Showing papers on "Dynamic Monte Carlo method published in 2001"

Quantum Monte Carlo simulations of solids

Abstract: This article describes the variational and fixed-node diffusion quantum Monte Carlo methods and how they may be used to calculate the properties of many-electron systems. These stochastic wave-function-based approaches provide a very direct treatment of quantum many-body effects and serve as benchmarks against which other techniques may be compared. They complement the less demanding density-functional approach by providing more accurate results and a deeper understanding of the physics of electronic correlation in real materials. The algorithms are intrinsically parallel, and currently available high-performance computers allow applications to systems containing a thousand or more electrons. With these tools one can study complicated problems such as the properties of surfaces and defects, while including electron correlation effects with high precision. The authors provide a pedagogical overview of the techniques and describe a selection of applications to ground and excited states of solids and clusters.

Multilevel Monte Carlo Methods

Abstract: We study Monte Carlo approximations to high dimensional parameter dependent integrals. We survey the multilevel variance reduction technique introduced by the author in [4] and present extensions and new developments of it. The tools needed for the convergence analysis of vector-valued Monte Carlo methods are discussed, as well. Applications to stochastic solution of integral equations are given for the case where an approximation of the full solution function or a family of functionals of the solution depending on a parameter of a certain dimension is sought.

Monte Carlo Methods in Bayesian Computation

Abstract: This is an important contribution to a modern area of applied probability that deals with nonstationary Markov chains in continuous time. This area is becoming increasingly useful in engineering, economics, communication theory, active networking, and so forth, where the Markov-chain system is subject to frequent  uctuations with clusters of states such that the chain  uctuates very rapidly among different states of a cluster but changes less rapidly from one cluster to another. The authors use the setting of singular perturbations, which allows them to study both weak and strong interactions among the states of the chain. This leads to simpliŽ cations through the averaging principle, aggregation, and decomposition. The main results include asymptotic expansions of the corresponding probability distributions, occupations measures, limiting normality, and exponential rates. These results give the asymptotic behavior of many controlled stochastic dynamic systems when the perturbation parameter tends to 0. The classical analytical method employs the asymptotic expansions of onedimensional distributions of the Markov chain as solutions to a system of singularly perturbed ordinary differential equations. Indeed, the asymptotic behavior of solutions of such equations is well studied and understood. A more probabilistic approach also used by the authors is based on the tightness of the family of probability measures generated by the singularly perturbed Markov chain with the corresponding weak convergence properties. Both of these methods are illustrated by practical dynamic optimization problems, in particular by hierarchical production planning in a manufacturing system. An important contribution is the last chapter, Chapter 10, which describes numerical methods to solve various control and optimization problems involving Markov chains. Altogether the monograph consists of three parts, with Part I containing necessary, technically rather demanding facts about Markov processes (which in the nonstationary case are deŽ ned through martingales.) Part II derives the mentioned asymptotic expansions, and Part III deals with several applications, including Markov decision processes and optimal control of stochastic dynamic systems. This technically demanding book may be out of reach of many readers of Technometrics. However, the use of Markov processes has become common for numerous real-life complex stochastic systems. To understand the behavior of these systems, the sophisticated mathematical methods described in this book may be indispensable.

Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table

Abstract: We present a method for carrying out long time scale dynamics simulations within the harmonic transition state theory approximation. For each state of the system, characterized by a local minimum on the potential energy surface, multiple searches for saddle points are carried out using random initial directions. The dimer method is used for the saddle point searches and the rate for each transition mechanism is estimated using harmonic transition state theory. Transitions are selected and the clock advanced according to the kinetic Monte Carlo algorithm. Unlike traditional applications of kinetic Monte Carlo, the atoms are not assumed to sit on lattice sites and a list of all possible transitions need not be specified beforehand. Rather, the relevant transitions are found on the fly during the simulation. A multiple time scale simulation of Al(100) crystal growth is presented where the deposition event, occurring on the time scale of picoseconds, is simulated by ordinary classical dynamics, but the time i...

Monte Carlo Experiments: Design and Implementation

Abstract: The use of Monte Carlo simulations for the empirical assessment of statistical estimators is becoming more common in structural equation modeling research. Yet, there is little guidance for the researcher interested in using the technique. In this article we illustrate both the design and implementation of Monte Carlo simulations. We present 9 steps in planning and performing a Monte Carlo analysis: (1) developing a theoretically derived research question of interest, (2) creating a valid model, (3) designing specific experimental conditions, (4) choosing values of population parameters, (5) choosing an appropriate software package, (6) executing the simulations, (7) file storage, (8) troubleshooting and verification, and (9) summarizing results. Throughout the article, we use as a running example a Monte Carlo simulation that we performed to illustrate many of the relevant points with concrete information and detail.

Extended ensemble monte carlo

Abstract: "Extended Ensemble Monte Carlo" is a generic term that indicates a set of algorithms, which are now popular in a variety of fields in physics and statistical information processing. Exchange Monte Carlo (Metropolis-Coupled Chain, Parallel Tempering), Simulated Tempering (Expanded Ensemble Monte Carlo) and Multicanonical Monte Carlo (Adaptive Umbrella Sampling) are typical members of this family. Here, we give a cross-disciplinary survey of these algorithms with special emphasis on the great flexibility of the underlying idea. In Sec. 2, we discuss the background of Extended Ensemble Monte Carlo. In Secs. 3, 4 and 5, three types of the algorithms, i.e., Exchange Monte Carlo, Simulated Tempering, Multicanonical Monte Carlo, are introduced. In Sec. 6, we give an introduction to Replica Monte Carlo algorithm by Swendsen and Wang. Strategies for the construction of special-purpose extended ensembles are discussed in Sec. 7. We stress that an extension is not necessary restricted to the space of energy or tempe...

Precise determination of the critical percolation threshold for the three- dimensional ''Swiss cheese'' model using a growth algorithm

Abstract: Precise values for the critical threshold for the three-dimensional “Swiss cheese” continuum percolation model have been calculated using extensive Monte Carlo simulations. These simulations used a growth algorithm and memory blocking scheme similar to what we used previously in three-dimensional lattice percolation. The simulations yield a value for the critical number density nc=0.652 960±0.000 005, which confirms recent work but extends the precision by two significant figures.

Evolutionary Monte Carlo for protein folding simulations

Abstract: We demonstrate that evolutionary Monte Carlo (EMC) can be applied successfully to simulations of protein folding on simple lattice models, and to finding the ground state of a protein. In all cases, EMC is faster than the genetic algorithm and the conventional Metropolis Monte Carlo, and in several cases it finds new lower energy states. We also propose one method for the use of secondary structures in protein folding. The numerical results show that it is drastically superior to other methods in finding the ground state of a protein.

Monte Carlo simulations of Wyoming sodium montmorillonite hydrates

Abstract: Monte Carlo simulations have been used to predict the interlayer basal separations of sodium-saturated Wyoming clays at constant stress (NPzzT ensemble) and at constant chemical potential (μVT ensemble). These simulations use the Ewald summation technique to incorporate long-range Coulombic interactions in the calculation of the total potential energy and the pressure tensor. A comparison is made between the use of one, two, and three sheets of clay. It is shown that, for small separations, at least two separate clay sheets must be used to avoid system-size effects. The stable interlamellar separations are determined by combining results from isostress–isothermal and grand canonical simulations. It is shown that, consistent with experiments, at the temperature and pressure studied here, the cations in the interlayer are hydrated, except at the smallest basal separations.

The computer simulation of microstructural evolution

Abstract: This paper reviews the kinetic Monte Carlo Potts model for simulating microstructural evolution. When properly implemented, that model provides a fast and flexible tool for evaluating a variety of materials systems in two and three dimensions, generating snapshots of the evolving microstructure with time. Examples of the model are provided, along with potential applications.

Fast Monte Carlo algorithm for supercooled soft spheres

Abstract: We present a nonlocal Monte Carlo algorithm with particle swaps that greatly accelerates thermalization of soft sphere binary mixtures in the glassy region. Our first results show that thermalization of systems of hundreds of particles is achievable, and find behavior compatible with a thermodynamic glass transition.

Spin correlations in Monte Carlo simulations

Abstract: We show that the algorithm originally proposed by Collins and Knowles for spin correlations in the QCD parton shower can be used in order to include spin correlations between the production and decay of heavy particles in Monte Carlo event generators. This allows correlations to be included while maintaining the step-by-step approach of the Monte Carlo event generation process. We present examples of this approach for both the Standard and Minimal Supersymmetric Standard Models. A merger of this algorithm and that used in the parton shower is discussed in order to include all correlations in the perturbative phase of event generation. Finally, we present all the results needed to implement this algorithm for the Standard and Minimal Supersymmetric Standard Models.

Aggregation-volume-bias Monte Carlo simulations of vapor-liquid nucleation barriers for Lennard-Jonesium

Abstract: A combination of the aggregation-volume-bias Monte Carlo algorithm and the umbrella sampling technique is applied to investigate homogeneous vapor–liquid nucleation. This combined approach is simple, general, and robust. Its efficiency is demonstrated for nucleation of Lennard-Jonesium, for which the precise calculation of the nucleation barriers takes only a few minutes at higher supersaturations to a few hours at lower supersaturations. Comparison of the simulation results to the classical nucleation theory (CNT) shows that CNT overestimates the barrier heights by a value nearly independent of the supersaturation, but provides a reasonable description of the critical cluster sizes.

Monte Carlo simulations of spin glasses at low temperatures

Abstract: We report the results of Monte Carlo simulations on several spin-glass models at low temperatures. By using the parallel tempering (exchange Monte Carlo) technique we are able to equilibrate down to low temperatures, for moderate sizes, and hence the data should not be affected by critical fluctuations. Our results for short-range models are consistent with a picture proposed earlier that there are large-scale excitations which cost only a finite energy in the thermodynamic limit, and these excitations have a surface whose fractal dimension is less than the space dimension. For the infinite range Viana-Bray model, our results obtained for a similar number of spins, are consistent with standard replica symmetry breaking.

A cluster Monte Carlo algorithm for 2-dimensional spin glasses

Abstract: A new Monte Carlo algorithm for 2-dimensional spin glasses is presented. The use of clusters makes possible global updates and leads to a gain in speed of several orders of magnitude. As an example, we study the 2-dimensional ±J Edwards-Anderson model. The new algorithm allows us to equilibrate systems of size 1002 down to temperature T = 0.1. Our main result is that the correlation length diverges as an exponential ( ξ∼e2βJ) and not as a power law as T↦Tc = 0.

Optimization of Ground- and Excited-State Wave Functions and van der Waals Clusters

Abstract: A quantum Monte Carlo method is introduced to optimize excited-state trial wave functions. The method is applied in a correlation function Monte Carlo calculation to compute ground- and excited-state energies of bosonic van der Waals clusters of up to seven particles. The calculations are performed using trial wave functions with general three-body correlations.

Monte Carlo update for chain molecules: Biased Gaussian steps in torsional space

Abstract: We develop a new elementary move for simulations of polymer chains in torsion angle space. The method is flexible and easy to implement. Tentative updates are drawn from a (conformation-dependent) Gaussian distribution that favors approximately local deformations of the chain. The degree of bias is controlled by a parameter b. The method is tested on a reduced model protein with 54 amino acids and the Ramachandran torsion angles as its only degrees of freedom, for different b. Without excessive fine tuning, we find that the effective step size can be increased by a factor of 3 compared to the unbiased b=0 case. The method may be useful for kinetic studies, too.

Population Monte Carlo algorithms

Abstract: We give a cross-disciplinary survey on “population” Monte Carlo algorithms.In these algorithms, a set of “walkers” or “particles” is used as a representation of a high-dimensional vector. The computation is carried out by a random walk and split/deletion of these objects. The algorithms are developed in various fields in physics and statistical sciences and called by lots of different terms — “quantum Monte Carlo”, “transfer-matrix Monte Carlo”, “Monte Carlo filter (particle filter)”, “sequential Monte Carlo” and “PERM” etc. Here we discuss them in a coherent framework. We also touch on related algorithms —genetic algorithms and annealed importance sampling.

Time Relaxed Monte Carlo Methods for the Boltzmann Equation

Abstract: A new family of Monte Carlo schemes is introduced for the numerical solution of the Boltzmann equation of rarefied gas dynamics. The schemes are inspired by the Wild sum expansion of the solution of the Boltzmann equation for Maxwellian molecules and consist of a novel time discretization of the equation. In particular, high order terms in the expansion are replaced by the equilibrium Maxwellian distribution. The two main features of the schemes are high order accuracy in time and asymptotic preservation. The first property allows to recover accurate solutions with time steps larger than those required by direct simulation Monte Carlo (DSMC), while the latter guarantees that for the vanishing Knudsen number, the numerical solution relaxes to the local Maxwellian. Conservation of mass, momentum, and energy are preserved by the scheme. Numerical results on several space homogeneous problems show the improvement of the new schemes over standard DSMC. Applications to a one-dimensional shock wave problem are also presented.

Monte Carlo Smoothing and Self-Organising State-Space Model

Abstract: A Monte Carlo method for nonlinear non-Gaussian filtering and smoothing and its application to self-organising state-space models are shown in this paper.

Direct Gibbs Ensemble Monte Carlo Simulations for Solid−Vapor Phase Equilibria: Applications to Lennard−Jonesium and Carbon Dioxide

Abstract: The Gibbs ensemble Monte Carlo method of Panagiotopoulos is extended to calculations of solid−vapor coexistence curves. As in the original Gibbs ensemble method, the new technique makes use of two simulation boxes that are in thermodynamic contact. However, the box that contains the solid phase is elongated along one axis and contains only a slab of solid material surrounded on both sides by vapor. Aggregation-volume-bias Monte Carlo moves are used to sample transfers from the solid to the vapor and vice versa in this box, whereas the usual particle swap moves are applied to transfers between the solid−vapor box and the other box that contains a bulk vapor phase. Volume moves for the solid−vapor box use separate displacements of individual cell lengths or of individual H-matrix elements. As one approaches the triple-point temperature from below, increased disorder at the solid−vapor interface is observed, and once the triple-point temperature is exceeded, the entire solid slab converts to a liquid. The us...

Smart Darting Monte Carlo

Abstract: The “Smart Walking” Monte Carlo algorithm is examined. In general, due to a bias imposed by the interbasin trial move, the algorithm does not satisfy detailed balance. While it has been shown that it can provide good estimates of equilibrium averages for certain potentials, for other potentials the estimates are poor. A modified version of the algorithm, Smart Darting Monte Carlo, which obeys the detailed balance condition, is proposed. Calculations on a one-dimensional model potential, on a Lennard-Jones cluster and on the alanine dipeptide demonstrate the accuracy and promise of the method for deeply quenched systems.