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A Galerkin strategy with Proper Orthogonal Decomposition for parameter-dependent problems – Analysis, assessments and applications to parameter estimation

Published online by Cambridge University Press:  07 October 2013

D. Chapelle ,
A. Gariah ,
P. Moireau  and
J. Sainte-Marie
D. Chapelle
Affiliation:
Inria Saclay Ile-de-France, Palaiseau, France.. dominique.chapelle@inria.fr
A. Gariah
Affiliation:
Inria Paris-Rocquencourt, Le Chesnay, France.
P. Moireau
Affiliation:
Inria Saclay Ile-de-France, Palaiseau, France.. dominique.chapelle@inria.fr
J. Sainte-Marie
Affiliation:
Inria Paris-Rocquencourt, Le Chesnay, France. Université Paris 6, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Paris, France. CETMEF, Margny-les-Compiègne, France.
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Abstract

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 Decompositionparameter variationsestimationFitzHugh − Nagumo equationscardiac modeling
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1821 - 1843
Copyright
© EDP Sciences, SMAI 2013

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