Some a posteriori error bounds for reduced-order modelling of (non-)parametrized linear systems

Feng, Lihong; Antoulas, Athanasios C.; Benner, Peter

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2127-2158.

We propose a posteriori error bounds for reduced-order models of non-parametrized linear time invariant (LTI) systems and parametrized LTI systems. The error bounds estimate the errors of the transfer functions of the reduced-order models, and are independent of the model reduction methods used. It is shown that for some special non-parametrized LTI systems, particularly efficiently computable error bounds can be derived. According to the error bounds, reduced-order models of both non-parametrized and parametrized systems, computed by Krylov subspace based model reduction methods, can be obtained automatically and reliably. Simulations for several examples from engineering applications have demonstrated the robustness of the error bounds.

Classification : 37M05, 65P99, 65L70, 65L80
Mots clés : Model order reduction, error estimation

R. Achar and M.S. Nakhla, Simulation of high-speed interconnects. Proc. IEEE 89 (2001) 693–728. | DOI

A.C. Antoulas, P. Benner and L. Feng, Model reduction by iterative error-system approximation (2014).

D. Amsallem and C. Farhat, An online method for interpolating linear parametric reduced-order models. SIAM J. Sci. Comput. 33 (2011) 2169–2198. | DOI | MR | Zbl

Z. Bai, R.D. Slone, W.T. Smith and Q. Ye, Error bound for reduced system model by Padé approximation via the Lanczos process. Comput.-Aid. Design Integr. Circuits Syst. 18 (1999) 133–141. | DOI

U. Baur and P. Benner, Model reduction for parametric systems using balanced truncation and interpolation. at-Automatisierungstechnik 57 (2009) 411–419. | DOI

U. Baur, C. Beattie, P. Benner and S. Gugercin, Interpolatory projection methods for parameterized model reduction. SIAM J. Sci. Comput. 33 (2011) 2489–2518. | DOI | MR | Zbl

U. Baur, P. Benner and L. Feng, Model order reduction for linear and nonlinear systems: a system-theoretic perspective. Archives Comput. Methods Eng. 21 (2014) 331–358. | DOI | MR | Zbl

T. Bechtold, E.B. Rudnyi and J.G. Korvink, Error indicators for fully automatic extraction of heat-transfer macromodels for MEMS. Micromech. Microeng. 15 (2005) 430–440. | DOI

P. Benner, System-theoretic methods for model reduction of large-scale systems: simulation, control, and inverse Problems. In vol. 35 of Proc. of MATHMOD 2009, 6th Vienna International Conference on Mathematical Modelling, ARGESIM Report (2009) 126–145.

P. Benner, S. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57 (2015) 483–531. | DOI | MR | Zbl

P. Benner and L. Feng, A robust algorithm for parametric model order reduction based on implicit moment matching. In: Reduced Order Methods for modeling and computational reduction, edited by A. Quateroni, G. Rozza. Vol. 9 of Springer MS&A series (2014) 159–185. | MR

A. Bodendiek and M. Bollhöfer, Adaptive-order rational Arnoldi-type methods in computational electromagnetism. BIT Numer. Math. 54 (2014) 357–380. | DOI | MR | Zbl

T. Bonin, H. Fassbender, A. Soppa and M. Zaeh, A fully adaptive rational global Arnoldi method for the model-order reduction of second-order MIMO systems with proportional damping. Math. Comput. Simul. 122 (2016) 1–19. | DOI | MR | Zbl

S. Boyaval, Mathematical modelling and numerical simulation in materials science. Ph.D. thesis, Université Paris-Est (2009).

Y. Choi, D. Amsallem and C. Farhat, Gradient-Based Constrained Optimization Using a Database of Linear Reduced-Order Models. Preprint (2015). | arXiv

L. Daniel, O.C. Siong, L.S. Chay, K.H. Lee and J. White, A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models. IEEE Trans. Comput.-Aid. Design Integr. Circuits Syst. 23 (2004) 678–693. | DOI

L. Feng, D. Koziol, E. Rudnyi and J.G. Korvink, Model order reduction for scanning electrochemical microscope: The treatment of nonzero initial condition. In vol. 3 of Proc. of Sensors 3 (2004) 1236–1239.

L. Feng, E.B. Rudnyi and J.G. Korvink, Preserving the film coefficient as a parameter in the compact thermal model for fast electrothermal simulation. IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems 24 (2005) 1838–1847. | DOI

L. Feng and P. Benner, A robust algorithm for parametric model order reduction. Proc. Appl. Math. Mech. 7 (2017) 1021501–1021502. | DOI

L. Feng, J.G. Korvink and P. Benner, A fully adaptive scheme for model order reduction based on moment-matching. IEEE Trans. Components, Packaging Manuf. Technol. 5 (2015) 1872–1884. | DOI

L. Feng, P. Benner and J.G. Korvink, Subspace recycling accelerates the parametric macro-modeling of MEMS. Inter. J. Numer. Methods Eng. 94 (2013) 84–110. | DOI | Zbl

R.W. Freund, Model reduction methods based on Krylov subspaces. Acta Numer. 12 (2003) 267–319. | DOI | MR | Zbl

E.J. Grimme, Krylov projection methods for model reduction. Ph.D. thesis, Univ. Illinois, Urbana Champaign (1997).

M. Grepl, Reduced-basis approximation and a posteriori error estimation for parabolic partial differential equations.Ph.D. thesis, Massachusetts Institute of Technology (2005).

M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: M2AN 39 (2005) 157–181. | DOI | Numdam | MR | Zbl

S. Gugercin, A. C. Antoulas and C. A. Beattie, ${\mathscr{H}}_2$ model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30 (2008) 609–638. | DOI | MR | Zbl

J. S. Hesthaven, B. Stamm and S. Zhang, Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods. ESAIM: M2AN 48 (2014) 259–283. | DOI | Numdam | MR | Zbl

J.S. Hesthaven, G. Rozza and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics. Springer (2016). | MR

U. Hetmaniuk, R. Tezaur and C. Farhat. An adaptive scheme for a class of interpolatory model reduction methods for frequency response problems. Inter. J. Numer. Methods Eng. 93 (2013) 1109–1124. | DOI | MR | Zbl

B. Haasdonk and M. Ohlberger. Efficient reduced models for parametrized dynamical systems by offline/online decomposition. In Proc. of MATHMOD 2009, 6th Vienna International Conference on Mathematical Modelling (2009). | MR | Zbl

D.B.P. Huynh, G. Rozza, S. Sen and A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. Comput. Rendus Math. 345 (2007) 473–478. | DOI | MR | Zbl

N. Jung, A.T. Patera, B. Haasdonk and B. Lohmann, Model order reduction and error estimation with an application to the parameter-dependent eddy current equation. Math. Comput. Model. Dynamical Syst. 17 (2011) 561–582. | DOI | MR | Zbl

H.-J. Lee, Ch.-Ch. Chu and W.-Sh. Feng, An adaptive-order rational Arnoldi method for model-order reductions of linear time-invariant systems. Lin. Algebra Appl. 415 (2006) 235–261. | DOI | MR | Zbl

T.H. Lee. The design of CMOS radio-frequency integrated circuits, 2nd ed. Cambridge UK: Cambridge University Press (2004).

S. Lefteriu, A.C. Antoulas and A.C. Ionita, Parametric model reduction in the Loewner framwork. In Proc. of 18th IFAC World Congress (2011) 12752–12756.

B.C. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automatic Control AC-26 (1981) 17–32. | DOI | MR | Zbl

H. Panzer, J. Mohring, R. Eid and B. Lohmann, Parametric model order reduction by matrix interpolation. at-Automatisierungstechnik 58 (2010) 475–484. | DOI

A. Odabasioglu, M. Celik and L.T. Pileggi, PRIMA: passive reduced-order interconnect macromodeling algorithm. IEEE Trans. Comput.-Aid. Design Integr. Circuits Syst. 17 (1998) 645–654. | DOI

D.V. Rovas, Reduced-basis output bound methods for parametrized partial differential equations. Ph.D. thesis, Massachusetts Institute of Technology (2003).

G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives Comput. Methods Eng. 15 (2008) 229–275. | DOI | MR | Zbl

E.B. Rudnyi and J.G. Korvink, Model order reduction for large scale engineering models developed in ANSYS. In vol. 3732 of Lect. Notes Comput. Sci. Springer Verlag (2006) 349–356.

S. Sen, Reduced basis approximation and a posteriori error estimation for non-coercive elliptic problems. Ph.D. thesis, Massachusetts Institute of Technology (2007).

T. Wolf, H. Panzer and B. Lohmann, Gramian-based error bound in model reduction by Krylov subspace methods. In Proc. of IFAC World Congress 44 (2011) 3587–3592.

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

Y.-T. Li, Z. Bai, Y.-F. Su and X. Zeng, Model order reduction of parameterized interconnect networks via a two-directional Arnoldi process. Comput.-Aid. Design Integr. Circuits Syst. 27 (2008) 1571–1582. | DOI

B. Salimbahrami, R. Eid and B. Lohmann, Model reduction by second order Krylov subspaces: extensions, stability and proportional damping. In Proc. of IEEE Conference on Computer Aided Control Systems Design (2006) 2997–3002.

A. Paul−Dubois−Taine and D. Amsallem. An adaptive and efficient greedy procedure for the optimal training of parametric reduced-order models. Inter. J. Numer. Methods Eng. 102 (2014) 1262–1292. | DOI | MR | Zbl

K. Urban and A.T. Patera, An improved error bound for reduced basis approximation of linear parabolic problems. Math. Comput. 83 (2014) 1599–1615. | DOI | MR | Zbl

M. Yano, A space-time Petrov-Galerkin certified reduced basis method: Application to the Boussinesq equations. SIAM J. Sci. Comput. 36 (2014) A232–A266. | DOI | MR | Zbl

M. Yano, A.T. Patera and K. Urban, A space-time hp-interpolation-based certified reduced basis method for Burgers’ equation. Math. Models Methods Appl. Sci. 24 (2014) 1903–1935. | DOI | MR | Zbl

Y. Zhang, L. Feng, S. Li and P. Benner, An efficient output error estimation for model order reduction of parametrized evolution equations. SIAM J. Sci. Comput. 37 (2015) B910–B936. | DOI | MR | Zbl