Showing papers on "Monte Carlo method published in 1987"

Model Selection and Akaike's Information Criterion (AIC): The General Theory and Its Analytical Extensions.

Abstract: During the last fifteen years, Akaike's entropy-based Information Criterion (AIC) has had a fundamental impact in statistical model evaluation problems. This paper studies the general theory of the AIC procedure and provides its analytical extensions in two ways without violating Akaike's main principles. These extensions make AIC asymptotically consistent and penalize overparameterization more stringently to pick only the simplest of the “true” models. These selection criteria are called CAIC and CAICF. Asymptotic properties of AIC and its extensions are investigated, and empirical performances of these criteria are studied in choosing the correct degree of a polynomial model in two different Monte Carlo experiments under different conditions.

Nonuniversal critical dynamics in Monte Carlo simulations

Abstract: A new approach to Monte Carlo simulations is presented, giving a highly efficient method of simulation for large systems near criticality. The algorithm violates dynamic universality at second-order phase transitions, producing unusually small values of the dynamical critical exponent.

Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble

Abstract: A methodology is presented for Monte Carlo simulation of fluids in a new ensemble that can be used to obtain phase coexistence properties of multicomponent systems from a single computer experiment. The method is based on performing a simulation simultaneously in two distinct physical regions of generally different densities and compositions. Three types of perturbations are performed, a random displacement of molecules that ensures equilibrium within each region, an equal and opposite change in the volume of the two regions that results in equality of pressures, and random transfers of molecules that equalize the chemical potentials of each component in the two regions. The method is applied to the calculation of the liquid-gas coexistence envelope for the pure Lennard-Jones (6, 12) fluid for several reduced temperatures from the vicinity of the triple point to close to the critical point (T* = 0·75 to T* = 1·30). Good overall agreement with previously available literature results is obtained, with some ...

Monte Carlo-minimization approach to the multiple-minima problem in protein folding.

Abstract: A Monte Carlo-minimization method has been developed to overcome the multiple-minima problem. The Metropolis Monte Carlo sampling, assisted by energy minimization, surmounts intervening barriers in moving through successive discrete local minima in the multidimensional energy surface. The method has located the lowest-energy minimum thus far reported for the brain pentapeptide [Met5]enkephalin in the absence of water. Presumably it is the global minimum-energy structure. This supports the concept that protein folding may be a Markov process. In the presence of water, the molecules appear to exist as an ensemble of different conformations.

Estimating the recombination parameter of a finite population model without selection.

Abstract: An estimator is proposed for the parameter C = 4Nc. where N is the population size and c is the recombination rate. The estimator is appropriate for use with sequence or restriction site data from random samples from within populations. Properties of the estimator are investigated for an infinite-sites neutral model using Monte Carlo simulations. The median and mode of the distribution of the estimator are close to the true value for all parameter values examined, but large data sets are required to obtain reliable estimates.

Option Pricing when the Variance Is Changing

Abstract: The Monte Carlo method is used to solve for the price of a call when the variance is changing stochastically.

Partial least squares analysis with cross‐validation for the two‐class problem: A Monte Carlo study

Abstract: A method for statistical analysis of two independent samples with respect to difference in location is investigated. The method uses the partial least squares projections to latent structures (PLS) with cross-validation. The relation to classical methods is discussed and a Monte Carlo study is performed to describe how the distribution of the test-statistic employed depends on the number of objects, the number of variables, the percentage variance explained by the first PLS-component and the percentage missing values. Polynomial approximations for the dependency of the 50 per cent and the 5 per cent levels of the test-statistic on these factors are given. The polynomial for the 50 per cent level is complicated, involving several first-, second- and third-degree terms, whereas the polynomial for the 5 per cent level is dependent only on the number of objects and the size of the first component. A separate Monte Carlo experiment indicates that a moderate difference in sample size does not affect the distribution of the test-statistic. The multi-sample location problem is also studied and the effect of increasing the number of samples on the test-statistic is shown in simulations.

A numerical investigation of nematic ordering based on a simple hard-rod model

Abstract: On the basis of a simple functional scaling, we have constructed the direct generalization of the Carnahan–Starling equation for the free energy of a hard sphere fluid to a nematic liquid crystal. The orientational degrees of freedom are completely decoupled from translational ones under that scaling, which is equivalent to the well‐known decoupling approximation for the pair correlation function of hard‐core fluids. For long rods the generalized free energy reduces exactly to Onsager’s in the low density limit. Numerical calculations have been performed for a variety of the length‐to‐diameter ratios of hard spherocylinders, with the aid of an accurate iterative solution to the nonlinear integral equation for the orientational distribution function. The predictions made by our calculations are in fairly good agreement with the results of Monte Carlo simulations on a system of rather short rods. We present the calculations of the jump in the order parameter, critical packing fraciton, and the other thermodynamic properties at the isotropic–nematic transition.

The Accuracy of Cross-Correlation Estimates of Quasar Emission-Line Region Sizes

Abstract: The use of cross-correlation analyses of emission-line and optical continuum fluxes to determine broad-line region (BLR) sizes in low-luminosity quasars is examined in detail. Uncertainties associated with this method are discussed, and analytic formulas for estimating errors and determining confidence levels are presented. It is pointed out that systematic correlated errors between continuum and line fluxes are particularly dangerous and can lead to BLR sizes that may be too small. The results of some Monte Carlo simulations which are matched as closely as possible to real situations are presented. They indicate that the very small BLR size estimated by the cross-correlation method is not highly reliable. 15 references.

Constant thermodynamic tension Monte Carlo studies of elastic properties of a two-dimensional system of hard cyclic hexamers

Abstract: A constant thermodynamic tension Monte Carlo method is introduced and applied studies of the elastic properties of a two-dimensional system of hard cyclic hexamers. Elastic compliances and elastic constants are determined at a number of different values of the pressure. The existence of the phase transition between a tilted and a straight phase is confirmed. The results obtained strongly suggest that the Poisson modulus can be negative in the tilted phase.

Solving Quadratic Assignment Problems by ‘Simulated Annealing’

Abstract: Recently, an interesting analogy between problems in combinatorial optimization and statistical mechanics has been developed and has proven useful in solving certain traditional optimization problems such as computer design, partitioning, component placement, wiring, and traveling salesman problems. The analogy has resulted in a methodology, termed “simulated annealing,” which, in the process of iterating to an optimum, uses Monte Carlo sampling to occasionally accept solutions to discrete optimization problems which increase rather than decrease the objective function value. This process is counter to the normal ‘steepest-descent’ algorithmic approach. However, it is argued in the analogy that by taking such controlled uphill steps, the optimizing algorithm need not get “stuck” on inferior solutions. This paper presents an application of the simulated annealing method to solve the quadratic assignment problem (QAP). Performance is tested on a set of “standard” problems, as well as some newly gen...

Overrelaxation and Monte Carlo simulation

Abstract: I study a simple variation of the algorithm of Metropolis et al. for simulating statistical systems. The trial changes in any given variable are taken from a region of phase space far from the old value but involving only small changes in energy. This results in correlation times which are short compared to the usual applications of the algorithm of Metropolis et al. Tests with SU(2) and SU(3) lattice gauge theories indicate substantial possible savings in computation time relative to standard approaches.

Variational Monte-Carlo Studies of Hubbard Model. I

Abstract: As a continuation of a previous paper [J. Phys. Soc. Jpn. 56 (1987) 1490], the variational Monte-Carlo method is extended to include the antiferromagnetic long-range order. The theory is based on the Gutzwiller-type correlation factor and its effect is exactly taken into account by the Monte-Carlo procedure. An application is made to the half-filled-band case of one-dimensional lattice (50 sites), two-dimensional square lattice (up to 20×20 sites) and three-dimensional simple cubic lattice (6×6×6 sites). The result is qualitatively different from previous studies relying on the random-phase-type “Gutzwiller approximation.” The variational energy for two and three dimensions is favorably compared with Hirsch's quantum Monte-Carlo data.

Predicting antibody hypervariable loop conformation. I. Ensembles of random conformations for ringlike structures.

Abstract: One approach to finding the conformation of minimum energy for a complicated molecule is to perform energy minimization, perhaps coupled to more exhaustive search procedures such as dynamics or Monte Carlo sampling, from many starting conformation. Where there are geometric constraints on the conformations, as in a ring molecule, or a variable loop starting and ending in known constant regions of one of a series of homologous proteins, rapidly generating many such starting conformations, all satisfying the constraints, has been a problem in the past. We have devised an algorithm, which we call random tweak, which performs this task in the context of a torsional description of a molecule, and have used it to model the backbones of the six CDRs (complementarity determining regions) of the immunoglobulin MCPC603. These range in size from 5 to 19 residues, and have from 8 to 36 variable dihedral angles. Ensembles of 100 properly closed backbone structures for each CDR were generated under several conditions of van der Waals screening internally and against the rest of the molecule, and ensembles of 1000 were generated for selected CDRs. These structure “libraries” reveal how the geometry at the base of a CDR and the topography of the surrounding protein surface restrict the region of space that a given CDR can occupy. In accord with simple notions of chain molecule statistics, the more highly extended a CDR at its base, the more similar the possible structures and the fewer that are necessary to span the conformational space. Energy minimization and molecular dynamics studies (reported elsewhere) using these libraries to furnish starting conformations show that, as the number of residues in a CDR goes from five to nine, the number of randomly generated structures necessary to ensure that low-lying energetic minima, such as the native conformation, will be found several times goes from a few tens to a few hundred. Some of the spatial features of an ensemble of random conformations are implicit in the histogram of the rms atomic displacements calculated for all the pairs in the ensemble. The random tweak method is carried out by setting each dihederal angle on the main chain of the variable fragment to a random value, then using an iterated linearized Lagrange multiplier technique to enforce the geometric constraints with the minimal conformational perturbation. The time required for the algorithm is linear in fragment length, and the resulting ability of the method to handle large loops makes it especially applicable to the modeling of homologous proteins. In most cases, hundreds of acceptable structures could be generated in a few hours on a VAX 11/780. Where van der Waals screening against fixed atoms need not be performed, as for isolated ring molecules, generation times go down by an order of magnitude or more.

Fluids in micropores. I. Structure of a simple classical fluid in a slit-pore

Abstract: Equilbrium properties of a rare‐gas fluid contained between two parallel fcc(100) planes of rigidly fixed rare‐gas atoms were computed by means of the grand‐canonical ensemble Monte Carlo method. The singlet distribution function ρ(1), and the pair‐correlation function g(2) in planes parallel to the solid layers, indicate that the structure of the pore fluid depends strongly on the distance h between the solid layers. As the separation increases from less than two atomic diameters, successive layers of fluid appear. The transitions between one and two layers and three and four layers are especially abrupt and are accompanied by changes in the character of g(2) from dense fluid‐like to solid‐like. Long‐range, in‐plane order in the fluid layers diminishes with increasing h, but is still evident in the contact layer (i.e., that nearest the solid layer) at h=16.5 atomic diameters, the largest separation considered. The structure of the contact layer reflects the solid‐layer structure and differs significantly...

Likelilood ratio gradient estimation: an overview

Abstract: The likelihood ratio method for gradient estimation is briefly surveyed. Two applications settings are described, namely Monte Carlo optimization and statistical analysis of complex stochastic systems. Steady-state gradient estimation is emphasized, and both regenerative and non-regenerative approaches are given. The paper also indicates how these methods apply to general discrete-event simulations; the idea is to view such systems as general state space Markov chains.

Computer simulations of sputtering

Abstract: Monte Carlo simulations have become a useful tool for studying ion radiation effects at or near surfaces or interfaces, such as sputtering, reflection, mixing, etc. The principal advantage of Monte Carlo calculations is that any physical process can be included directly. Also multielement and multilayer targets, even complex geometries, can be treated exactly in order to simulate realistic cases. The present paper will concentrate mainly on sputtering calculations with the Monte Carlo code TRIM, which treats ion and recoil transport in amorphous matter. It is based — as are analytic theories and most other Monte Carlo codes — on binary collisions with target atoms initially at rest. Over the past few years, the basic physical input has been greatly improved. Both the interatomic potentials and the electronic stopping powers proved to be of crucial importance even for the lowest energies occurring in recoil cascades. With the Kr-C or universal potential and the recent ZBL electronic stopping, which includes the Z oscillations, and a planar surface binding energy set equal to the heat of sublimation, realistic sputtering predictions could be obtained for most metals — without the use of any adjustable parameters.

Monte Carlo evaluation of kerma at a point for photon transport problems.

Abstract: Estimation of collision kerma at a geometric point arising from scattered photons is a potentially important application of Monte Carlo simulation, especially in the presence of steep flux gradients. We examine the usual method of extracting point-kerma estimates from randomly generated photon trajectories which consists of tallying the energy lost by photon collisions occurring in the vicinity of the point of interest. Several other methods derived from the equivalence of track length per unit volume and flux are evaluated as to accuracy and efficiency. Finally, a next-flight estimator is discussed in which the expected contribution of each simulated photon collision to kerma at the point of interest is calculated regardless of proximity of the collision to the point. All of these techniques are shown to involve a trade-off between statistical precision and spatial resolution: increasing the number of contributing collisions requires averaging kerma over a larger volume. Based upon both analytic models and realistic Monte Carlo simulations, use of next-flight and track-length estimators is shown to improve simulation efficiencies by factors of 2 to 20 compared to analog scoring. Practical guidelines as to choice of estimator and successful implementation are presented.

Critical properties of the Flory–Huggins lattice model of polymer mixtures

Abstract: Monte Carlo simulations are presented for a model of a symmetrical polymer mixture on the simple cubic lattice, modeling both polymers A, B by self‐avoiding walks of NA=NB=N steps. If a pair of nearest‐neighbor sites is taken by monomers of the same species, an energy e is won. In the Monte Carlo algorithm local motions of the chains are considered (allowing for 20% vacancies to ensure enough chain mobility) as well as transformations of A chains into B chains and vice versa, since the simulation applies the grand‐canonical ensemble where the chemical potential difference rather than the volume fraction is fixed. The phase diagram, the excess specific heat, and the structure factor in the long‐wavelength limit are obtained for N=4, 8, 16, and 32 using finite L×L×L lattices with L ranging from 8 to 20. Analyzing these results with finite size scaling techniques, both critical exponents and critical amplitudes are estimated. Although the exponents are consistent with those of the three‐dimensional Ising mod...

Computer-simulation of channeling in single-crystals

Abstract: A Monte Carlo program for the calculation of channeling phenomena is described. The program combines the binary collision model and the multistring approximation. The energy loss due to electronic excitation is taken into account, with the use of the model of Dettmann and Robinson for the inner-shell electrons and the theory of Pines for valence electrons. The output of the Monte Carlo program may be used for the determination of the impurity sites in single crystals, via a set of auxiliary programs, that enable that calculation of the impurity yield and the analysis of experimental channeling dips. As an application, the site determination of iodine in silicon is described. Another application is the simulation of RBS spectra of planar channeling ions. Simulated and experimental spectra are compared for 1 MeV ions in the (110), (111) and (100) planes of silicon. A reasonable agreement was found. The possible causes of the remaining deviations are discussed.

Sampling Turbulence in the Stratified Ocean: Statistical Consequences of Strong Intermittency

Abstract: Turbulence and turbulent mixing in the ocean are strongly intermittent in amplitude, space and time. The degree of intermittency is measured by the “intermittency factor” σ2, defined as either σ2lnϵ, the variance of the logarithm of the viscous dissipation rate ϵ, or σ2lnχ, the variance of the logarithm of the temperature dissipation rate χ. Available data suggest that the cumulative distribution functions of ϵ and χ in stratified layers are approximately lognormal with large σ2 values in the range 3–7. Departures from lognormality are remarkably similar to those for Monte Carlo generated lognormal distributions contaminated with simulated noise and undersampling effects. Confidence limits for the maximum likelihood estimator of the mean of a lognormal random variable are determined by Monte Carlo techniques and by theoretical modeling. They show that such large σ2 values cause large uncertainty in estimates of the mean unless the number of data samples is extremely large. To obtain estimates of ...

Overrelaxed heat-bath and Metropolis algorithms for accelerating pure gauge Monte Carlo calculations.

Abstract: The lattice-gauge-theory generalization of Adler's over-relaxed heat-bath algorithm and an over-relaxed Metropolis update are shown to accelerate the decorrelation of physical observables. The heat-bath's microcanonical limit is especially attractive. Numerical tests for pure gauge SU(3) (${4}^{4}$ lattice, \ensuremath{\beta}=5.6) show, for example, that over-relaxation reduces the Polyakov-loop magnitude autocorrelation time for the Cabibbo-Marinari algorithm from 28 sweeps to 9, increasing computational efficiency threefold. These approaches are applicable to a large class of physical systems.