  We adapt and study a variance reduction approach for the homogenization of
elliptic equations in divergence form. The approach, borrowed from atomistic
simulations and solid-state science [von Pezold et al, Physical Review B 2010;
Wei et al, Physical Review B 1990; Zunger et al, Physical Review Letters 1990],
consists in selecting random realizations that best satisfy some statistical
properties (such as the volume fraction of each phase in a composite material)
usually only obtained asymptotically.
  We study the approach theoretically in some simplified settings
(one-dimensional setting, perturbative setting in higher dimensions), and
numerically demonstrate its efficiency in more general cases.
