Wang-Landau Research Articles

Phase transition of Laplacian roughening model on a triangular lattice using Wang–Landau Monte Carlo simulation and Fisher zeros

Phase transition of the discrete Laplacian roughening model was investigated on a triangular lattice of various sizes by using the Wang–Landau (WL) Monte Carlo simulation and finite-size scaling analysis of partition-function zeros. The density of states was calculated using the WL method, and the internal energy and specific heat were calculated as a function of temperature. A single transition was observed at T c = 1.855(5) with a new universality class of the correlation exponent ν = 0.70(1) and the specific heat exponent α = 0.60(4). The results were confirmed by the finite size scaling analysis of the first zeros for the partition function. The critical exponents satisfy the scaling relation α = 2 − dν very well.

Flat-histogram method comparison on the two-dimensional Ising model

We compare the convergence of several flat-histogram methods applied to the two-dimensional Ising model, including the recently introduced stochastic approximation with a dynamic update factor (SAD) method. We compare this method to the Wang-Landau (WL) method, the 1/t variant of the WL method, and standard stochastic approximation Monte Carlo (SAMC). In addition, we consider a procedure WL followed by a "production run" with fixed weights that refines the estimation of the entropy. We find that WL followed by a production run does converge to the true density of states, in contrast to pure WL. Three of the methods converge robustly: SAD, 1/t-WL, and WL followed by a production run. Of these, SAD does not require a priori knowledge of the energy range. This work also shows that WL followed by a production run performs superior to other forms of WL while ensuring both ergodicity and detailed balance.

An Adaptive Image Watermarking Method Combining SVD and Wang-Landau Sampling in DWT Domain

To keep a better trade-off between robustness and imperceptibility is difficult for traditional digital watermarks. Therefore, an adaptive image watermarking method combining singular value decomposition (SVD) and the Wang–Landau (WL) sampling method is proposed to solve the problem. In this method, the third-level approximate sub-band obtained by applying the three-level wavelet transform is decomposed by SVD to obtain the principal component, which is firstly selected as the embedded position. Then, the information is finally embedded into the host image by the scaling factor. The Wang–Landau sampling method is devoted to finding the best embedding coefficient through the proposed objective evaluation function, which is a global optimization algorithm. The embedding strength is adaptively adjusted by utilizing the historical experience, which overcomes the problem of falling into local optimization easily in many traditional optimization algorithms. To affirm the reliability of the proposed method, several image processing attacks are applied and the experimental results are given in detail. Compared with other existing related watermarking techniques based on both qualitative and quantitative evaluation parameters, such as peak signal to noise ratio (PSNR) and normalized cross-correlation (NC), this method has been proven to achieve a trade-off between robustness and invisibility.

Wang-Landau algorithm: An adapted random walk to boost convergence

The Wang-Landau (WL) algorithm is a recently developed stochastic algorithm computing densities of states of a physical system, and also performing numerical integration in high dimensional spaces. Since its inception, it has been used on a variety of (bio-)physical systems, and in selected cases, its convergence has been proved. The convergence speed of the algorithm is tightly tied to the connectivity properties of the underlying random walk.As such, we propose an efficient random walk that uses geometrical information to circumvent the following inherent difficulties: avoiding overstepping strata, toning down concentration phenomena in high-dimensional spaces, and accommodating multidimensional distributions. These improvements are especially well suited to improve calculations on a per basin basis – included anharmonic ones.Experiments on various models stress the importance of these improvements to make WL effective in challenging cases. Altogether, these improvements make it possible to compute density of states for regions of the phase space of small biomolecules.

Global Thermodynamic Properties of Complex Spin Systems Calculated from Density of States and Indirectly by Thermodynamic Integration Method

Evaluation of global thermodynamic properties such as the entropy or the free energy of complex systems featuring a high degree of frustration or disorder is often desirable. Nevertheless, they cannot be measured directly in standard Monte Carlo simulation. Therefore, they are either evaluated indirectly from the directly measured quantities, for example by the thermodynamic integration method (TIM), or by applying more sophisticated simulation methods, such as the Wang-Landau (WL) algorithm, which can directly sample density of states. In the present investigation we compare the performance of the WL and TIM methods for the calculation of the entropy of an Ising antiferromagnetic system on a Kagome lattice – a typical example of a complex spin system with high geometrical frustration resulting in a non-zero residual entropy the value of which is exactly known. It is found that the easier to implement TIM can yield results of comparable accuracy with that of the more involved WL method.

Wlamc Scheme To Monomodal Brain Icon Registering

An advanced method to get the better of the challenges of medical icon registering is anticipated, using Wang Landau Adaptive Monte Carlo (WLAMC) approach. The Wang-Landau (WL) technique is a distinct Adaptive Monte Carlo (AMC) procedure that has engendered a lot of curiosity in the medical icon registering (IR) technique owing to some remarkable simulation implementations. The multiresolution centered process for registering multimodal imageries by using AMC structure is the efficient existing approach for icon registering anywhere arbitrary finding aspirants are caused as of an all dimensional outcome spread out of probable geometric conversions on each of the repetition by using a testing approach. The remedy applicants generated are appraised instituted upon the Pearson category-Seven inaccuracy amongst the stage instants of the pictures in deciding the key contender by the deepest faulty left over. Even the AMC approach is efficient, it has some drawbacks and it can be eliminated with this proposed WLAMC approach. The experiments are performed on the real time medical imageries and the comparison of results illustrates that the approach put forward performs considerably well for monomodal brain images than the previous icon registering techniques.

Numerical simulations study of a spin-1 Blume–Emery–Griffiths model on a square lattice

We use the Monte Carlo simulation technique to study the critical behavior of a three-state spin model, with bilinear and biquadratic nearest-neighbor interactions, known as the Blume–Emery–Griffiths model (BEG), in a square lattice. In order to characterize this model, we study the phase diagram, in which we identify three different phases: ferromagnetic, paramagnetic and quadrupolar, the later with one sublattice filled with spins and the other with vacancies. We perform our studies by using two algorithms: Metropolis update (MU) and Wang–Landau (WL). The critical scaling behavior of the model is complementary studied by applying results obtained by using both algorithms, while tricritical points and the tricritical scaling behavior is analyzed by means of WL measuring the joint density of states and using the method of field mixing in conjunction with finite-size scaling. Furthermore, motivated by the decoupling between spins observed within the quadrupolar phase, we further generalize the BEG model in order to study the behavior of the system by adding a next nearest neighbour (NNN) interaction between spins. We found that by increasing the strength of the (ferromagnetic) NNN interaction, a new ferromagnetic phase takes over that contains both quadrupolar and ferromagnetic order.

Wang–Landau sampling: Saving CPU time

In this work we propose an improvement to the Wang–Landau (WL) method that allows an economy in CPU time of about 60% leading to the same results with the same accuracy. We used the 2D Ising model to show that one can initiate all WL simulations using the outputs of an advanced WL level from a previous simulation. We showed that up to the seventh WL level (f6) the simulations are not biased yet and can proceed to any value that the simulation from the very beginning would reach. As a result the initial WL levels can be simulated just once. It was also observed that the saving in CPU time is larger for larger lattice sizes, exactly where the computational cost is considerable. We carried out high-resolution simulations beginning initially from the first WL level (f0) and another beginning from the eighth WL level (f7) using all the data at the end of the previous level and showed that the results for the critical temperature Tc and the critical static exponents β and γ coincide within the error bars. Finally we applied the same procedure to the 1/2-spin Baxter–Wu model and the economy in CPU time was of about 64%.

Optimal updating magnitude in adaptive flat-distribution sampling.

We present a study on the optimization of the updating magnitude for a class of free energy methods based on flat-distribution sampling, including the Wang-Landau (WL) algorithm and metadynamics. These methods rely on adaptive construction of a bias potential that offsets the potential of mean force by histogram-based updates. The convergence of the bias potential can be improved by decreasing the updating magnitude with an optimal schedule. We show that while the asymptotically optimal schedule for the single-bin updating scheme (commonly used in the WL algorithm) is given by the known inverse-time formula, that for the Gaussian updating scheme (commonly used in metadynamics) is often more complex. We further show that the single-bin updating scheme is optimal for very long simulations, and it can be generalized to a class of bandpass updating schemes that are similarly optimal. These bandpass updating schemes target only a few long-range distribution modes and their optimal schedule is also given by the inverse-time formula. Constructed from orthogonal polynomials, the bandpass updating schemes generalize the WL and Langfeld-Lucini-Rago algorithms as an automatic parameter tuning scheme for umbrella sampling.

Control of accuracy in the Wang-Landau algorithm.

The Wang-Landau (WL) algorithm has been widely used for simulations in many areas of physics. Our analysis of the WL algorithm explains its properties and shows that the difference of the largest eigenvalue of the transition matrix in the energy space from unity can be used to control the accuracy of estimating the density of states. Analytic expressions for the matrix elements are given in the case of the one-dimensional Ising model. The proposed method is further confirmed by numerical results for the one-dimensional and two-dimensional Ising models and also the two-dimensional Potts model.

Combining Wang–Landau sampling algorithm and heuristics for solving the unequal-area dynamic facility layout problem

The dynamic facility layout problem (DFLP) is the problem of placing facilities in a certain plant floor for multiple stages so that facilities do not overlap and the sum of the material handling and rearrangement costs are minimized. We describe a model, where the facilities have unequal-areas and the layout for each stage is produced on the continuous plant floor. The most difficulty of solving this problem consists in the lack of a powerful optimization method. Wang–Landau (WL) sampling algorithm is an improved Monte Carlo method, and has been successfully applied to solve many optimization problems. In this paper, we combine the WL sampling algorithm and some heuristic strategies to solve the unequal-area DFLP. In the WL sampling algorithm, a vacant point strategy is applied to update layout at one stage. To prevent overlapping of facilities and reduce the empty space among facilities, a pushing strategy and a pressuring strategy are applied. We have tested the proposed algorithm on four groups of cases and the computational results show that the proposed algorithm is effective in solving the unequal-area DFLP.

Multi-objective layout optimization of a satellite module using the Wang-Landau sampling method with local search

The layout design of satellite modules is considered to be NP-hard. It is not only a complex coupled system design problem but also a special multi-objective optimization problem. The greatest challenge in solving this problem is that the function to be optimized is characterized by a multitude of local minima separated by high-energy barriers. The Wang-Landau (WL) sampling method, which is an improved Monte Carlo method, has been successfully applied to solve many optimization problems. In this paper we use the WL sampling method to optimize the layout of a satellite module. To accelerate the search for a global optimal layout, local search (LS) based on the gradient method is executed once the Monte-Carlo sweep produces a new layout. By combining the WL sampling algorithm, the LS method, and heuristic layout update strategies, a hybrid method called WL-LS is proposed to obtain a final layout scheme. Furthermore, to improve significantly the efficiency of the algorithm, we propose an accurate and fast computational method for the overlapping depth between two objects (such as two rectangular objects, two circular objects, or a rectangular object and a circular object) embedding each other. The rectangular objects are placed orthogonally. We test two instances using first 51 and then 53 objects. For both instances, the proposed WL-LS algorithm outperforms methods in the literature. Numerical results show that the WL-LS algorithm is an effective method for layout optimization of satellite modules.

Nonconvergence of the Wang-Landau algorithms with multiple random walkers

This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and 1/t algorithms. The classical algorithms are modified by the use of m-independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number π by multiple dimensional integration is used in the continuum approximation. In each case, the error is obtained separately for each walker at a fixed time, t; then, the average over m walkers is performed. It is observed that the error goes as 1/sqrt[m]. However, if the number of walkers increases above a certain critical value m>m_{x}, the error reaches a constant value (i.e., it saturates). This occurs for both algorithms; however, it is shown that for a given system, the 1/t algorithm is more efficient and accurate than the similar version of the WL algorithm. It follows that it makes no sense to increase the number of walkers above a critical value m_{x}, since it does not reduce the error in the calculation. Therefore, the number of walkers does not guarantee convergence.

Static critical behavior of the [formula omitted]-states Potts model: High-resolution entropic study

Here we report a precise computer simulation study of the static critical properties of the two-dimensional q-states Potts model using very accurate data obtained from a modified Wang–Landau (WL) scheme proposed by Caparica and Cunha-Netto (2012). This algorithm is an extension of the conventional WL sampling, but the authors changed the criterion to update the density of states during the random walk and established a new procedure to windup the simulation run. These few changes have allowed a more precise microcanonical averaging which is essential to a reliable finite-size scaling analysis. In this work we used this new technique to determine the static critical exponents β, γ, and ν, in an unambiguous fashion. The static critical exponents were determined as β=0.10811(77), γ=1.4459(31), and ν=0.8197(17), for the q=3 case, and β=0.0877(37), γ=1.3161(69), and ν=0.7076(10), for the q=4 Potts model. A comparison of the present results with conjectured values and with those obtained from other well established approaches strengthens this new way of performing WL simulations.

Intrinsic convergence properties of entropic sampling algorithms

.We study the convergence of the density of states and thermodynamic properties in three flat-histogram simulation methods, the Wang–Landau (WL) algorithm, the 1/t algorithm, and tomographic sampling (TS). In the first case the refinement parameter f is rescaled (f → f/2) each time the flat-histogram condition is satisfied, in the second f ∼ 1/t after a suitable initial phase, while in the third f is constant (t corresponds to Monte Carlo time). To examine the intrinsic convergence properties of these methods, free of any complications associated with a specific model, we study a featureless entropy landscape, such that for each allowed energy E = 1, ..., L, there is exactly one state, that is, g(E) = 1 for all E. Convergence of sampling corresponds to g(E, t) → const. as t → ∞, so that the standard deviation σg of g over energy values is a measure of the overall sampling error. Neither the WL algorithm nor TS converge: in both cases σg saturates at long times. In the 1/t algorithm, by contrast, σg decays . Modified TS and 1/t procedures, in which f ∝ 1/tα, converge for α values between 0 < α ≤ 1. There are two essential facets to convergence of flat-histogram methods: elimination of initial errors in g(E) and correction of the sampling noise accumulated during the process. For a simple example, we demonstrate analytically, using a Langevin equation, that both kinds of errors can be eliminated, asymptotically, if f ∼ 1/t α with 0 < α ≤ 1. Convergence is optimal for α = 1. For α ≤ 0 the sampling noise never decays, while for α > 1 the initial error is never completely eliminated.

Applications of the Wang-Landau algorithm to phase transitions of a single polymer chain

The Wang-Landau (WL) algorithm is a Monte Carlo simulation technique providing a direct computation of the density of states of a many-body system. The temperature independent density of states function encodes all thermodynamic information about a system, and thus, its construction allows for a very efficient determination of phase behavior. Here we describe the application of the WL approach to continuum interaction-site polymer chains and compute single-chain phase diagrams for three specific models (flexible and semiflexible homopolymer chains built from square-well sphere monomers and a flexible AB-heteropolymer comprised of alternating square-well and hard-sphere monomers). We provide details on the implementation of the WL algorithm, the underlying Monte Carlo move set, and the subsequent thermodynamic and structural analysis required to characterize phase behavior.

A comparison of the performance of Wang–Landau-Transition-Matrix algorithm with Wang–Landau algorithm for the determination of the joint density of states for continuous spin models

Monte Carlo simulation has been performed in a one-dimensional Lebwohl–Lasher model and two-dimensional XY-model using the Wang–Landau (WL) and the Wang–Landau-Transition-Matrix (WLTM) Monte Carlo methods. Random walk has been performed in the two-dimensional space comprising of energy-order parameter and energy-correlation function and the joint density of states (JDOS) were obtained. From the JDOS the order parameter, susceptibility and correlation function are calculated. Agreement between the results obtained from the two algorithms is very good. The work shows that for the purpose of determination of JDOS, the WLTM method is a good alternative to the WL algorithm.

Perturbation method to calculate the density of states

Monte Carlo switching moves ("perturbations") are defined between two or more classical Hamiltonians sharing a common ground-state energy. The ratio of the density of states (DOS) of one system to that of another is related to the ensemble averages of the microcanonical acceptance probabilities of switching between these Hamiltonians, analogously to the case of Bennett's acceptance ratio method for the canonical ensemble [C. H. Bennett, J. Comput. Phys. 22, 245 (1976)]. Thus, if the DOS of one of the systems is known, one obtains those of the others and, hence, the partition functions. As a simple test case, the vapor pressure of an anharmonic Einstein crystal is computed, using the harmonic Einstein crystal as the reference system in one dimension; an auxiliary calculation is also performed in three dimensions. As a further example of the algorithm, the energy dependence of the ratio of the DOS of the square-well and hard-sphere tetradecamers is determined, from which the temperature dependence of the constant-volume heat capacity of the square-well system is calculated and compared with canonical Metropolis Monte Carlo estimates. For these cases and reference systems, the perturbation calculations exhibit a higher degree of convergence per Monte Carlo cycle than Wang-Landau (WL) sampling, although for the one-dimensional oscillator the WL sampling is ultimately more efficient for long runs. Last, we calculate the vapor pressure of liquid gold using an empirical Sutton-Chen many-body potential and the ideal gas as the reference state. Although this proves the general applicability of the method, by its inherent perturbation approach the algorithm is suitable for those particular cases where the properties of a related system are well known.

Numerical estimate for boiling points via Wang–Landau simulations

We show how molecular simulations, based on the Wang–Landau (WL) sampling method, can be used to determine the boiling points of a wide range of fluids. The simulations presented here rely on an efficient sampling of the isothermal–isobaric ensemble. This circumvents the difficult deletion/insertion of molecules in dense liquid phases, such as those encountered at the standard boiling point. Using a combination of the WL sampling and hybrid Monte Carlo simulations, we show how we can determine the standard boiling point of atomic fluids, such as liquid metals. We then apply WL configurational bias Monte Carlo simulations to determine the boiling points for a series of long n-alkane chains (icosane, tetracosane and triacontane). The results obtained in this work are in good agreement with the available experimental data and simulation results.

A Wang-Landau study of a lattice model for lipid bilayer self-assembly

The Wang-Landau (WL) Monte Carlo method has been applied to simulate the self-assembly of a lipid bilayer on a 3D lattice. The WL method differs from conventional Monte Carlo methods in that a complete density of states is obtained directly for the system, from which properties, such as the free energy, can be derived. Furthermore, from a single WL simulation, continuous curves of the average energy and heat capacity can be determined, which provide a complete picture of the phase behavior. The lipid model studied consists of 3 or 5 coarse-grained segments on lattices of varying sizes, with the empty lattice sites representing water. A bilayer structure is found to form at low temperatures, with phase transitions to clusters as temperature increases. For 3-segment chains, varying lattice sizes were studied, with the observation that the ratio of chain number to lattice area (i.e., area per lipid) affects the phase transition temperature. At small ratios, only one phase transition occurs between the bilayer and cluster phases, while at high lipid ratios the phase transition occurs in a two-step process with a stable intermediate phase. This second phase transition was not observed in conventional Metropolis Monte Carlo simulations on the same model, demonstrating the advantage of being able to perform a complete scan of the whole temperature range with the WL method. For longer 5-segment chains similar phase transitions are also observed with changes in temperature. In the WL method, due to the extensive nature of the energy, the number of energy bins required to represent the density of states increases as the system size increases and so limits its practical application to larger systems. To improve this, an extension of the WL algorithm, the statistical-temperature Monte Carlo method that allows simulations with larger energy bin sizes, has recently been proposed and is implemented in this work for the 3-segment lattice model. The results obtained are in good agreement with the original WL method and appear to be independent of the energy bin size used.