  We introduce a new micro-macro Markov chain Monte Carlo method (mM-MCMC) with
indirect reconstruction to sample invariant distributions of molecular dynamics
systems that exhibit a time-scale separation between the microscopic (fast)
dynamics, and the macroscopic (slow) dynamics of some low-dimensional set of
reaction coordinates. The algorithm enhances exploration of the state space in
the presence of metastability by allowing larger proposal moves at the
macroscopic level, on which a conditional accept-reject procedure is applied.
Only when the macroscopic proposal is accepted, the full microscopic state is
reconstructed from the newly sampled reaction coordinate value and is subjected
to a second accept/reject procedure. The computational gain stems from the fact
that most proposals are rejected at the macroscopic level, at low computational
cost, while microscopic states, once reconstructed, are almost always accepted.
This paper discusses an indirect method to reconstruct microscopic samples from
macroscopic reaction coordinate values, that can also be applied in cases where
direct reconstruction is cumbersome. The indirect reconstruction method
generates a microscopic sample by performing a biased microscopic simulation,
starting from the previous microscopic sample and driving the microscopic state
towards the proposed reaction coordinate value. We show numerically that the
mM-MCMC scheme with indirect reconstruction can significantly extend the range
of applicability of the mM-MCMC method.
