  In this paper we present a variational technique that handles coarse-graining
and passing to a limit in a unified manner. The technique is based on a duality
structure, which is present in many gradient flows and other variational
evolutions, and which often arises from a large-deviations principle. It has
three main features: (A) a natural interaction between the duality structure
and the coarse-graining, (B) application to systems with non-dissipative
effects, and (C) application to coarse-graining of approximate solutions which
solve the equation only to some error. As examples, we use this technique to
solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck
equation and the small-noise limit of randomly perturbed Hamiltonian systems
with one and with many degrees of freedom.
