Extended ensemble Monte Carlo

Extended Ensemble Monte Carlo is a generic term that indicates a set of algorithms, which are now 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) axe typical members of this family. Here, we give a cross-disciplinary survey of these algorithms with special emphasis on the great flexibility of the underlying idea. In Sec. 2, we discuss the background of Extended Ensemble Monte Carlo. In Sees. 3, 4 and 5, three types of the algorithms, i.e., Exchange Monte Carlo, Simulated Tempering, Multicanonical Monte Carlo, are introduced. In Sec. 6, we give an introduction to Replica Monte Carlo algorithm by Swendsen and Wang. Strategies for the construction of special-purpose extended ensembles are discussed in Sec. 7. We stress that an extension is not necessary restricted to 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 offsets the increasing cost of the sampling procedure. Multivariate (multicomponent) extensions are also useful in many examples. In Sec. 8, we give a survey on extended ensembles with a state space whose dimensionality is dynamically varying. In the appendix, we discuss advantages and disadvantages of three types of extended ensemble algorithms.