  We consider the inverse problem of reconstructing an effective model for a
prototypical diffusion process in strongly heterogeneous media based on coarse
measurements. The approach is motivated by quasi-local numerical effective
forward models that are provably reliable beyond periodicity assumptions and
scale separation. The goal of this work is to show that an identification of
the matrix representation related to these effective models is possible. On the
one hand, this provides a reasonable surrogate in cases where a direct
reconstruction is unfeasible due to a mismatch between the coarse data scale
and the microscopic quantities to be reconstructed. On the other hand, the
approach allows us to investigate the requirement for a certain non-locality in
the context of numerical homogenization. Algorithmic aspects of the inversion
procedure and its performance are illustrated in a series of numerical
experiments.
