  The effective large-scale properties of materials with random heterogeneities
on a small scale are typically determined by the method of representative
volumes: A sample of the random material is chosen - the representative volume
- and its effective properties are computed by the cell formula. Intuitively,
for a fixed sample size it should be possible to increase the accuracy of the
method by choosing a material sample which captures the statistical properties
of the material particularly well: For example, for a composite material
consisting of two constituents, one would select a representative volume in
which the volume fraction of the constituents matches closely with their volume
fraction in the overall material. Inspired by similar attempts in material
science, Le Bris, Legoll, and Minvielle have designed a selection approach for
representative volumes which performs remarkably well in numerical examples of
linear materials with moderate contrast. In the present work, we provide a
rigorous analysis of this selection approach for representative volumes in the
context of stochastic homogenization of linear elliptic equations. In
particular, we prove that the method essentially never performs worse than a
random selection of the material sample and may perform much better if the
selection criterion for the material samples is chosen suitably.
