  A new strategy based on numerical homogenization and Bayesian techniques for
solving multiscale inverse problems is introduced. We consider a class of
elliptic problems which vary at a microscopic scale, and we aim at recovering
the highly oscillatory tensor from measurements of the fine scale solution at
the boundary, using a coarse model based on numerical homogenization and model
order reduction. We provide a rigorous Bayesian formulation of the problem,
taking into account different possibilities for the choice of the prior
measure. We prove well-posedness of the effective posterior measure and, by
means of G-convergence, we establish a link between the effective posterior and
the fine scale model. Several numerical experiments illustrate the efficiency
of the proposed scheme and confirm the theoretical findings.
