Project for a Postdoctoral Researcher on “ Variance Reduction Approaches for Random Homogenization ”

Consider heterogeneous elastic materials, modelled by a random stationary microstructure. The homogenized properties are deterministic. To compute them, one needs to solve auxiliary problems (the so-called corrector problems), which are set on the entire space. In practice, these corrector problems need to be approximated, and a usual approach is to consider a large but finite domain, on which the corrector problem (complemented by appropriate boundary conditions, such as periodic boundary conditions) is solved. As a consequence of this truncation, the apparent homogenized properties that one can compute are random. A natural quantity of interest is then the expectation of the apparent homogenized properties (i.e. the average over all possible realizations of the microstructure). A brute force, standard Monte Carlo approach then consists in considering several microstructures, and averaging over these. In this project, our aim is to use more advanced numerical approaches from the applied probability community, and more precisely control variate variance reduction approaches, which are based on using a surrogate model. It turns out that, in the computational mechanics community, many reduced models have been proposed to approximate homogenized quantities: self-consistent models, Hashin-Shtrikman bounds, Mori-Tanaka like schemes,. . . These schemes (which are based on analytical formulaes) typically make use of a small number of statistical informations on the microstructure: for a composite materials, the volumic fraction, the shape and the orientation of the inclusions, . . . It is only recently that high-order correlations (such as threeand four-points correlations) have been taken into account to build more precise estimators.