Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 731-757.

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.

DOI : 10.1051/m2an/2011053
Classification : 65M60, 35A35, 35B45
Mots clés : POD, Galerkin approximation error estimates, non-linear parabolic problems, cardiac models
@article{M2AN_2012__46_4_731_0,
     author = {Chapelle, Dominique and Gariah, Asven and Sainte-Marie, Jacques},
     title = {Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {731--757},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {4},
     year = {2012},
     doi = {10.1051/m2an/2011053},
     mrnumber = {2891468},
     zbl = {1273.65125},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2011053/}
}
TY  - JOUR
AU  - Chapelle, Dominique
AU  - Gariah, Asven
AU  - Sainte-Marie, Jacques
TI  - Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2012
SP  - 731
EP  - 757
VL  - 46
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2011053/
DO  - 10.1051/m2an/2011053
LA  - en
ID  - M2AN_2012__46_4_731_0
ER  - 
%0 Journal Article
%A Chapelle, Dominique
%A Gariah, Asven
%A Sainte-Marie, Jacques
%T Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 731-757
%V 46
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2011053/
%R 10.1051/m2an/2011053
%G en
%F M2AN_2012__46_4_731_0
Chapelle, Dominique; Gariah, Asven; Sainte-Marie, Jacques. Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 731-757. doi : 10.1051/m2an/2011053. http://archive.numdam.org/articles/10.1051/m2an/2011053/

[1] D. Amsallem and C. Farhat, Interpolation method for adapting reduced-order models and application to aeroelasticity. AIAA J. 46 (2008) 1803-1813.

[2] A. Astolfi, Model reduction by moment matching for linear and nonlinear systems. IEEE Trans. Automat. Cont. 55 (2010) 2321-2336. | MR | Zbl

[3] K.J. Bathe, Finite Element Procedures. Prentice Hall (1996). | Zbl

[4] 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).

[5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland (1987). | MR | Zbl

[6] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 8 (1975) 77-84. | Numdam | MR | Zbl

[7] 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.

[8] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology 5 (1992). | MR | Zbl

[9] B.F. Feeny and R. Kappagantu, On the physical interpretation of proper orthogonal modes in vibrations. J. Sound Vib. 211 (1998) 607-616.

[10] T.M. Flett, Differential Analysis. Cambridge University Press (1980). | MR | Zbl

[11] S. Gugercin and A.C. Athanasios, A survey of model reduction by balanced truncation and some new results. Int. J. Control 77 (2004) 748-766. | MR | Zbl

[12] 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. | MR | Zbl

[13] M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319-345. | MR | Zbl

[14] P. Holmes, J. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996). | MR | Zbl

[15] M. Kahlbacher and S. Volkwein, Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems. Discussiones Mathematicae : Differential Inclusions, Control and Optimization 27 (2007) 95-117. | MR | Zbl

[16] D.-D. Kosambi, Statistics in function space, J. Indian Math. Soc. (N.S.) 7 (1943) 76-88. | MR | Zbl

[17] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117-148. | MR | Zbl

[18] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40 (2002) 492-515 (electronic). | MR | Zbl

[19] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems. ESAIM : M2AN 42 (2008) 1-23. | Numdam | MR | Zbl

[20] Y. Maday, A.T. Patera and G. Turinici, A priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. J. Sci. Comput. 17 (2002) 437-446. | MR | Zbl

[21] C. Prud'Homme, D.V. Rovas, K. Veroy and A.T. Patera, A mathematical and computational framework for reliable real-time solution of parametrized partial differential equations. ESAIM : M2AN 36 (2002) 747-771. Programming. | Numdam | MR | Zbl

[22] 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). | MR | Zbl

[23] D.V. Rovas, L. Machiels and Y. Maday, Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423-445. | MR | Zbl

[24] 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 : application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15 (2008) 229-275. | MR

[25] J. Sainte-Marie, D. Chapelle, R. Cimrman and M. Sorine, Modeling and estimation of the cardiac electromechanical activity. Comput. Struct. 84 (2006) 1743-1759. | MR

[26] T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation. Linear Algebra Appl. 415 (2006) 262-289. | MR | Zbl

[27] K. Veroy, 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. | MR | Zbl

[28] K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (2002) 2323-2330.

Cité par Sources :