We present two works related to inverse problems (and parameters identification) in the framework of stochastic homogenization of linear elliptic PDEs.

In the first work (joint with W. Minvielle, M. Simon and A. Obliger), we consider the case when the highly oscillatory random coefficients are given by some probability law, the form of which is given, but with unknown parameters. We discuss the identification of these parameters, on the basis of some observed macroscopic quantities, such as the homogenized coefficient.

In the second work (joint with C. Le Bris and K. Li), we consider an elliptic problem with highly oscillatory (possibly stochastic) coefficients, and we show how to approximate it using a problem of the same type, but with constant coefficients. These constant coefficients are defined by an optimization procedure. We illustrate the links between this particular approach and the classical theory of homogenization. On some illustrating examples we show the potential practical interest of the approach.