Consider the two problems of simulating observations and estimating expectations and normalizing constants for multiple distributions. First, we present a self-adjusted mixture sampling method, which ...accommodates both adaptive serial tempering and a generalized Wang-Landau algorithm. The set of distributions are combined into a labeled mixture, with the mixture weights depending on the initial estimates of log normalizing constants (or free energies). Then, observations are generated by Markov transitions, and free energy estimates are adjusted online by stochastic approximation. We propose two stochastic approximation schemes by Rao-Blackwellization of the scheme commonly used, and derive the optimal choice of a gain matrix, resulting in the minimum asymptotic variance for free energy estimation, in a simple and feasible form. Second, we develop an offline method, locally weighted histogram analysis, for estimating free energies and expectations, using all the simulated data from multiple distributions by either self-adjusted mixture sampling or other sampling algorithms. This method can be computationally much faster, with little sacrifice of statistical efficiency, than a global method currently used, especially when a large number of distributions are involved. We provide both theoretical results and numerical studies to demonstrate the advantages of the proposed methods.
By mixing the target posterior distribution with a surrogate distribution, of which the normalizing constant is tractable, we propose a method for estimating the marginal likelihood using the ...Wang-Landau algorithm. We show that a faster convergence of the proposed method can be achieved via the momentum acceleration. Two implementation strategies are detailed: (i) facilitating global jumps between the posterior and surrogate distributions via the multiple-try Metropolis (MTM); (ii) constructing the surrogate via the variational approximation. When a surrogate is difficult to come by, we describe a new jumping mechanism for general reversible jump Markov chain Monte Carlo algorithms, which combines the MTM and a directional sampling algorithm. We illustrate the proposed methods on several statistical models, including the log-Gaussian Cox process, the Bayesian Lasso, the logistic regression, and the g-prior Bayesian variable selection.
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We investigate the Laplacian roughening model on kagome and honeycomb lattices. By using the Wang–Landau algorithm, we obtain the density of states, and calculate both the specific heat and the ...partition function zeros. Critical exponents and transition temperature are measured by means of the finite-size scaling analysis. The results indicate strong evidences that the system undergoes a single first-order phase transition on both lattices. On triangular and square lattices, a single second-order transition was found in our previous studies. It seems that the type of transition in the Laplacian roughening model depends on the substrate lattice structures.
•The discrete Laplacian roughening model is studied on kagome and honeycomb lattices.•We calculate the specific heat and the partition function zeros.•Our results strongly support a single first-order transition.•The type of transition seems to depend on the substrate lattice structures.
The thermodynamic and critical properties, and phase transitions of two-dimensional Ising model on a square lattice with competing interactions are investigated by the Monte Carlo method. Estimations ...are made for the magnitude relations of the next-nearest-neighbor and nearest-neighbor exchange interactions r=J2/J1 in the value ranges of 0.1≤r≤1.0. The anomalies of thermodynamic observables are shown to be present in this model on the interval 0.45≤r≤0.5. The phase diagram for the dependence of the critical temperature on a value of next-nearest neighbor interaction is plotted. A phase transition for all values in the interval 0.45≤r≤0.5 is shown to be a second order. Our data show that the temperature of the heat capacity maximum at r=0.5 tends to a finite value. The static critical exponents of the heat capacity α, susceptibility γ, order parameter β, correlation length ν, and the Fisher exponent η are calculated by means of the finite-size scaling theory. It is found that the change in next-nearest neighbor interaction value in the range 0.7≤r≤1.0 leads to nonuniversal critical behavior.
•The thermodynamic and critical properties, and phase transitions of two-dimensional Ising model on a square lattice with competing interactions are investigated by the Monte Carlo method.•Estimations are made for the magnitude relations of the next-nearest-neighbor and nearest-neighbor exchange interactions r=J2/J1 in the value ranges of 0.1≤r≤1.0.•The anomalies of thermodynamic parameters are shown to be present in this model on the interval 0.45≤r≤0.5.•The phase diagram for the dependence of the critical temperature on a value of next-nearest neighbor interaction is plotted.•A phase transition for all values in the interval 0.45≤r≤0.5 is shown to be a second order.•Our data show that the temperature of the heat capacity maximum at r=0.5 tends to a finite value.•The static critical exponents of the heat capacity α, susceptibility γ, order parameter β, correlation length ν, and the Fisher exponent η are calculated by means of the finite-size scaling theory.•It is found that the change in next-nearest neighbor interaction value in the range 0.7≤r≤1.0 leads to nonuniversal critical behavior.
The Wang–Landau algorithm is used to study the magnetic properties of the Ising model on the Shastry–Sutherland lattice in order to understand the interesting magnetization plateaus observed in TmB4. ...The simulated results demonstrate that the equilibrium state of the model produces only the 1/3 and 1/2 magnetization plateaus at low temperatures even when the random-exchange term or the long-range interactions are taken into account. This confirms our earlier conclusion (Huang et al., 2013) 20 that those fractional plateaus observed in experiments may be due to the magnetization dynamics.
•The magnetic behaviors of TmB4 are investigated using the Wang–Landau method.•The equilibrium state only produces the 1/3 and 1/2 magnetization plateaus.•Those fractional plateaus must arise from the non-equilibrium magnetization.
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•Wang–Landau method to calculate density of states of a protein folding–unfolding transition of villin headpiece (HP35).•Two dimensional potential of mean force profile for geometric ...parameters at different temperatures.
A computational study, using the Wang–Landau algorithm, is carried out for the 35 residue mini-protein villin headpiece (HP35) to investigate some equilibrium aspects of its folding–unfolding transition in water. The force field used is ECEPP/3 and a solvent-accessible surface area method is used to describe the interaction with water. The density of states and the conditional probabilities of some physically interesting geometrical descriptors of the molecule at various energies are calculated by analyzing the high-statistics data generated by the simulations. All canonical ensemble averages are subsequently calculated by using these data. The computed quantities include specific heat, radius of gyration, number of native contacts, number of helical residues and the potential of mean force for three pairs of geometric descriptors. High precision and a systematic check on the convergence of the computation are the two key points of our methodology.
Modern physical networks, for example in communication and transportation, can be interpreted as directed graphs. Network models are used to identify the probability that given nodes are connected, ...and therefore the effect of a failure at a given link. This is essential for network design, optimization, and reliability. In this study, we investigated three alternative ensembles for estimating network reliability using the Wang–Landau algorithm. The first performed random walks on a structure function having two possible states: connected and disconnected. The second used random walks on a reliability polynomial. The third combined random walks with the average of connecting probabilities. The accuracy and limitations of the three ensembles were compared by estimating the reliability of three network models: a bridge network, a ladder-type network, and a dodecahedron network. The simulation results showed that the use of a random walk on a structure function failed to produce estimates when applied to highly reliable networks in any of the three network types. The other two approaches performed efficiently for bridge or ladder-type networks at any level of network reliability. The random walk on a probability space using the 1∕t algorithm was the only ensemble that was able to yield accurate estimates for a dodecahedron network, though even this failed at the highest level of network reliability. The other two methods failed to converge within 108 Monte Carlo trials. The use of the average of connecting probabilities required a shorter computation time when applied to a large network. Methods that can reduce variance for large, highly reliable networks require further investigation.
Critical behavior of three-dimensional classical frustrated antiferromagnets with a collinear spin ordering and with an additional twofold degeneracy of the ground state is studied. We consider two ...lattice models, whose continuous limit describes a single phase transition with a symmetry class differing from the class of non-frustrated magnets as well as from the classes of magnets with non-collinear spin ordering. A symmetry breaking is described by a pair of independent order parameters, which are similar to order parameters of the Ising and O(N) models correspondingly. Using the renormalization group method, it is shown that a transition is of first order for non-Ising spins. For Ising spins, a second order phase transition from the universality class of the O(2) model may be observed. The lattice models are considered by Monte Carlo simulations based on the Wang–Landau algorithm. The models are a ferromagnet on a body-centered cubic lattice with the additional antiferromagnetic exchange interaction between next-nearest-neighbor spins and an antiferromagnet on a simple cubic lattice with the additional interaction in layers. We consider the cases N = 1, 2, 3 and in all of them find a first-order transition. For the N = 1 case we exclude possibilities of the second order or pseudo-first order of a transition. An almost second order transition for large N is also discussed.
•A magnet on a body-centered cubic lattice with an additional interaction between next-nearest-neighbor spins.•A magnet on a simple cubic lattice with an additional interaction between next-nearest-neighbor spins in layers.•Monte Carlo simulations based on the Wang–Landau algorithm.•The renormgroup approach based on the 4-e expansions.•A first-order transition simultaneously in continuous and discrete order parameters.
The phase diagram of a general biquadratic Hamiltonian model of biaxial nematic liquid crystals was investigated both analytically through mean-field approximation and with computer simulations. ...However, their largely concurrent predictions are not borne out by experimentation, and the issue is still debated. We revisited this problem with Monte Carlo simulations based on the computation of density of states of the system through entropic sampling procedure, traversing through the relevant model parameter space (S) along representative trajectories. Our recent work indicated that the competing roles of different contributions in the Hamiltonian over significant regions of S could be the underlying entropic reason defying the earlier predictions. We find that our data differ from the reported results qualitatively, specifically as the trajectories approach the so-called partly repulsive regions of S. The complex free-energy profiles that we obtain in such cases, as a function of system order parameters, indicate entropic barriers to the development of the biaxial order to the expected degree. Significant increase in the influence of the intermolecular interactions between the uniaxial and biaxial tensors, at the expense of contributions from pure biaxial couplings, is indicated to be the inhibiting factor for the onset of the biaxial phase.
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