Abstract
This review treats the mathematical and algorithmic foundations of non-reversible Markov chains in the context of event-chain Monte Carlo (ECMC), a continuous-time lifted Markov chain that employs the factorized Metropolis algorithm. It analyzes a number of model applications and then reviews the formulation as well as the performance of ECMC in key models in statistical physics. Finally, the review reports on an ongoing initiative to apply ECMC to the sampling problem in molecular simulation, i.e., to real-world models of peptides, proteins, and polymers in aqueous solution.
Highlights
Markov-chain Monte Carlo (MCMC) is an essential tool for the natural sciences
The distribution π {t} depends on the initial distribution, but for any choice of π {0}, for an irreducible and aperiodic transition matrix, the total variation distance is smaller than an exponential bound Cαt with α ∈ (0, 1)
The conductance yields the considerable diffusive-to-ballistic speedup that may be reached by a non-reversible lifting if the collapsed Markov chain is itself close to the ∼ 1/ 2 upper bound of Equations (11) and (13)
Summary
Markov-chain Monte Carlo (MCMC) is an essential tool for the natural sciences. It is the subject of a research discipline in mathematics. The Metropolis or the heatbath (Gibbs-sampling) algorithms are popular choices They generally compute acceptance probabilities from the changes in the total potential (the system’s energy) and mimic the behavior of physical systems in the thermodynamic equilibrium. Event-chain Monte Carlo (ECMC) [7, 8] is a family of local, non-reversible MCMC algorithms developed over the last decade. This remains within the framework of (memoryless) Markov chains.
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