Simple Monte Carlo Method Research Articles

Multiple Testing of General Contrasts Using Logical Constraints and Correlations

Use of logical constraints among hypotheses and correlations among test statistics can greatly improve the power of step-down tests. An algorithm for uncovering these logically constrained subsets in a given dataset is described. The multiple testing results are summarized using adjusted p values, which incorporate the relevant dependence structures and logical constraints. These adjusted p values are computed consistently and efficiently using a generalized least squares hybrid of simple and control-variate Monte Carlo methods, and the results are compared to alternative stepwise testing procedures.

Relation of Gutzwiller Variational State with Matrix Model

It is shown that mean values of electron operators in variational Gutzwiller state are equal to mean values of corresponding classical quantities calculated by means of a hermitian matrix model. In cases with small number of electrons in the system this property enables exact calculation of the mean values. In case of large number of electrons a simple and effective Monte Carlo method is formulated (within matrix model).

Estimating the age of the common ancestor of a sample of DNA sequences.

We present a simple Monte Carlo method for estimating the age of the most recent common ancestor (MRCA) of a sample of DNA sequences. We show that Templeton's (1993) estimator of the age of the MRCA based on the maximum number of nucleotide differences between two sequences in a sample is inaccurate, and we demonstrate the new method by reanalyzing a sample of DNA sequences from human Y chromosomes and a sample of human Alu sequences.

A Chromatin Switch

Cellular and molecualr biological approaches that study eukaryotic gene regulation have led to separate models which describe structure and mechanism with differing precision. Using principles of combinatorial and cooperative interactions inherent in both models, we have extended concepts derived from a “genetic switch” for prokaryotes to a chromatin switch for eukaryotes composed of DNA, transactivators, nucleosomes and the nuclear matrix. We present a consensus model for gene regulation that uses a simple Monte Carlo method for simulating condensation and extension of chromatin. Such a chromatin switch can be modulated by known biochemical and molecular modifications, and the transactivator binding sites or enhancers within DNA domains can be organized into a hierarchy to control cell cycling and differentiation.

Simultaneous wavelet estimation and deconvolution of reflection seismic signals

The problem of simultaneous wavelet estimation and deconvolution is investigated with a Bayesian approach under the assumption that the reflectivity obeys a Bernoulli-Gaussian distribution. Unknown quantities, including the seismic wavelet, the reflection sequence, and the statistical parameters of reflection sequence and noise are all treated as realizations of random variables endowed with suitable prior distributions. Instead of deterministic procedures that can be quite computationally burdensome, a simple Monte Carlo method, called Gibbs sampler, is employed to produce random samples iteratively from the joint posterior distribution of the unknowns. Modifications are made in the Gibbs sampler to overcome the ambiguity problems inherent in seismic deconvolution. Simple averages of the random samples are used to approximate the minimum mean-squared error (MMSE) estimates of the unknowns. Numerical examples are given to demonstrate the performance of the method.

Shape of Film Grown on Microsize Trenches and Holes by Chemical Vapor Deposition: 3-Dimensional Monte Carlo Simulation

A semi-microscale 3-dimensional film growth simulation code using a simple Monte Carlo method was developed. This code predicts the step coverage on a trench and a hole of arbitrary shapes and requires much smaller computer memory size and less calculation time than the Direct Simulation Monte Carlo (DSMC) method. The simulation code was evaluated by comparing the results with experimental results of a zirconia ( ZrO2) film grown on a hole. The experiments and/or 3-dimensional simulations indicated that the film grown on the side and bottom walls of a hole is thinner than that of a 2-dimensional trench, and complete occlusion of a hole is more difficult compared with a trench with opening width equal to the hole diameter. The surface reaction rate constant is the most important factor in the occlusion process. When the reaction rate constant is small, the hole is occluded with a thin film. However, when the reaction rate constant is large, there remain a void inside and a small unfilled pinhole through the thick film grown on the top.

Applicability of one-dimensional diffusion model for step coverage analysis—Comparison with a simple Monte Carlo method

Applicability of a one-dimensional reaction-diffusion model (ORDM) as a tool for analyzing the gas–solid phase transition rate process during chemical vapor deposition is tested by comparing its results with those predicted by a simple Monte Carlo method. The results reveal that the ORDM is applicable when the conditions of Knudsen number (Kn)<0.03, aspect ratio (As)≳1.0, and reactive sticking coefficient (η)<0.1 are fulfilled. At large Kn and large η, the ORDM gives quite an erroneous result. The ORDM is also applicable to predict the grown film shape when η<0.1, Kn ≫1.0, and As≳2.0.

Algorytmy stochastyczne w optymalizacji dyskretnej przy zaburzonych wartościach funkcji

The paper deals with stochastic methods for searching approximately global minimum of function defined on discrete set. A measure of quality of solution is defined to compare different algorithms. Simple Monte Carlo method is analysed as main algorithm for which formulas dealing with the measure of quality are derived(two cases: exact values and noisy values of function). This Monte Carlo method is used as a base in simulation experiments for comparing other stochastic algorithms. The second part of the paper analyses asymptotic properties of the generalised simulated annealing algorithms. Theory of Markov chains is used in modelling this class of algorithms. Theorems about convergence of the records of algorithms to set of optima with probability one are presented in the case of function having random noisy values. The paper also reviews known results in the field of simulated annealing type algorithms for function with randomly perturbated values.

A Monte Carlo Simulation for a Uniform Ladder of Energy Levels: Statistical Thermodynamic Properties

Monte Carlo (MC) computer programs are useful for simulating processes that occur randomly in nature, handling problems in which fluctuations are important, or investigating problems that are mathematically intractable. A simple MC method is employed here that can be carried out by undergraduate physical chemistry students. The MC method illustrates several important concepts of statistical mechanics.

ON RELIABILITY-BASED STRUCTURAL OPTIMISATION USING STOCHASTIC QUASIGRADIENT METHODS/ZUR ZUVERLÄSSIGKEITSTHEORETISCH GESTÜTZTEN TRAGWERKS-OPTIMIERUNG MIT VERFAHREN DER STOCHASTISCHEN QUASIGRA-DIENTEN

Methodical aspects of the reliability-based structural optimisation using stochastic quasigradient methods are considered. For an example of the simply supported reinforced concrete beam, the employment of the Lagrange multiplier method that belongs to the class of stochastic quasigradient methods is demonstrated. The classical optimum design goal to minimise structural cost or weight under the constraint on the structural failure probability is taken for consideration. Optimisation problems solved with the Lagrangemultiplier method are formulated in form of general stochastic programming problem. The mathematical expectation of the concrete volume reduced with respect to the in-place cost of the beam materials is taken as the objective function. Constraint function is the limitation placed on the beam failure probability. The beam is considered as a series structural system. Values of the prescribed allowable failure probability belongs to the interval in which the estimation of the failure probabilities by the simple Monte-Carlomethod is possible with an acceptable confidence. The time-independent case as well as the time-dependent one is considered in the optimisation problems. The generalisation on the time-dependent case is undertaken through the introduction into the constraint function of the quasi-linear distribution law of the random variables. In the time-dependent case, the objective function is associated with beginning and the constraint function with end of the service period. An expression of the stochastic gradient based on the differentiation under the integral sign is used for calculations with the Lagrange multiplier method. The stochastic gradient used is computationally more effective in comparison with stochastic finite-difference formulae usual in stochastic quasigradient methods because it requires only one computation of the structure in search iteration of the optimisation process. Three rules based on statistical argumentation are used for the stopping of the seat according to the procedure of the Lagrange multiplier method. The optimising of the beam shows that the Lagrange multiplier method is applicable for the optimal design of structures in that cases when the structural reliability can be estimated by means of the simple Monte-Carlo method. Additional research is needed for integration in the Lagrange multiplier method of statistical simulation techniques for the estimation of small structural failure probabilities.

Hydroconsolidation and subsidence of loess: Studies from China, Russia, North America and Europe: In memory of Jan Sajgalik

Various approaches to the widespread problem of the hydroconsolidation and subsidence of loess have been suggested. These include considerations of rheology, thermodynamics, phase movements, particle packing, interparticle bonding, pore structure and distribution, catastrophe theory, topology, and simple structural frameworks. Chinese, North American and most European investigators tend to concentrate on mechanisms of loess collapse. The Russian literature, however, retains an extra dimension. Two approaches, the ‘syngenetic’ and the ‘epigenetic’ approach, to the formation of subsiding loess have been defined in the literature. Most investigators follow a syngenetic approach which appears to be a consequence of the aeolian idea of loess deposition. Some Russian writers, in contrast, promote an epigenetic approach in which collapsibility can develop in an originally noncollapsible material, which can then suffer from hydroconsolidation and subsidence. The basis of the phenomenon is a change in the packing structure of the major loess particles, and this can be modelled using simple Monte Carlo methods to develop appropriate structures. This paper aims to review the work done on this important subject. Serious investigation of hydroconsolidation and subsidence of loess began in the early nineteen-forties, fifty years ago, and this has been reported in a piecemeal manner. A detailed, critical review of this diverse work is now overdue and this is presented herein in the light of recent work in the United Kingdom. An attempt is made to describe the process in a phenomenological and a structural sense. Inherent in this, the role of N. Ya Denisov as ‘subsidence pioneer’ is considered.

Causal Stochastic Simulation of Dissolved Oxygen Depletion in Rivers Receiving Combined Sewer Overflows

A method for stochastic analysis of the effect of combined sewer overflows on the dissolved oxygen concentration in receiving rivers is developed. The method is based on repeated operation of the water quality model MOUSE-DOSMO. Each operation of this model covers a series of overflow events calculated by the MOUSE-SAMBA runoff model based on a historical rainfall record. For each event selected input data and model parameters in both models are drawn from fundamental statistical distributions by a simple Monte Carlo method. These data and parameters vary from event to event in each series. Each operation of the MOUSE-DOSMO model results in an extreme event statistics on dissolved oxygen minimum values -while repeated operation of the model yields extreme event percentiles by means of which a probability-based assessment of the model result may be performed.

Relaxation theory for curve-crossing corrections to electronic absorption line shapes in condensed phases

A quantum mechanical relaxation theory is developed to enable approximate computation of electronic absorption line shapes of condensed phase systems where nonadiabatic coupling effects are important. At the simplest level, these computations require a time kernel (termed a memory kernel) which can be obtained from a sequence of wave packet propagations, each carried out on a single Born–Oppenheimer potential surface. Complications associated with the need to evolve wave packets on several nonadiabatically coupled surfaces are thereby avoided. Moreover, for many condensed phase problems the memory kernel can be computed via semiclassical techniques which rely on classical trajectories and simple Monte Carlo methods. The promise of the theory is demonstrated by numerical applications to the spectroscopic spin boson model [R. D. Coalson, J. Chem. Phys. 86, 995 (1987)], a nontrivial multimode model of electronic absorption lineshapes involving two nonadiabatically coupled excited state surfaces. The relevant quantum dynamics for the spectroscopic spin boson model can be computed exactly via path integration techniques. In this way, the accuracy of the proposed relaxation theory can be benchmarked, and the applicability of various semiclassical prescriptions for computing the memory kernel ascertained.

Estimation of the probability distribution of end-group distances of chain-linked electron donor-acceptor molecules and radical ion pairs: a Monte-Carlo approach

Probability distributions of center-to-center and edge-to-edge distances of chain-linked electron donor-acceptor molecules have been calculated applying a simple Monte-Carlo method. The donor is N,N-dimethylaniline, and the acceptor is pyrene. The linkages, in one study, are different flexible polymethylene chains, (CH 2 ) n , with n=5-16. In the systems with n=5-10, the appearance of an odd-even effect in the calculated population of conformations of short distances below 10 A correlates well with the fluorescence lifetimes measured recently

Chern-simons on a lattice

Let u be a continuous SU (2)-valued lattice gauge field on a triangulation Λ of a manifold X of dimension ≥3. Let ξ be the corresponding principal SU (2)-bundle over X. It is shown how to choose a piecewise smooth connection ω in ξ with the following property. The Chern-Simons character χ of the gauge field (ξ,ω) can be computed as the sum of local contributions ^χ(σ), one for each 3-simplex σ of Λ, where ^χ(σ) is the sum of one term that can be exactly calculated from u, and other terms (volumes of elementary regions in the 3-sphere) which can be estimated in terms of u by simple Monte-Carlo methods. For any 3-cycle Z = Σ i a i σ i , the sum Σ i a i ^ χ( σ i ), reduced mod Z, is gauge-invariant.

Center-of-mass Monte Carlo simulation of neutron scattering experiments

A simple Monte Carlo method for simulating neutron-scattering experiments is presented. The method is based on performing isotropic collision kinematics in the center-of-mass coordinate system and the utilization of special particle-fluence estimators. The developed algorithm is verified against the general-purpose Monte Carlo code MORSE-CG for two diverse problems. The use of the method to aid in the development and design of neutron scatterometers for gauging applications is also demonstrated.

Monte Carlo simulation of the hard-sphere fluid with quantum correction and estimate of its free energy

A simple Monte Carlo (MC) method is introduced to treat the quantum-mechanical hard-sphere fluid. It uses the relative two-body Slater sum (instead of the classical Boltzmann factor) as a quantum-mechanical probability in the relative two-body configurational space. The MC calculations have been carried out for the hard-sphere fluid with quantum corrections at various densities and for various thermal wavelengths. The results are consistent with the interpretation that, in the quantum-mechanical picture, hard spheres repel each other at distances slightly greater (by an amount of the order of the thermal wavelength λT) than the contact collision distance of the classical treatment. The radial distribution functions of the classical and quantum-mechanical simulations differ considerably. The relative two-body free energy has been obtained from the relative two-body Slater sum, and the total free energy has been estimated (in preliminary calculations) by using the classical expression but replacing the classical hard-sphere diameter by the effective diameter. A good analytical fit to the MC results has been obtained for the equation of state of the quantum-mechanical hard-sphere fluid; the total free energy has been obtained (in extensive calculations) by integrating over the resulting pressures and densities. The pressure, internal energy, and free energy all increase with an increase in the thermal wavelength, i.e., with increasing quantum effect. From the equation of state, the second virial coefficients at various thermal wavelengths are in good agreement with theoretical values in the literature, and the third and fourth virial coefficients are predicted. Evidence has been found for a momentum condensation; the corresponding relative two-body entropy is negative. The heat capacity due to the relative two-particle kinetic energy has been found to be proportional to T−1/2 (approximately) whereas, classically, it is zero.

Monte carlo method in calculation of the melting temperatures of silicate glasses

Numerical constants for the various oxides for calculating the melting temperature of silicate glasses are re-determined by using a simple Monte Carlo method. These constants are used to calculate the melting temperatures of 60 glass compositions; the deviations of the calculated values from the practical melting temperatures are small compared with those obtained with an earlier method.

CALCULATION OF NONLINEAR CONFIDENCE AND PREDICTION INTERVALS FOR GROUND‐WATER FLOW MODELS1

ABSTRACT: A method is derived to efficiently compute nonlinear confidence and prediction intervals on any function of parameters derived as output from a mathematical model of a physical system. The method is applied to the problem of obtaining confidence and prediction intervals for manually‐calibrated ground‐water flow models. To obtain confidence and prediction intervals resulting from uncertainties in parameters, the calibrated model and information on extreme ranges and ordering of the model parameters within one or more independent groups are required. If random errors in the dependent variable are present in addition to uncertainties in parameters, then calculation of prediction intervals also requires information on the extreme range of error expected. A simple Monte Carlo method is used to compute the quantiles necessary to establish probability levels for the confidence and prediction intervals. Application of the method to a hypothetical example showed that includsion of random errors in the dependent variable in addition to uncertainties in parameters can considerably widen the prediction intervals.

An analysis of five simulation methods for determining the number of replications in a complex Monte Carlo study

In this article we summarize the results of a simulation study that compares and contrasts five different methods for selecting the number of replications necessary to accurately estimate a group of parameters utilizing the simple Monte Carlo method. The five methods are: an extension to several dimensions of Stein's two-stage procedure, an extension of Hall's three-stage procedure, a purely-sequential procedure that continuously monitors the simulation output, a continuously-monitoring procedure with a correction term, and a maximum eigenvalue procedure due to Srivastava. This study varies three design factors: the dimension of the simulated response (two or three), the number of replications necessary to be accurate (small, moderate and large sample sizes), and the kind of accuracy sets (rectangular or spherical). The simulation methods are compared on the basis of their probability of being accurate and on the basis of their efficiency, as measured by the distribution of the number of replications needed to be accurate. Typical distributional and density estimates are presented in both table and graph form.