Google Scholar 

  51. Stavrakakis, K., Wittig, T., Ackermann, W., Weiland, T.: Three dimensional geometry variations of FIT systems for model order reduction. In: 2010 URSI International Symposium on Electromagnetic Theory, pp. 788–791 (2010)

    Google Scholar 

  52. Stavrakakis, K., Wittig, T., Ackermann, W., Weiland, T.: Model order reduction methods for multivariate parameterized dynamical systems obtained by the finite integration theory. In: 2011 URSI General Assembly and Scientifc Symposium, p. 4 (2011)

    Google Scholar 

  53. Stavrakakis, K., Wittig, T., Ackermann, W., Weiland, T.: Parametric model order reduction by neighbouring subspaces. In: Michielsen, B., Poirier, J.-R. (eds.) Scientific Computing in Electrical Engineering SCEE 2010. Series Mathematics in Industry, vol. 16, pp. 443–451. Springer, Berlin/New York (2012)

    Chapter  Google Scholar 

  54. Stroud, A.: Approximate Calculation of Multiple Integrals. Prentice Hall, Englewood Cliffs (1971)

    MATH  Google Scholar 

  55. SUMO (SUrrogate MOdeling) Lab. IBCN research group of the Department of Information Technology (INTEC), Ghent University (2012). http://www.sumo.intec.ugent.be/

  56. ter Maten, E.J.W., Heijmen, T.G.A., Lin, A., El Guennouni, A.: Optimization of electronic circuits. In: Cutello, V., Fotia, G., Puccio, L. (eds.) Applied and Industrial Mathematics in Italy II, Selected Contributions from the 8th SIMAI Conference. Series on Advances in Mathematics for Applied Sciences, vol. 75, pp. 573–584. World Scientific Publishing Co. Pte. Ltd., Singapore (2007)

    Google Scholar 

  57. ter Maten, E.J.W., Pulch, R., Schilders, W.H.A., Janssen, H.H.J.M.: Efficient calculation of Uncertainty Quantification. In: Fontes, M., Günther, M., Marheineke, N. (eds) Progress in Industrial Mathematics at ECMI 2012, Series Mathematics in Industry Vol. 19, Springer, pp. 361–370 (2014)

    Google Scholar 

  58. Ugryumova, M.V.: Applications of model order reduction for IC modeling. Ph.D.-Thesis, TU Eindhoven (2011). http://alexandria.tue.nl/extra2/711015.pdf

  59. Volkwein, S.: Model reduction using proper orthogonal decomposition (2008). http://www.uni-graz.at/imawww/volkwein/POD.pdf

  60. Xiu, D.: Numerical integration formulas of degree two. Appl. Numer. Math. 58, 1515–1520 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  61. Xiu, D.: Fast numerical methods for stochastic computations: a review. Commun. Comput. Phys. 5(2–4), 242–272 (2009)

    MathSciNet  Google Scholar 

  62. Xiu, D.: Numerical Methods for Stochastic Computations – A Spectral Method Approach. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

References for Section 5.4

  1. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. Advances in Design and Control. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  2. Antoulas, A.C.: A new result on passivity preserving model reduction. Syst. Control Lett. 54, 361–374 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)

    Book  MATH  Google Scholar 

  4. Daniel, L., Siong, O.C., Chay, L.S., Lee, K.H., White, J.: A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models. IEEE Trans. Comput. Aided Des. 23(5), 678–693 (2004)

    Article  Google Scholar 

  5. Feng, L., Benner, P.: A robust algorithm for parametric model order reduction based on implicit moment matching. PAMM Proc. Appl. Math. Mech. 7, 1021501–1021502 (2007)

    Article  Google Scholar 

  6. Günther, M.: Partielle differential-algebraische Systeme in der numerischen Zeitbereichsanalyse elektrischer Schaltungen. Nr. 343 in Fortschritt-Berichte VDI Serie 20. VDI, Düsseldorf (2001)

    Google Scholar 

  7. Günther, M., Feldmann, U.: CAD based electric circuit modeling in industry I: mathematical structure and index of network equations. Surv. Math. Ind. 8, 97–129 (1999)

    MATH  Google Scholar 

  8. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  9. Li, Y., Bai, Z., Su, Y., Zeng, X.: Parameterized model order reduction via a two-directional Arnoldi process. In: Proceedings of the IEEE/ACM International Conference on Computer-Aided Design (ICCAD), pp. 868–873 (2007)

    Google Scholar 

  10. Li, Y.-T., Bai, Z., Su, Y., Zeng, X.: Model order reduction of parameterized interconnect networks via a two-directional Arnoldi process. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 27(9), 1571–1582 (2008)

    Article  Google Scholar 

  11. Mohaghegh, K.: Linear and nonlinear model order reduction for numerical simulation of electric circuits. Ph.D.-Thesis, Bergische Universität Wuppertal, Germany. Available at Logos Verlag, Berlin (2010)

    Google Scholar 

  12. Mohaghegh, K., Pulch, R., Striebel, M., ter Maten, J.: Model order reduction for semi-explicit systems of differential algebraic systems. In: Troch, I., Breitenecker, F. (eds) Proceedings MATHMOD 09 Vienna – Full Papers CD Volume, pp. 1256–1265 (2009)

    Google Scholar 

  13. Mohaghegh, K., Pulch, R., ter Maten, J.: Model order reduction using singularly perturbed systems, provisionally accepted for J. of Applied Numerical Mathematics (APNUM) (2015)

    Google Scholar 

  14. Phillips, J., Silveira, L.M.: Poor’s man TBR: a simple model reduction scheme. In: Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE), vol. 2, pp. 938–943 (2004)

    Article  Google Scholar 

  15. Schwarz, D.E., Tischendorf, C.: Structural analysis of electric circuits and consequences for MNA. Int. J. Circuit Theory Appl. 28, 131–162 (2000)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Politehnica University of Bucharest, Spl.Independentei 313, 060042, Bucharest, Romania

    Gabriela Ciuprina & Daniel Ioan

  2. INESC ID/IST - TU Lisbon, Rua Alves Redol 9, 1000-029, Lisbon, Portugal

    Jorge Fernández Villena & L. Miguel Silveira

  3. European Space & Technology Centre, Keplerlaan 1, 299, 2200, AG Noordwijk, The Netherlands

    Zoran Ilievski

  4. Institute of Mechanics and Applied Computer Science, Kazimierz Wielki University, ul. Kopernika 1, 85-074, Bydgoszcz, Poland

    Sebastian Kula

  5. Applied Mathematics/Numerical Analysis, Bergische Universität Wuppertal, Gaußstraße 20, D-42119, Wuppertal, Germany

    E. Jan W. ter Maten (Chair)

  6. Department of Mathematics and Computer Science, CASA, Eindhoven University of Technology, 513, 5600, Eindhoven, The Netherlands

    E. Jan W. ter Maten (Chair)

  7. Multiscale in Mechanical and Biological Engineering (M2BE), Aragón Institute of Engineering Research (I3A), University of Zaragoza, María de Luna, 3, E-50018, Zaragoza, Spain

    Kasra Mohaghegh

  8. Institut für Mathematik und Informatik, Ernst Moritz Arndt Universität Greifswald, Walther-Rathenau-Straße 47, D-17487, Greifswald, Germany

    Roland Pulch

  9. Research Centre of Excellence “Micro- and nanosystems for radiofrequency and photonics”, IMT Bucharest – National Institute for Research and Development in Microtechnologies, 126A (32B), Erou Iancu Nicolae str., 077190, Bucharest, Romania

    Alexandra Ştefănescu

  10. ZF Lenksysteme GmbH, Richard-Bullinger-Straße 77, D-73527, Schwäbisch Gmünd, Germany

    Michael Striebel

  11. Department of Mathematics and Computer Science, CASA, Eindhoven University of Technology, 51, 5600, Eindhoven, The Netherlands

    Wil H. A. Schilders

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    Michael Günther

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Ciuprina, G. et al. (2015). Parameterized Model Order Reduction. In: Günther, M. (eds) Coupled Multiscale Simulation and Optimization in Nanoelectronics. Mathematics in Industry(), vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46672-8_5

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