Stochastic Homogenization: An introduction to some recent variants and to numerical approaches

The series of lectures will overview some recent contributions on several theoretical aspects and numerical approaches in stochastic homogenization. After an introduction to the elementary aspects of stochastic homogenization, a variant of the classical theory will be presented. It has been introduced in [6], and further studied in [11, 16]. The relation between stochastic homogenization problems and other multiscale problems in materials science [7] will be emphasized. Several numerical approaches will be presented: some for genuinely stochastic problems (where variance issues are a practical concern and need to be addressed for efficiency purposes, [9, 10, 13]), and some for approximations of stochastic problems when the random character is only a perturbation of a deterministic model [1, 2, 3, 12, 14]. Most of these contributions are summarized in [15, 4]. Further details will be available in [5, 11, 17]. The series of talks will be centered around a simple, linear elliptic situation, since the focus is