  We introduce a new micro-macro Markov chain Monte Carlo method (mM-MCMC) 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. We
analytically show convergence and discuss the rate of convergence of the
proposed algorithm, and numerically illustrate its efficiency on a number of
standard molecular test cases. We also investigate the effect of the choice of
different numerical parameters on the efficiency of the resulting mM-MCMC
method.
