Extended Ensemble Monte Carlo

" Extended Ensemble Monte Carlo " is a generic term that indicates a set of algorithms which are now being popular in a variety of fields in physics and statistical information processing. Exchange Monte Carlo (Metropolis-Coupled Chain, Parallel Tempering), Simulated Tempering (Expanded Ensemble Monte Carlo), and Multicanonical Monte Carlo (Adaptive Umbrella Sampling) are typical members of this family. Here we give a cross-disciplinary survey on these algorithms with special emphasis on the great flexibility of the idea behind them. In Sec. 2, we discuss the background of Extended Ensemble Monte Carlo. In Sec. 3, 4 and 5, three types of the algorithms, i.e., Exchange Monte Carlo, Simulated Tempering, Multicanonical Monte Carlo, are introduced. Strategies for the construction of special-purpose extended ensembles are discussed in Sec. 6. We stress that an extension is not necessary restricted to that in the space of energy or temperature. Even unphysical (unrealizable) configurations can be included in the ensemble , if the resultant fast mixing of the Markov chain offset the cost. Multivariate (multi-component) extensions are also useful in many examples. In the next two sections, we pick up two topics of current interest: In Sec. 7, extended ensembles with a state space whose di-mensionality is dynamically varying are discussed. In Sec. 8, we give an introduction to Replica Monte Carlo algorithm by Swendsen and Wang, which interpolates Extended Ensemble Monte Carlo and Cluster Monte Carlo algorithms. In the appendix, we discuss advantages and disadvantages of three types of extended ensemble algorithms introduced in Sec. 3–Sec. 5.