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            <funder>The work of FL and WM is partially supported by ONR under Grant N00014-12-1-0383.</funder>
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              <p>In porous media physics, calibrating model parameters through experiments is a challenge. This process is plagued with errors that come from modelling, measurement and computation of the macroscopic observables through random homogenization -- the forward problem -- as well as errors coming from the parameters fitting procedure -- the inverse problem. In this work, we address these issues by considering a least-square formulation to identify parameters of the microscopic model on the basis on macroscopic observables. In particular, we discuss the selection of the macroscopic observables which we need to know in order to uniquely determine these parameters. To gain a better intuition and explore the problem without a too high computational load, we mostly focus on the one-dimensional case. We show that the Newton algorithm can be efficiently used to robustly determine optimal parameters, even if some small statistical noise is present in the system.</p>
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              <p>En physique des milieux poreux, calibrer certains paramètres d’un modèle microscopique sur la base d’expériences donnant accès à des grandeurs macroscopiques est un enjeu majeur. Cette démarche est entachée d’erreurs de modèle, de mesure et de calculs dans la procédure d’homogénéisation : le problème direct est biaisé. La résolution du problème inverse, lorsqu’il s’agit d’estimer les paramètres à partir des observations, engendre aussi des erreurs. Nous considérons ici une formulation “moindres carrés” du problème, cherchant à minimiser l’erreur entre les quantités macroscopiques observées et celles calculées via l’homogénéisation aléatoire. Nous discutons en particulier de la nature des informations macroscopiques nécessaires pour déterminer de manière univoque les paramètres de la densité de probabilité des propriétés microscopiques. Afin d’explorer plus facilement cette question, nous nous intéressons ici essentiellement au cas unidimensionel. Nous montrons que le problème peut être résolu de manière efficace par l’algorithme de Newton, même en présence d’un petit bruit statistique.</p>
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