We address the issue of parameter variations in POD approximations of time-dependent problems, without any specific restriction on the form of parameter dependence. Considering a parabolic model problem, we propose a POD construction strategy allowing us to obtain some a priori error estimates controlled by the POD remainder - in the construction procedure - and some parameter-wise interpolation errors for the model solutions. We provide a thorough numerical assessment of this strategy with the FitzHugh - Nagumo 1D model. Finally, we give detailed illustrations of the approach in two parameter estimation applications, the first in a variational estimation framework with the FitzHugh - Nagumo model, and the second with a beating heart mechanical model for which we employ a sequential estimation method to characterize model parameters using real image data in a clinical case.
@article{M2AN_2013__47_6_1821_0,
author = {Chapelle, D. and Gariah, A. and Moireau, P. and Sainte-Marie, J.},
title = {A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems - Analysis, assessments and applications to parameter estimation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {47},
year = {2013},
pages = {1821-1843},
doi = {10.1051/m2an/2013090},
mrnumber = {3123378},
zbl = {1295.65096},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2013__47_6_1821_0}
}
Chapelle, D.; Gariah, A.; Moireau, P.; Sainte-Marie, J. A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems - Analysis, assessments and applications to parameter estimation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 47 (2013) pp. 1821-1843. doi : 10.1051/m2an/2013090. http://gdmltest.u-ga.fr/item/M2AN_2013__47_6_1821_0/
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