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

ESAIM: Mathematical Modelling and Numerical Analysis

Volumn 46, Issue 4, 2012, Pages 731-757

  Chapelle, Dominique ^a   Gariah, Asven ^a   Sainte Marie, Jacques ^a,b  

^a INRIA   (France)

^b 2 Boulevard Gambetta   (France)

Author keywords

Cardiac models; Galerkin approximation error estimates; Non linear parabolic problems; POD

Indexed keywords

ELECTROPHYSIOLOGY; GALERKIN METHODS;

CARDIAC MODEL; GALERKIN APPROXIMATION ERROR ESTIMATES; GALERKIN APPROXIMATIONS; NON-LINEAR PARABOLIC PROBLEMS; PARABOLIC EQUATIONS; PRIORI ERROR ESTIMATE; PROPER ORTHOGONAL DECOMPOSITIONS; WAVE-LIKE EQUATIONS;

ERRORS;

EID: 84857271792     PISSN: 28227840     EISSN: 28047214     Source Type: Journal    
DOI: 10.1051/m2an/2011053     Document Type: Article

References (28)

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).
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.
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).
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).
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.
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.
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.
14. P. Holmes, J. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996).
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.
16. D.-D. Kosambi, Statistics in function space, J. Indian Math. Soc. (N.S.) 7 (1943) 76-88.
17. K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117-148.
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).
19. K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems. ESAIM: M2AN 42 (2008) 1-23.
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.
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.
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.
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.
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.
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.
28. K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (2002) 2323-2330.