Coupled coarse graining and Markov Chain Monte Carlo for lattice systems

TL;DR

This paper introduces an efficient Markov Chain Monte Carlo method for sampling equilibrium distributions in lattice models, combining coarse-graining techniques with Metropolis algorithms to improve computational efficiency while maintaining accuracy.

Contribution

The paper develops a coupled coarse-graining and Metropolis MCMC algorithm that reduces computational costs and preserves mixing properties for lattice systems.

Findings

Algorithm reduces energy difference computations.

Comparable mixing times to classical Metropolis.

Effective in one-dimensional Ising models.

Abstract

We propose an efficient Markov Chain Monte Carlo method for sampling equilibrium distributions for stochastic lattice models, capable of handling correctly long and short-range particle interactions. The proposed method is a Metropolis-type algorithm with the proposal probability transition matrix based on the coarse-grained approximating measures introduced in a series of works of M. Katsoulakis, A. Majda, D. Vlachos and P. Plechac, L. Rey-Bellet and D.Tsagkarogiannis,. We prove that the proposed algorithm reduces the computational cost due to energy differences and has comparable mixing properties with the classical microscopic Metropolis algorithm, controlled by the level of coarsening and reconstruction procedure. The properties and effectiveness of the algorithm are demonstrated with an exactly solvable example of a one dimensional Ising-type model, comparing efficiency of the single spin-flip Metropolis dynamics and the proposed coupled Metropolis algorithm.