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.
Keywords: proper orthogonal decomposition, parameter variations, estimation, Fitzhugh − Nagumo equations, cardiac modeling
@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 },
pages = {1821--1843},
year = {2013},
publisher = {EDP-Sciences},
volume = {47},
number = {6},
doi = {10.1051/m2an/2013090},
mrnumber = {3123378},
zbl = {1295.65096},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013090/}
}
TY - JOUR AU - Chapelle, D. AU - Gariah, A. AU - Moireau, P. AU - Sainte-Marie, J. TI - A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems - Analysis, assessments and applications to parameter estimation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1821 EP - 1843 VL - 47 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013090/ DO - 10.1051/m2an/2013090 LA - en ID - M2AN_2013__47_6_1821_0 ER -
%0 Journal Article %A Chapelle, D. %A Gariah, A. %A Moireau, P. %A Sainte-Marie, J. %T A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems - Analysis, assessments and applications to parameter estimation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1821-1843 %V 47 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2013090/ %R 10.1051/m2an/2013090 %G en %F 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 , Tome 47 (2013) no. 6, pp. 1821-1843. doi: 10.1051/m2an/2013090
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