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Galerkin approximationwith proper orthogonal decomposition :new error estimates and illustrative examples

Published online by Cambridge University Press:  03 February 2012

Dominique Chapelle ,
Asven Gariah  and
Jacques Sainte-Marie
Dominique Chapelle
Affiliation:
Inria Rocquencourt Laboratoire Saint-Venant Domaine de Voluceau Rocquencourt, B.P. 105, 78153 Le Chesnay, France. jacques.sainte-marie@inria.fr
Asven Gariah
Affiliation:
Inria Rocquencourt Laboratoire Saint-Venant Domaine de Voluceau Rocquencourt, B.P. 105, 78153 Le Chesnay, France. jacques.sainte-marie@inria.fr
Jacques Sainte-Marie
Affiliation:
Inria Rocquencourt Laboratoire Saint-Venant Domaine de Voluceau Rocquencourt, B.P. 105, 78153 Le Chesnay, France. jacques.sainte-marie@inria.fr CETMEF, 2 boulevard Gambetta, 60200 Compiègne, France
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Abstract

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

Keywords

PODGalerkin approximation error estimatesnon-linear parabolic problemscardiac models
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 4 , July 2012 , pp. 731 - 757
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

Amsallem, D. and Farhat, C., Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46 (2008) 1803–1813. Google Scholar
Astolfi, A., Model reduction by moment matching for linear and nonlinear systems. IEEE Trans. Automat. Cont. 55 (2010) 2321–2336. Google Scholar
K.J. Bathe, Finite Element Procedures. Prentice Hall (1996).
R. Chabiniok, D. Chapelle, P.-F. Lesault, A. Rahmouni and J.-F. Deux, Validation of a biomechanical heart model using animal data with acute myocardial infarction, in MICCAI Workshop on Cardiovascular Interventional Imaging and Biophysical Modelling (CI2BM09) (2009).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1987).
Clément, P., Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 8 (1975) 77–84. Google Scholar
L. Daniel, C.S. Ong, S.C. Low, H.L. Lee and J. White, A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.23 (2004) 678–693.
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology 5 (1992).
Feeny, B.F. and Kappagantu, R., On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211 (1998) 607–616. Google Scholar
T.M. Flett, Differential Analysis. Cambridge University Press (1980).
Gugercin, S. and Athanasios, A.C., A survey of model reduction by balanced truncation and some new results. Int. J. Control 77 (2004) 748–766. Google Scholar
M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems : Error estimates and suboptimal control, inDimension Reduction of Large-Scale Systems, edited by T.J. Barth, M. Griebel, D.E. Keyes, R.M. Nieminen, D. Roose, T. Schlick, P. Benner, D.C. Sorensen and V. Mehrmann. Lect. Notes Comput. Sci. Eng. 45 (2005) 261–306.
Hinze, M. and Volkwein, S., Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319–345. Google Scholar
P. Holmes, J. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996).
Kahlbacher, M. and Volkwein, S., Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems. Discussiones Mathematicae : Differential Inclusions, Control and Optimization 27 (2007) 95–117. Google Scholar
Kosambi, D.-D., Statistics in function space, J. Indian Math. Soc. (N.S.) 7 (1943) 76–88. Google Scholar
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117–148. Google Scholar
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 492–515 (electronic). Google Scholar
Kunisch, K. and Volkwein, S., Proper orthogonal decomposition for optimality systems. ESAIM : M2AN 42 (2008) 1–23. Google Scholar
Maday, Y., Patera, A.T. and Turinici, G., A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437–446. Google Scholar
Prud’homme, C., Rovas, D.V., Veroy, K. and Patera, A.T., A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. ESAIM : M2AN 36 (2002) 747–771. Programming. Google Scholar
P.-A. Raviart and J.-M. Thomas, Introduction à l’Analyse Numérique des Equations aux Dérivées Partielles. Collection Mathématiques Appliquées pour la Maîtrise (in French), Masson (1983).
Rovas, D.V., Machiels, L. and Maday, Y., Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423–445. Google Scholar
Rozza, G., Huynh, D.B.P. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations : application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15 (2008) 229–275. Google Scholar
Sainte-Marie, J., Chapelle, D., Cimrman, R. and Sorine, M., Modeling and estimation of the cardiac electromechanical activity. Comput. Struct. 84 (2006) 1743–1759. Google Scholar
Stykel, T., Balanced truncation model reduction for semidiscretized Stokes equation. Linear Algebra Appl. 415 (2006) 262–289. Google Scholar
Veroy, K., C. Prud’homme and A.T. Patera, Reduced-basis approximation of the viscous Burgers equation : rigorous a posteriori error bounds. C. R. Math. Acad. Sci. Paris 337 (2003) 619–624. Google Scholar
Willcox, K. and Peraire, J., Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (2002) 2323–2330. Google Scholar