  In this paper, we propose upper and lower error bounding techniques for
reduced order modelling applied to the computational homogenisation of random
composites. The upper bound relies on the construction of a reduced model for
the stress field. Upon ensuring that the reduced stress satisfies the
equilibrium in the finite element sense, the desired bounding property is
obtained. The lower bound is obtained by defining a hierarchical enriched
reduced model for the displacement. We show that the sharpness of both error
estimates can be seamlessly controlled by adapting the parameters of the
corresponding reduced order model.
