Galerkin approximation with proper orthogonal decomposition : new error estimates and illustrative examples
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, p. 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 : https://doi.org/10.1051/m2an/2011053
Classification:  65M60,  35A35,  35B45
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {4},
     year = {2012},
     pages = {731-757},
     doi = {10.1051/m2an/2011053},
     zbl = {1273.65125},
     mrnumber = {2891468},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, pp. 731-757. doi : 10.1051/m2an/2011053. http://www.numdam.org/item/M2AN_2012__46_4_731_0/

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