Determining efficient temperature sets for the simulated tempering method

In statistical physics, the efficiency of tempering approaches strongly depends on ingredients such as the number of replicas $R$, reliable determination of weight factors and the set of used temperatures, ${\mathcal T}_R = \{T_1, T_2, \ldots, T_R\}$. For the simulated tempering (SP) in particular -- useful due to its generality and conceptual simplicity -- the latter aspect (closely related to the actual $R$) may be a key issue in problems displaying metastability and trapping in certain regions of the phase space. To determine ${\mathcal T}_R$'s leading to accurate thermodynamics estimates and still trying to minimize the simulation computational time, here it is considered a fixed exchange frequency scheme for the ST. From the temperature of interest $T_1$, successive $T$'s are chosen so that the exchange frequency between any adjacent pair $T_r$ and $T_{r+1}$ has a same value $f$. By varying the $f$'s and analyzing the ${\mathcal T}_R$'s through relatively inexpensive tests (e.g., time decay toward the steady regime), an optimal situation in which the simulations visit much faster and more uniformly the relevant portions of the phase space is determined. As illustrations, the proposal is applied to three lattice models, BEG, Bell-Lavis, and Potts, in the hard case of extreme first-order phase transitions, always giving very good results, even for $R=3$. Also, comparisons with other protocols (constant entropy and arithmetic progression) to choose the set ${\mathcal T}_R$ are undertaken. The fixed exchange frequency method is found to be consistently superior, specially for small $R$'s. Finally, distinct instances where the prescription could be helpful (in second-order transitions and for the parallel tempering approach) are briefly discussed.Comment: 10 pages, 14 figure

Feedback-optimized parallel tempering Monte Carlo

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.Comment: 12 pages, 14 figure

Population annealing: Theory and application in spin glasses

Population annealing is an efficient sequential Monte Carlo algorithm for simulating equilibrium states of systems with rough free energy landscapes. The theory of population annealing is presented, and systematic and statistical errors are discussed. The behavior of the algorithm is studied in the context of large-scale simulations of the three-dimensional Ising spin glass and the performance of the algorithm is compared to parallel tempering. It is found that the two algorithms are similar in efficiency though with different strengths and weaknesses.Comment: 16 pages, 10 figures, 4 table

Number of Magic Squares From Parallel Tempering Monte Carlo

There are 880 magic squares of size 4 by 4, and 275,305,224 of size 5 by 5. It seems very difficult if not impossible to count exactly the number of higher order magic squares. We propose a method to estimate these numbers by Monte Carlo simulating magic squares at finite temperature. One is led to perform low temperature simulations of a system with many ground states that are separated by energy barriers. The Parallel Tempering Monte Carlo method turns out to be of great help here. Our estimate for the number of 6 by 6 magic squares is 0.17745(16) times 10**20.Comment: 8 pages, no figure

An efficient Monte Carlo method for calculating ab initio transition state theory reaction rates in solution

In this article, we propose an efficient method for sampling the relevant state space in condensed phase reactions. In the present method, the reaction is described by solving the electronic Schr\"{o}dinger equation for the solute atoms in the presence of explicit solvent molecules. The sampling algorithm uses a molecular mechanics guiding potential in combination with simulated tempering ideas and allows thorough exploration of the solvent state space in the context of an ab initio calculation even when the dielectric relaxation time of the solvent is long. The method is applied to the study of the double proton transfer reaction that takes place between a molecule of acetic acid and a molecule of methanol in tetrahydrofuran. It is demonstrated that calculations of rates of chemical transformations occurring in solvents of medium polarity can be performed with an increase in the cpu time of factors ranging from 4 to 15 with respect to gas-phase calculations.Comment: 15 pages, 9 figures. To appear in J. Chem. Phy

Taming a non-convex landscape with dynamical long-range order: memcomputing Ising benchmarks

Recent work on quantum annealing has emphasized the role of collective behavior in solving optimization problems. By enabling transitions of clusters of variables, such solvers are able to navigate their state space and locate solutions more efficiently despite having only local connections between elements. However, collective behavior is not exclusive to quantum annealers, and classical solvers that display collective dynamics should also possess an advantage in navigating a non-convex landscape. Here, we give evidence that a benchmark derived from quantum annealing studies is solvable in polynomial time using digital memcomputing machines, which utilize a collection of dynamical components with memory to represent the structure of the underlying optimization problem. To illustrate the role of memory and clarify the structure of these solvers we propose a simple model of these machines that demonstrates the emergence of long-range order. This model, when applied to finding the ground state of the Ising frustrated-loop benchmarks, undergoes a transient phase of avalanches which can span the entire lattice and demonstrates a connection between long-range behavior and their probability of success. These results establish the advantages of computational approaches based on collective dynamics of continuous dynamical systems

Optimization by Record Dynamics

Large dynamical changes in thermalizing glassy systems are triggered by trajectories crossing record sized barriers, a behavior revealing the presence of a hierarchical structure in configuration space. The observation is here turned into a novel local search optimization algorithm dubbed Record Dynamics Optimization, or RDO. RDO uses the Metropolis rule to accept or reject candidate solutions depending on the value of a parameter akin to the temperature, and minimizes the cost function of the problem at hand through cycles where its `temperature' is raised and subsequently decreased in order to expediently generate record high (and low) values of the cost function. Below, RDO is introduced and then tested by searching the ground state of the Edwards-Anderson spin-glass model, in two and three spatial dimensions. A popular and highly efficient optimization algorithm, Parallel Tempering (PT) is applied to the same problem as a benchmark. RDO and PT turn out to produce solution of similar quality for similar numerical effort, but RDO is simpler to program and additionally yields geometrical information on the system's configuration space which is of interest in many applications. In particular, the effectiveness of RDO strongly indicates the presence of the above mentioned hierarchically organized configuration space, with metastable regions indexed by the cost (or energy) of the transition states connecting them.Comment: 14 pages, 12 figure