  Constructing atom-resolved states from low-resolution data is of practical
importance in many areas of science and engineering. This problem is addressed
in this paper in the context of multiscale factorization methods for molecular
dynamics. These methods capture the crosstalk between atomic and coarse-grained
scales arising in macromolecular systems. This crosstalk is accounted for by
Trotter factorization, which is used to separate the all-atom from the
coarse-grained phases of the computation. In this approach, short molecular
dynamics runs are used to advance in time the coarse-grained variables, which
in turn guide the all-atom state. To achieve this coevolution, an all-atom
microstate consistent with the updated coarse-grained variables must be
recovered. This recovery is cast here as a non-linear optimization problem that
is solved with a quasi-Newton method. The approach yields a Boltzmann-relevant
microstate whose coarse-grained representation and some of its fine-scale
features are preserved. Embedding this algorithm in multiscale factorization is
shown to be accurate and scalable for simulating proteins and their assemblies.
