Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples

Dominique Chapelle; Asven Gariah; Jacques Sainte-Marie

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 4, page 731-757
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Chapelle, Dominique, Gariah, Asven, and Sainte-Marie, Jacques. "Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 731-757. <http://eudml.org/doc/277849>.

@article{Chapelle2012,
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.},
author = {Chapelle, Dominique, Gariah, Asven, Sainte-Marie, Jacques},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {POD; Galerkin approximation error estimates; non-linear parabolic problems; cardiac models; proper orthogonal decomposition; Galerkin approximation; eror estimates; wave-like equations; numerical experiments},
language = {eng},
month = {2},
number = {4},
pages = {731-757},
publisher = {EDP Sciences},
title = {Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples},
url = {http://eudml.org/doc/277849},
volume = {46},
year = {2012},
}

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
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 4
SP - 731
EP - 757
AB - 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.
LA - eng
KW - POD; Galerkin approximation error estimates; non-linear parabolic problems; cardiac models; proper orthogonal decomposition; Galerkin approximation; eror estimates; wave-like equations; numerical experiments
UR - http://eudml.org/doc/277849
ER -

References

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  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.  
  3. K.J. Bathe, Finite Element Procedures. Prentice Hall (1996).  Zbl0994.74001
  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).  
  6. P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.8 (1975) 77–84.  Zbl0368.65008
  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 Technology5 (1992).  
  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).  Zbl0442.34002
  11. S. Gugercin and A.C. Athanasios, A survey of model reduction by balanced truncation and some new results. Int. J. Control77 (2004) 748–766.  Zbl1061.93022
  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.  Zbl1079.65533
  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.  Zbl1191.49040
  14. P. Holmes, J. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996).  Zbl0890.76001
  15. M. Kahlbacher and S. Volkwein, Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems. Discussiones Mathematicae : Differential Inclusions, Control and Optimization27 (2007) 95–117.  Zbl1156.35020
  16. D.-D. Kosambi, Statistics in function space, J. Indian Math. Soc. (N.S.)7 (1943) 76–88.  Zbl0063.03317
  17. K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math.90 (2001) 117–148.  Zbl1005.65112
  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).  Zbl1075.65118
  19. K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems. ESAIM : M2AN42 (2008) 1–23.  Zbl1141.65050
  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.  Zbl1014.65115
  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 : M2AN36 (2002) 747–771. Programming.  
  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).  
  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.  Zbl1101.65099
  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.  Zbl1304.65251
  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.  
  26. T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation. Linear Algebra Appl.415 (2006) 262–289.  Zbl1102.65075
  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. Paris337 (2003) 619–624.  Zbl1036.65075
  28. K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition. AIAA J.40 (2002) 2323–2330.  

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