Feedback-optimized parallel tempering Monte Carlo

Abstract

We introduce an algorithm to systematically improve the efficiency of parallel tempering Monte Carlo simulations by optimizing the simulated temperature set. Our approach is closely related to a recently introduced adaptive algorithm that optimizes the simulated statistical ensemble in generalized broad-histogram Monte Carlo simulations. Conventionally, a temperature set is chosen in such a way that the acceptance rates for replica swaps between adjacent temperatures are independent of the temperature and large enough to ensure frequent swaps. In this paper, we show that by choosing the temperatures with a modified version of the optimized ensemble feedback method we can minimize the round-trip times between the lowest and highest temperatures which effectively increases the efficiency of the parallel tempering algorithm. In particular, the density of temperatures in the optimized temperature set increases at the "bottlenecks" of the simulation, such as phase transitions. In turn, the acceptance rates are now temperature dependent in the optimized temperature ensemble. We illustrate the feedback-optimized parallel tempering algorithm by studying the two-dimensional Ising ferromagnet and the two-dimensional fully-frustrated Ising model, and briefly discuss possible feedback schemes for systems that require configurational averages, such as spin glasses.

The ALPS project release 1.3: Open-source software for strongly correlated systems

“…-Implementation of parallel tempering [38] for the classical and quantum Monte Carlo algorithms including feedback-optimized temperature sets [39]. -A QMC algorithm in the valence bond representation for quantum spin systems [40] -A continuum QMC program for bosons, implementing the continuous space worm algorithm [41].…”

Enhanced Sampling in the Well-Tempered Ensemble

“…First we consider the performance of WTE in the single replica mode. We simulate the two dimensional ferromagnetic Ising model for which an exact solution exists [30] and on which a large number of methods have been tested [31,32]. The Hamiltonian for this model is: H = −J <i,j> S i S j .…”

“…This is measured in terms of round-trip time t γ , which is the time needed for a configuration in the coldest replica to reach the hottest temperature and come back [31]. It can be seen in Empirically, the ratio between the smallest energy difference between successive β i and the largest energy fluctuation measured in the unbiased ensemble provides a good estimate for the optimal γ.…”

ClassicalJ1-J2Heisenberg model on the kagome lattice

“…We employ an improved parallel tempering (PT) approach, [17][18][19] based on the ALPS libraries and parapack. 21 We have tested several optimizations from the literature and finally used the following method: A suitable set of temperatures is chosen, then auto-correlation-times are measured to optimize the frequency of exchange steps between neighbouring replicas.…”