David Aristoff,  CSU Math

The parallel replica method for Markov chains

Markov chains are simple random structures widely used in many areas of applied and computational math, physics and statistics. Typically, simulations of Markov chains are used to understand complicated but deterministic quantities. Of course, it is essential that the Markov chains can be simulated efficiently. We present a very general algorithm for improving the efficiency of Markov chain simulations. In many cases of practical interest, the chains tend to get "stuck" in certain subsets of configuration space. Our algorithm uses many replicas of the chain, simulated in parallel, to help it get "unstuck". We discuss some applications in materials science and in Markov chain Monte Carlo. ed in nature correspond to low energy isometric immersions of an imposed geometry.