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arXiv:1305.4907 · Physical Review E

Comparative study of selected parallel tempering methods

Anastasios Malakis, Theodoros Papakonstantinou

Published Jul 30, 2013• 11 citations
arXivHTMLPDF

Abstract

We review several parallel tempering schemes and examine their main ingredients for accuracy and efficiency. The present study covers two selection methods of temperatures and several choices for the exchange of replicas, including a recent novel all-pair exchange method. We compare the resulting schemes and measure specific heat errors and efficiency using the two-dimensional (2D) Ising model. Our tests suggest that an earlier proposal for using numbers of local moves related to the canonical correlation times is one of the key ingredients for increasing efficiency, and protocols using cluster algorithms are found to be very effective. Some of the protocols are also tested for efficiency and ground state production in 3D spin-glass models where we find that a simple nearest-neighbor approach using a local n-fold-way algorithm is the most effective. Finally, we present evidence that the asymptotic limits of the ground state energy for the isotropic case and for an anisotropic case of the 3D spin-glass model are very close and may even coincide.

DOI
10.1103/physreve.88.013312

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We review several parallel tempering schemes and examine their main ingredients for accuracy and efficiency. The present study covers two selection methods of temperatures and several choices for the exchange of replicas, including a recent novel all-pair exchange method. We compare the resulting schemes and measure specific heat errors and efficiency using the two-dimensional (2D) Ising model. Our tests suggest that an earlier proposal for using numbers of local moves related to the canonical correlation times is one of the key ingredients for increasing efficiency, and protocols using cluster algorithms are found to be very effective. Some of the protocols are also tested for efficiency and ground state production in 3D spin-glass models where we find that a simple nearest-neighbor approach using a local n-fold-way algorithm is the most effective. Finally, we present evidence that the asymptotic limits of the ground state energy for the isotropic case and for an anisotropic case of the 3D spin-glass model are very close and may even coincide.
We introduce a new update scheme to systematically improve the efficiency of parallel tempering Monte Carlo simulations by taking into account the temperature dependence of autocorrelation times.In contrast to previous attempts the temperatures are not dynamically adjusted but chosen in such a way that the acceptance rate for proposed exchanges of all adjacent replica is about 50%.We show that by adapting the number of sweeps between the parallel tempering moves to the canonical autocorrelation time, the average round-trip time of a replica between the lowest and the highest temperatures is significantly decreased and, therefore, the efficiency of the parallel tempering algorithm is considerably improved.We illustrate the new algorithm with results for the two-dimensional Ising model and propose a toy model to find the optimal parameter set for the parallel tempering routine within no time.
The efficiency of parallel tempering Monte Carlo is studied for a two-dimensional Ising system of length L with N=L^2 spins. An external field is used to introduce a difference in free energy between the two low temperature states.\nIt is found that the number of replicas R_opt that optimizes the parallel tempering algorithm scales as the square root of the system size N. For two symmetric low temperature states, the time needed for equilibration is observed to grow as L^2.18. If a significant difference in free energy is present between the two states, this changes to L^1.02.\nIt is therefore established that parallel tempering is sped up by a factor of roughly L if an asymmetry is introduced between the low temperature states. This confirms previously made predictions for the efficiency of parallel tempering. These findings should be especially relevant when using parallel tempering for systems like spin glasses, where no information about the degeneracy of low temperature states is available prior to the simulation.
Because of its complexity, the 3D Ising model has not been given an exact analytic solution so far, as well as the 2D Ising in non zero external field conditions. In real materials the phase transition creates a discontinuity. We analysed the Ising model that presents similar discontinuities. We use Monte Carlo methods with a single spin change or a spin cluster change to calculate macroscopic quantities, such as specific heat and magnetic susceptibility. We studied the differences between these methods. Local MC algorithms (such as Metropolis) perform poorly for large lattices because they update only one spin at a time, so it takes many iterations to get a statistically independent configuration. More recent spin cluster algorithms use clever ways of finding clusters of sites that can be updated at once. The single cluster method is probably the best sequential cluster algorithm. We also used the entropic sampling method to simulate the density of states. This method takes into account all possible configurations, not only the most probable. The entropic method also gives good results in the 3D case. We studied the usefulness of distributed computing for Ising model. We established a parallelization strategy to explore Metropolis Monte Carlo simulation and Swendsen-Wang Monte Carlo simulation of this spin model using the data parallel languages on different platform. After building a computer cluster we made a Monte Carlo estimation of 2D and 3D Ising thermodynamic properties and compare the results with the sequential computing. In the same time we made quantitative analysis such as speed up and efficiency for different sets of combined parameters (e.g. lattice size, parallel algorithms, chosen model).
Some recent progress in Monte Carlo simulations of spin glasses will be presented. The problem of slow dynamics at low temperatures is partially alleviated by use of the parallel tempering (replica exchange) method. A useful technique to check for equilibration (applicable only for a Gaussian distribution) will be discussed. It will be argued that a finite size scaling analysis of the scaled correlation length of the system is a good approach with which to investigate phase transitions in spin glasses. This method will be used to study two questions: (i) whether there is a phase transition in zero field in the Heisenberg spin glass in three dimensions, and (ii) whether there is phase transition in a magnetic field in an Ising spin glass, also in three dimensions.
Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.