Improving the acceptance in Monte Carlo simulations: Sampling through intermediate states

In this paper we discuss a variation of the multiple-proposal approach to Markov chain-based Monte Carlo methods. We show that by means of intermediate states, used as pivots between the departure configuration and a number of possible arrival configurations, detailed balance is maintained and the acceptance of trial moves is significantly improved with respect to the standard Metropolis Monte Carlo approach. When applied to the case of a particle in a one-dimensional rough potential, this method results in an improved accuracy in the sampling of the distribution function. For all densities of a three-dimensional bulk system of Lennard-Jones particles, it requires less Monte Carlo steps to reach equilibrium and it allows for using wider displacement steps than Metropolis Monte Carlo. In particular, for low-intermediate densities the maximum displacement step can be set to exceptionally large values. We show how biases can be introduced to further reduce the number of operations necessary to construct the trial states while still fulfilling the detailed balance requirement. Although this method is more computationally demanding than (standard) single-proposal Monte Carlo, it samples the configuration space faster. Moreover, its multiple-proposal nature makes it amenable to parallel implementation, while requiring about half the number of operations required by multiple-try Monte Carlo.