A hybrid transport-diffusion Monte Carlo method for frequency-dependent radiative-transfer simulations

Discrete Diffusion Monte Carlo (DDMC) is a technique for increasing the efficiency of Implicit Monte Carlo radiative-transfer simulations in optically thick media. In DDMC, particles take discrete steps between spatial cells according to a discretized diffusion equation. Each discrete step replaces many smaller Monte Carlo steps, thus improving the efficiency of the simulation. In this paper, we present an extension of DDMC for frequency-dependent radiative transfer. We base our new DDMC method on a frequency-integrated diffusion equation for frequencies below a specified threshold, as optical thickness is typically a decreasing function of frequency. Above this threshold we employ standard Monte Carlo, which results in a hybrid transport-diffusion scheme. With a set of frequency-dependent test problems, we confirm the accuracy and increased efficiency of our new DDMC method.     

Comparison between semiclassical and full quantum transport analysis of THz quantum cascade lasers

We implement and compare two theoretical models for stationary electron transport in quantum cascade lasers and Stark ladders. The first one, the nonequilibrium Green's function method is a very general scheme to include coherent quantum mechanics and incoherent scattering with phonons and device imperfections self-consistently. However, it is numerically very demanding and cannot be used for systematic device parameter scans. For this reason, we also implement the approximate, numerically efficient ensemble Monte Carlo method and assess its applicability on the above mentioned transport problems. We identify a transport regime in which results of both methods quantitatively agree. In this regime, the ensemble Monte Carlo method is well suited to propose design improvements.     

An Analysis of Polynomial Chaos Approximations for Modeling Single-Fluid-Phase Flow in Porous Medium Systems

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.     

Multilevel coarse graining and nano-pattern discovery in many particle stochastic systems

In this work we propose a hierarchy of Markov chain Monte Carlo methods for sampling equilibrium properties of stochastic lattice systems with competing short and long range interactions. Each Monte Carlo step is composed by two or more sub-steps efficiently coupling coarse and finer state spaces. The method can be designed to sample the exact or controlled-error approximations of the target distribution, providing information on levels of different resolutions, as well as at the microscopic level. In both strategies the method achieves significant reduction of the computational cost compared to conventional Markov chain Monte Carlo methods. Applications in phase transition and pattern formation problems confirm the efficiency of the proposed methods.     

A dynamical mean field theory for the study of surface diffusion constants

We present a combined analytical and numerical approach based on the Mori projection operator formalism and Monte Carlo simulations to study surface diffusion within the lattice-gas model. In the present theory, the average jump rate and the susceptibility factor appearing are evaluated through Monte Carlo simulations, while the memory functions are approximated by the known results for a Langmuir gas model. This leads to a dynamical mean field theory (DMF) for collective diffusion, while approximate correlation effects beyond DMF are included for tracer diffusion. We apply our formalism to three very different strongly interacting systems and compare the results of the new approach with those of usual Monte Carlo simulations. We find that the combined approach works very well for collective diffusion, whereas for tracer diffusion the influence of interactions on the memory effects is more prominent.     

基于SALOME的蒙卡计算模型CAD反转可视化

基于开源SALOME平台,采用以体代面思想和栅元层次多叉树方法,开展蒙卡计算模型CAD反向转换及三维可视化研究。基于本文方法开发了CAD反转可视化程序模块SALOME-MC,模块可实现蒙卡计算模型几何建模、材料建模和三维可视化等功能。选取三种典型反应堆蒙卡计算模型对本文方法和程序进行测试验证,测试结果表明,本文方法和程序可妥善处理蒙卡计算模型的复杂几何体与大规模重复结构,并精准地实现蒙卡计算模型CAD三维反转可视化,证明SALOME-MC的蒙卡计算模型反转能力和可视化效果的正确性与可靠性,提高了蒙卡计算模型几何建模效率和展示度。     

Chemical potential,Helmholtz free energy and entropy of argon with kinetic Monte Carlo simulation

We present a method based on kinetic Monte Carlo (kMC) to determine the chemical potential, Helmholtz free energy and entropy of a fluid within the course of a simulation. The procedure requires no recourse to auxiliary methods to determine the chemical potential, such as the implementation of a Widom scheme in Metropolis Monte Carlo simulations, as it is determined within the course of the simulation. The equation for chemical potential is proved, for the first time in the literature, to have a direct connection with inverse Widom potential theory in using real molecules rather than ghost molecules. We illustrate this new procedure by several examples, including fluid argon and adsorption of argon as a non-uniform fluid on a graphite surface and in slit pores.     

Monte Carlo simulation of quantum statistical lattice models

In this article we review recent developments in computational methods for quantum statistical lattice problems. We begin by giving the necessary mathematical basis, the generalized Trotter formula, and discuss the computational tools, exact summations and Monte Carlo simulation, that will be used to examine explicit examples. To illustrate the general strategy, the method is applied to an analytically solvable, non-trivial, model: the one-dimensional Ising model in a transverse field. Next it is shown how to generalized Trotter formula most naturally leads to different path-integral representations of the partition function by considering one-dimensional fermion lattice models. We show how to analyze the different representations and discuss Monte Carlo simulation results for one-dimensional fermions. Then Monte Carlo work on one- and two-dimensional spin-$\text{1}\text{2}$ models based upon the Trotter formula approach is reviewed and the more dedicated Handscomb Monte Carlo method is discussed. We consider electron-phonon models and discuss Monte Carlo simulation data on the Molecular Crystal Model in one, two and three dimensions and related one-dimensional polaron models. Exact numerical results are presented for free fermions and free bosons in the canonical ensemble. We address the main problem of Monte Carlo simulations of fermions in more than one dimension: the cancellation of large contributions. Free bosons on a lattice are compared with bosons in a box and the effects of finite size on Bose-Einstein condensation are discussed.     

Unitary approximation for radiative high energy scattering processes: Application to Bhabha scattering

We describe how to apply the unitary approximation to high energy scattering processes which involve the radiation of arbitrarily many photons. We compare our results to calculations of inclusive quantities using leading log resummation techniques well known from QCD. In contrast to the latter method the unitary approximation is also appropriate for differential cross sections. The Monte Carlo implementation of the method is discussed and detailed results are given for the case of radiative Bhabha scattering.     

The influence of step-step interactions on step wandering

We derive relationships between the amount of step wandering and the strength of step-step interactions to aid interpretation of scanning tunneling microscopy images of steps on surfaces. We make contact with well-established results for the statistical mechanics of interfacial wandering. In particular, we use the analogy between a step meandering in a potential and a quantum-mechanical particle moving in a one-dimensional potential well. We also set out an approximate procedure for computing the terrace-width distribution for non-interacting steps using free-fermion techniques, and show using Monte Carlo that the resulting temperature-independent distribution is a good aproximation until remarkably high temperatures.     

Simulation of coherent properties of photons in scattering medium for optical coherence tomography

Monte Carlo model of optical coherence tomography is developed for simulation of photon transport in half infinite homogenous media. The procedure is accelerated by scaling the baseline data from standard Monte Carlo calculation in turbid media with arbitrary optical parameters. Gaussian beam is modeled by hyperboloid of one sheet for actual condition to obtain distribution of photons on sample surface. Depth dependence coherent signal and photons distribution are calculated in this way, which is important to reconstruction of optical parameters by inverse Monte Carlo. Numerical results have verified this method in turbid medium of different optical parameters with acceptable relative errors.     

Some Spin Glass Ideas Applied to the Clique Problem

In this paper we introduce a new algorithm to study some NP-complete problems. This algorithm is a Markov Chain Monte Carlo (MCMC) inspired by the cavity method developed in the study of spin glass. We will focus on the maximum clique problem and we will compare this new algorithm with several standard algorithms on some DIMACS benchmark graphs and on random graphs. The performances of the new algorithm are quite surprising. Our effort in this paper is to be clear as well to those readers who are not in the field.     

Classical diffusion in random environments: A new numerical method

We present a numerical method for classical lattice diffusion processes in a random environment. The special merits of the presented procedure in comparison with Monte Carlo methods are in the economy of computer time and storage. As an example for the potential of the method we present results for excitation dynamics in disordered polymer chains.     

Accelerated Monte Carlo for Optimal Estimation of Time Series

Asbstract By casting stochastic optimal estimation of time series in path integral form, one can apply analytical and computational techniques of equilibrium statistical mechanics. In particular, one can use standard or accelerated Monte Carlo methods for smoothing, filtering and/or prediction. Here we demonstrate the applicability and efficiency of generalized (nonlocal) hybrid Monte Carlo and multigrid methods applied to optimal estimation, specifically smoothing. We test these methods on a stochastic diffusion dynamics in a bistable potential. This particular problem has been chosen to illustrate the speedup due to the nonlocal sampling technique, and because there is an available optimal solution which can be used to validate the solution via the hybrid Monte Carlo strategy. In addition to showing that the nonlocal hybrid Monte Carlo is statistically accurate, we demonstrate a significant speedup compared with other strategies, thus making it a practical alternative to smoothing/filtering and data assimilation on problems with state vectors of fairly large dimensions, as well as a large total number of time steps.     

Transition Matrix Monte Carlo Method

We present a formalism of the transition matrix Monte Carlo method. A stochastic matrix in the space of energy can be estimated from Monte Carlo simulation. This matrix is used to compute the density of states, as well as to construct multi-canonical and equal-hit algorithms. We discuss the performance of the methods. The results are compared with single histogram method, multi-canonical method, and other methods. In many aspects, the present method is an improvement over the previous methods.     

Accurate, efficient, and simple forces computed with quantum Monte Carlo methods

Computation of ionic forces using quantum Monte Carlo methods has long been a challenge. We introduce a simple procedure, based on known properties of physical electronic densities, to make the variance of the Hellmann-Feynman estimator finite. We obtain very accurate geometries for the molecules H(2), LiH, CH(4), NH(3), H(2)O, and HF, with a Slater-Jastrow trial wave function. Harmonic frequencies for diatomics are also in good agreement with experiment. An antithetical sampling method is also discussed for additional reduction of the variance.     

Statistical hadronization and hadronic micro-canonical ensemble II

We present a Monte Carlo calculation of the micro-canonical ensemble of the ideal hadron-resonance gas including all known states up to a mass of about 1.8 GeV and full quantum statistics. The micro-canonical average multiplicities of the various hadron species are found to converge to the canonical ones for moderately low values of the total energy, around 8 GeV, thus bearing out previous analyses of hadronic multiplicities in the canonical ensemble. The main numerical computing method is an importance sampling Monte Carlo algorithm using the product of Poisson distributions to generate multi-hadronic channels. It is shown that the use of this multi-Poisson distribution allows for an efficient and fast computation of averages, which can be further improved in the limit of very large clusters. We have also studied the fitness of a previously proposed computing method, based on the Metropolis Monte Carlo algorithm, for event generation in the statistical hadronization model. We find that the use of the multi-Poisson distribution as proposal matrix dramatically improves the computation performance. However, due to the correlation of subsequent samples, this method proves to be generally less robust and effective than the importance sampling method.Received: 9 July 2004, Revised: 21 July 2004, Published online: 9 November 2004     

Advanced multicanonical Monte Carlo methods for efficient simulations of nucleation processes of polymers

The investigation of freezing transitions of single polymers is computationally demanding, since surface effects dominate the nucleation process. In recent studies we have systematically shown that the freezing properties of flexible, elastic polymers depend on the precise chain length. Performing multicanonical Monte Carlo simulations, we faced several computational challenges in connection with liquid–solid and solid–solid transitions. For this reason, we developed novel methods and update strategies to overcome the arising problems. We introduce novel Monte Carlo moves and two extensions to the multicanonical method.     

Monte Carlo-based tail exponent estimator

In this paper we propose a new approach to estimation of the tail exponent in financial stock markets. We begin the study with the finite sample behavior of the Hill estimator under α-stable distributions. Using large Monte Carlo simulations, we show that the Hill estimator overestimates the true tail exponent and can hardly be used on samples with small length. Utilizing our results, we introduce a Monte Carlo-based method of estimation for the tail exponent. Our proposed method is not sensitive to the choice of tail size and works well also on small data samples. The new estimator also gives unbiased results with symmetrical confidence intervals. Finally, we demonstrate the power of our estimator on the international world stock market indices. On the two separate periods of 2002-2005 and 2006-2009, we estimate the tail exponent.