Analysis and optimization of population annealing

Abstract

Population annealing is an easily parallelizable sequential Monte Carlo algorithm that is well suited for simulating the equilibrium properties of systems with rough free-energy landscapes. In this work we seek to understand and improve the performance of population annealing. We derive several useful relations between quantities that describe the performance of population annealing and use these relations to suggest methods to optimize the algorithm. These optimization methods were tested by performing large-scale simulations of the three-dimensional (3D) Edwards-Anderson (Ising) spin glass and measuring several observables. The optimization methods were found to substantially decrease the amount of computational work necessary as compared to previously used, unoptimized versions of population annealing. We also obtain more accurate values of several important observables for the 3D Edwards-Anderson model.

Estimating the density of states of frustrated spin systems

“…The disorder average of ρ t at β=3 is found to be 49, 135, 420, 663 and 840 for k=3, 4, 5, 6, and 7, respectively, indicating that while for the main part of the distribution the hardnesses in EPA and parallel tempering are strongly correlated, for the tails of the distribution the hardness in EPA increases more gently than that found in parallel tempering. As is demonstrated elsewhere, these intrinsic hardness measures can be used to make PA simulations adaptive to the sample hardness [60,63]. We note that the planted samples of section 4.1 have an average ρ t of≈2000 (see appendix F), indicating that planted samples of this type are much harder than random ones.…”

“…The new algorithm introduced here, which we call EPA, is not based on Markov chains but on the sequential Monte Carlo method. PA was first studied in [53,54] and more recently developed further in [55][56][57][58][59][60][61]. It is based on the initialization of a population of replicas drawn from the equilibrium distribution at high temperatures, which is then subsequently cooled to lower and lower temperatures.…”

“…The method is not very sensitive to the precise value of α * , and we choose α * =0.86 in the runs below. While the algorithm described above is just PA [55] improved by adaptive temperature steps [58,60], the possibility of sampling the entropy arises from a combination of the method with multi-histogram techniques [62]. An estimator of the free energy follows directly from the resampling factors [55],…”

“…For the DOS estimators studied here a strategy to avoid bias hence consists of only including histograms from the well-equilibrated regime in the estimate (6). Here, equilibration can be ensured by monitoring heuristic criteria such as R 0 >100ρ t [56,60] that can be evaluated on the fly to determine a stopping temperature. Figure B1.…”

Numerical observation of a glassy phase in the three-dimensional Coulomb glass

“…In this paper we investigate the phase diagram of the CG model using Monte Carlo simulations in three spatial dimensions. For the finite-temperature simulations we make use of the population annealing Monte Carlo (PAMC) algorithm [53][54][55][56][57] which enables us to thermalize for a broad range of disorder values down to unprecedented low temperatures previously inaccessible. In addition, we argue that the detection of a glass phase requires a four-replica correlation length, as commonly used in spin-glass simulations in a field [58,59].…”