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