Publication Cover
Mathematical and Computer Modelling of Dynamical Systems
Methods, Tools and Applications in Engineering and Related Sciences
Volume 17, 2011 - Issue 2
1,081
Views
75
CrossRef citations to date
0
Altmetric
Articles

Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition

&
Pages 145-161 | Received 02 Dec 2009, Accepted 02 Dec 2009, Published online: 12 Jan 2011

References

  • Grepl , M.A. , Maday , Y. , Nguyen , N.C. and Patera , A.T. 2007 . Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations . M2AN, Math. Model. Numer. Anal. , 41 ( 3 ) : 575 – 605 .
  • Grepl , M.A. and Patera , A.T. 2005 . A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations . M2AN, Math. Model. Numer. Anal. , 39 ( 1 ) : 157 – 181 .
  • Haasdonk , B. and Ohlberger , M. 2008 . Reduced basis method for finite volume approximations of parametrized linear evolution equations . M2AN, Math. Model. Numer. Anal. , 42 ( 2 ) : 277 – 302 .
  • Prud'homme , C. , Rovas , D.V. , Veroy , K. , Machiels , L. , Maday , Y. , Patera , A.T. and Turinici , G. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods . J. Fluids Eng. , 124 ( 2002 ) 70 – 80 .
  • Rovas , D.V. , Machiels , L. and Maday , Y. 2006 . Reduced basis output bound methods for parabolic problems . IMA J. Numer. Anal. , 26 ( 3 ) : 423 – 445 .
  • Tonn , T. and Urban , K. 2006 . “ A reduced-basis method for solving parameter-dependent convection-diffusion problems around rigid bodies ” . In ECCOMAS CFD 2006 Proceedings , Edited by: Wesseling , P. , Onate , E. and Periaux , J. The Netherlands : Delft University of Technology .
  • Patera , A.T. and Rozza , G. Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations . MIT, 2007. To appear in MIT Pappalardo Graduate Monographs in Mechanical Engineering. ,
  • Rozza , G. , Huynh , D.B.P. and Patera , A.T. 2008 . Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics . Arch. Comput. Meth. Eng. , 15 ( 3 ) : 229 – 275 .
  • Balmes , E. 1996 . Parametric families of reduced finite element models, theory and applications . Mech. Syst. Signal Process. , 10 ( 4 ) : 381 – 394 .
  • Daniel , L. , Siong , O. , Chay , L. , Lee , K. and White , J. 2004 . Multi-parameter moment-matching modelreduction approach for generating geometrically parameterized interconnect performance models . IEEE Trans. Comput. Aided Des. Integrated Circ. Syst. , 23 ( 5 ) : 678 – 693 .
  • Feng , L. and Benner , P. A robust algorithm for parametric model order reduction . Proceedings in Applied Mathematics and Mechanics . Vol. 7 , pp. 1021501 – 1021502 . John Wiley and Sons .
  • Rudnyi , E.B. , Moosmann , C. , Greiner , A. , Bechtold , T. and Korvink , J.G. 2006 . Parameter preserving model reduction for MEMS system-level simulation and design . Proceedings of MATHMOD 2006 ,
  • Weile , S. , Michielssen , E. , Grimme , E. and Gallivan , K. 1999 . A method for generating rational interpolant reduced order models of two- parameter linear systems . Appl. Math. Lett. , 12 ( 5 ) : 93 – 102 .
  • Farle , O. , Hill , V. , Ingelström , P. and Dyczij-Edlinger , R. 2006 . Ordnungsreduktion linearer zeitinvarianter Finite-Elemente-Modelle mit multivariater polynomieller Parametrisierung (model order reduction of linear finite element models parameterized by polynomials in several variables) . Automatisierungstechnik , 54 : 161 – 169 .
  • Bond , B. and Daniel , L. Parameterized model order reduction of nonlinear dynamical systems . Proceedings of the IEEE Conference on Computer-Aided Design .
  • Baur , U. and Benner , P. 2008 . Parametrische Modellreduktion mit d¨unnen Gittern . GMA-Fachausschuss 1.30, Modellbildung, Identifizierung und Simulation in der Automatisierungstechnik, Salzburg , : 262 – 271 .
  • Lohmann , B. and Eid , R. Efficient order reduction of parametric and nonlinear models by superposition of locally reduced models . Methoden und Anwendungen der Regelungstechnik. Erlangen-Münchener Workshops 2007 und 2008 . Shaker Verlag, Aachen. pp. 27 – 36 .
  • Bai , Z. , Slone , R.D. , Smith , W.T. and Ye , Q. 1999 . Error bound for reduced system model by Padé approximation via the Lanczos process . IEEE Trans. Comput. Aided Des. Integrated Circ. Syst. , 18 ( 2 ) : 133 – 141 .
  • Bechtold , T. , Rudnyi , E.B. and Korvink , J.G. Error indicators for fully automatic extraction of heattransfer macromodels for MEMS . J. Micromech. Microeng. , 15 ( 2005 ) 430 – 440 .
  • Jung , N. , Rain , O. , Schumacher , K. , Eid , R. and Lohmann , B. 2009 . Error estimators for projection-based order reduction methods . MoRePaS 09 Book of Abstracts , : 41 16–18 September 2004, Münster
  • Tröltzsch , F. and Volkwein , S. 2009 . POD a-posteriori error estimation for linear-quadratic optimal control problems . Comput. Optim. Appl. , 44 ( 1 ) : 83 – 115 .
  • Minh , H.B. 2009 . Model Reduction in a Behavioural Framework , The Netherlands : Ph.D. thesis, Rijksuniversiteit Groningen .
  • Antoulas , A.C. 2005 . Approximation of Large–Scale Dynamical Systems , Philadelphia, PA : SIAM Publications .
  • Haasdonk , B. and Ohlberger , M. Adaptive basis enrichment for the reduced basis method applied to finite volume schemes . Proceedings of 5th International Symposium on Finite Volumes for Complex Applications . pp. 471 – 478 . France : Aussois .
  • Moosmann , C. 2008 . ParaMOR – Model Order Reduction for parameterized MEMS applications , Germany : Ph.D. thesis, IMTEK, University of Freiburg .
  • Barrault , M. , Maday , Y. , Nguyen , N.C. and Patera , A.T. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations . C. R. Math. Acad. Sci. Paris Series I, , 339 ( 2004 ) 667 – 672 .
  • Meyer , M. and Matthies , H.G. Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods . Comput. Mech. , 31 ( 2003 ) 179 – 191 .
  • Bui-Thanh , T. 2007 . Model-Constrained Optimization Methods for Reduction of Parameterized Large-Scale Systems , Ph.D. thesis, Massachusetts Institute of Technology .
  • Volkwein , S. and Kunisch , K. Optimal snapshot location for computing POD basis functions . Esaim. Math. Model Numer. Anal. , 44 ( 2010 ) 509 – 529 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.