Some problems related to statistical error in stochastic homogenization.

Summary The work of this thesis focused on the development of numerical techniques for the homogenization of equations whose coefficients exhibit random small-scale heterogeneities. The difficulties related to the solution of such partial differential equations can be solved thanks to the theory of stochastic homogenization. The solution of an equation with random and oscillating coefficients at the finest scale of the problem is then substituted by the solution of an equation with constant coefficients. However, a difficulty remains: the calculation of these so-called homogenized coefficients are defined by an ergodic mean, which cannot be reached in practice. Only random approximations of these deterministic quantities can be computed, and the error committed during the approximation is important. These issues are developed in detail in Chapter 1 which serves as an introduction. The purpose of Chapter 2 of this thesis is to reduce the error of this approximation in a nonlinear case, by reducing the variance of the estimator by the method of antithetic variables. In Chapter 3, we show how to obtain a better variance reduction by the method of control variables. This approach is based on an approximate model, available in the case studied. It is more invasive and less generic, and we study it in a linear case. In Chapter 4, again in a linear case, we introduce a selection method to reduce the error. Finally, Chapter 5 deals with the analysis of an in-pour problem, where we search for parameters at the finest scale, knowing only a few macroscopic quantities, for example the homogenized coefficients of the model.