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, Volume 47 (2013) no. 6, p. 1821-1843

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.

DOI : https://doi.org/10.1051/m2an/2013090
Classification:  65M60,  35A35,  35B45,  93E10
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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {6},
     year = {2013},
     pages = {1821-1843},
     doi = {10.1051/m2an/2013090},
     zbl = {1295.65096},
     mrnumber = {3123378},
     language = {en},
     url = {http://www.numdam.org/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, Volume 47 (2013) no. 6, pp. 1821-1843. doi : 10.1051/m2an/2013090. http://www.numdam.org/item/M2AN_2013__47_6_1821_0/

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