Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
A Novel Approach to Model Order Reduction for Coupled Multiphysics Problems Read chapter Read first chapter

2014 | OriginalPaper | Chapter

2. Case Study: Parametrized Reduction Using Reduced-Basis and the Loewner Framework

Authors : Antonio C. Ionita, Athanasios C. Antoulas

Published in: Reduced Order Methods for Modeling and Computational Reduction

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this case study, we compare two methods for model reduction of parametrized systems, namely, Reduced-Basis and Loewner rational interpolation.
While having the same goal of constructing reduced-order models for large-scale parameter-dependent systems, the two methods follow fundamentally different approaches. On the one hand, the well known Reduced-Basis method takes a time domain approach, using offline snapshots of the full-order system combined with a rigorous error bound. On the other hand, the recently introduced Loewner matrix framework takes a frequency-domain approach that constructs rational interpolants of transfer function measurements, and has the flexibility of allowing different reduced-orders for each of the frequency and parameter variables.
We apply the two methods to a parametrized partial differential equation modeling the transient temperature evolution near the surface of a cylinder immersed in fluid. Then, we compare the resulting reduced-order models with the full-order finite element system by running both time- and frequency-domain simulations.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter A Novel Approach to Model Order Reduction for Coupled Multiphysics Problems
next chapter Comparison of Some Reduced Representation Approximations
Literature
1.
Antoulas, A.C.: Approximation of large-scale dynamical systems. Advances in Design and Control, DC-06. SIAM, Philadelphia (2005, second printing: Summer 2008)
2.
Antoulas, A.C., Anderson, B.D.O.: On the scalar rational interpolation problem. IMA J. of Mathematical Control and Information 3, 61–88 (1986)CrossRefMATH
3.
Antoulas, A.C., Ionita, A.C., Lefteriu, S.: On two-variable rational interpolation. Linear Algebra and its Applications 436(8), 2889–2915 (2012). DOI 10.1016/j.laa.2011.07.017CrossRefMATHMathSciNet
4.
Berrut, J.P., Trefethen, L.N.: Barycentric Lagrange Interpolation. SIAM Review 46(3), 501–517 (2004). DOI 10.1137/S0036144502417715CrossRefMATHMathSciNet
5.
Chahlaoui, Y., Van Dooren, P.: A collection of benchmark examples for model reduction of linear time invariant dynamical systems. SLICOT Working Note 2002–2 (2002)
6.
Eising, R.: Realization and stabilization of 2-d systems. Automatic Control, IEEE Transactions on 23(5), 793–799 (1978). DOI 10.1109/TAC.1978.1101861CrossRefMATHMathSciNet
7.
Fornasini, E., Marchesini, G.: Doubly-indexed dynamical systems: State-space models and structural properties. Theory of Computing Systems 12, 59–72 (1978). DOI 10.1007/BF01776566CrossRefMATHMathSciNet
8.
Glover, K.: All optimal Hankel-norm approximations of linear multivariable systems and their L∞-error bounds. International Journal of Control 39(6), 1115–1193 (1984). DOI 10.1080/00207178408933239CrossRefMATHMathSciNet
9.
Grimme, E.J.: Krylov projection methods for model reduction. Ph.D. Thesis, ECE Dept., U. of Illinois, Urbana, Champaign, IL, USA(1997)
10.
Gu, G., Aravena, J.L., Zhou, K.: On minimal realization of 2-D systems. IEEE Transactions on Circuits and Systems 38(10), 1228–1233 (1991). DOI 10.1109/31.97545CrossRef
11.
Gugercin, S., Antoulas, A.C., Beattie, C.: ℋ2 Model Reduction for Large-Scale Linear Dynamical Systems. SIAM Journal on Matrix Analysis and Applications 30(2), 609–638 (2008). DOI 10.1137/060666123CrossRefMATHMathSciNet
12.
Haasdonk, B.: Reduzierte-Basis-Methoden. Preprint 2011/004 — Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Stuttgart (2011)
13.
Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Mathematical Modelling and Numerical Analysis 42(02), 277–302 (2008). DOI 10.1051/m2an:2008001CrossRefMATHMathSciNet
14.
Haasdonk, B., Ohlberger, M.: Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Mathematical and Computer Modelling of Dynamical Systems 17(2), 145–161 (2011). DOI 10.1080/13873954.2010.514703CrossRefMATHMathSciNet
15.
Ionita A.C., Antoulas A.C.: Data-driven parametrized model reduction in the Loewner framework. Submitted to SIAM Journal on Scientific Computing (2013)
16.
Kahaner, D., Moler, K., Nash, S.: Numerical Methods and Software. Prentice Hall, NJ, USA (1989)MATH
17.
Kung, S.Y., Levy, B.C., Morf, M., Kailath, T.: New results in 2-D systems theory, part II: 2-D state-space models-Realization and the notions of controllability, observability, and minimality. Proceedings of the IEEE 65(6), 945–961 (1977). DOI 10.1109/PROC.1977.10592CrossRef
18.
Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T., Rovas, D.V.: Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. Comptes Rendus de l’Académie des Sciences, Series I, Mathematics 331(2), 153–158 (2000). DOI 10.1016/S0764-4442(00)00270-6
19.
Mayo, A.J., Antoulas, A.C.: A framework for the solution of the generalized realization problem. Linear Algebra and its Applications 425(2–3), 634–662 (2007). DOI 10.1016/j.laa.2007.03.008CrossRefMATHMathSciNet
20.
Moler, C., Van Loan, C.: Nineteen Dubious Ways to Compute the Exponential of a Matrix. SIAM Review 20(4), 801–836 (1978). DOI 10.1137/1020098CrossRefMATHMathSciNet
21.
Moler, C., Van Loan, C.: Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review 45(1), 3–49 (2003). DOI 10.1137/S00361445024180CrossRefMATHMathSciNet
22.
Patera, A.T., Rozza, G.: Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering. Cambridge, MA (2007). Available from http://​augustine.​mit.​edu/​methodology/​methodology_​book.​htm
23.
Prudhomme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T., Turinici, G.: Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods. Journal of Fluids Engineering 124(1), 70 (2002). DOI 10.1115/1.1448332CrossRef
24.
Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. Journal of Mathematics in Industry 1(1), 3 (2011). DOI 10.1186/2190-5983-1-3CrossRefMathSciNet
25.
Roesser, R.: Adiscrete state-space model for linear image processing. Automatic Control, IEEE Transactions on 20(1), 1–10 (1975). DOI 10.1109/TAC.1975.1100844CrossRefMATHMathSciNet
26.
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd Ed. SIAM (2003)
27.
Volkwein, S.: Proper orthogonal decomposition and singular value decomposition. Technical Report SFB-153, Institut für Mathematik, Universität Graz. Appeared also as part of the author’s Habilitationsschrift, Institut für Mathematik, Universität Graz, Graz (2001) (1999)
Metadata
Title
Case Study: Parametrized Reduction Using Reduced-Basis and the Loewner Framework
Authors
Antonio C. Ionita
Athanasios C. Antoulas
Copyright Year
2014
Publisher
Springer International Publishing
Book
Reduced Order Methods for Modeling and Computational Reduction

Print ISBN: 978-3-319-02089-1

Electronic ISBN: 978-3-319-02090-7

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-02090-7_2

Premium Partner

    Image Credits