  This paper addresses the complexity reduction of stochastic homogenisation of
a class of random materials for a stationary diffusion equation. A
cost-efficient approximation of the correctors is built using a method designed
to exploit quasi-periodicity. Accuracy and cost reduction are investigated for
local perturbations or small transformations of periodic materials as well as
for materials with no periodicity but a mesoscopic structure, for which the
limitations of the method are shown. Finally, for materials outside the scope
of this method, we propose to use the approximation of homogenised quantities
as control variates for variance reduction of a more accurate and costly Monte
Carlo estimator (using a multi-fidelity Monte Carlo method). The resulting cost
reduction is illustrated in a numerical experiment with a control variate from
weakly stochastic homogenisation for comparison, and the limits of this
variance reduction technique are tested on materials without periodicity or
mesoscopic structure.
