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

¹ Inria Saclay Ile-de-France, Palaiseau, France.
dominique.chapelle@inria.fr
² Inria Paris-Rocquencourt, Le Chesnay, France.
³ Université Paris 6, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, Paris, France.
⁴ CETMEF, Margny-les-Compiègne, France.

Received: 21 May 2012

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.