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

A micro-macro Markov chain Monte Carlo method for molecular dynamics using reaction coordinate proposals II: indirect reconstruction