Conventional Monte Carlo Method Research Articles

Bayesian assessment of uncertainty in aerosol size distributions and index of refraction retrieved from multiwavelength lidar measurements

We investigate the assessment of uncertainty in the inference of aerosol size distributions from backscatter and extinction measurements that can be obtained from a modern elastic/Raman lidar system with a Nd:YAG laser transmitter. To calculate the uncertainty, an analytic formula for the correlated probability density function (PDF) describing the error for an optical coefficient ratio is derived based on a normally distributed fractional error in the optical coefficients. Assuming a monomodal lognormal particle size distribution of spherical, homogeneous particles with a known index of refraction, we compare the assessment of uncertainty using a more conventional forward Monte Carlo method with that obtained from a Bayesian posterior PDF assuming a uniform prior PDF and show that substantial differences between the two methods exist. In addition, we use the posterior PDF formalism, which was extended to include an unknown refractive index, to find credible sets for a variety of optical measurement scenarios. We find the uncertainty is greatly reduced with the addition of suitable extinction measurements in contrast to the inclusion of extra backscatter coefficients, which we show to have a minimal effect and strengthens similar observations based on numerical regularization methods.

Assessment of high dimensional model representation techniques for reliability analysis

This paper presents an assessment of efficient response surface techniques based on the High Dimensional Model Representation (HDMR) and the Factorized High Dimensional Model Representation (FHDMR). The HDMR is a general set of quantitative model assessment and analysis tools for capturing the high-dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher order variable correlations are weak and if the response function is dominantly of an additive nature, allowing the physical model to be captured by the first few lower order terms. But, if the multiplicative nature of the response function is dominant, then all the right hand side components of the HDMR must be used to be able to obtain the best result. However, if the HDMR requires all components, which means 2 N of them, to get a desired accuracy, making the method very expensive in practice, then the FHDMR can be used. The component functions of the FHDMR are determined by using the component functions of the HDMR. This paper presents the formulation of the FHDMR based response surface approximation of a limit state/performance function which is dominantly multiplicative in nature. It is a given that conventional methods for reliability analysis are computationally very demanding, when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently HDMR/FHDMR based response surface techniques can capture complex model output uncertainty. As a part of this effort, the efficacy of the HDMR, which is recently applied to reliability analysis, is also demonstrated. The response surface is constructed using the moving least squares interpolation formula by including constant, first-order, and second-order terms of the HDMR and the FHDMR. Once the response surface form is defined, the failure probability can be obtained by statistical simulation. Results of seven numerical examples involving structural/solid-mechanics/geo-technical engineering problems indicate that the failure probability obtained using the FHDMR based response surface method for a limit state/performance function that is dominantly multiplicative in nature, provides a significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations.

A direct solution method for carrier transport using Markov process representation

A direct solution method for semi-classical carrier transport problems by representing the transport process as a Markov process (also known as Continuous-Time Markov chain) is introduced to evaluate the carrier mobility in semiconductors. Comparison of the results from our direct method to conventional Monte Carlo method shows that this method outperforms the Monte Carlo method in both computational efficiency and accuracy.

Detection of trans-Planckian effects in the cosmic microwave background

Quantum gravity effects are expected to modify the primordial density fluctuations produced during inflation and leave their imprint on the cosmic microwave background observed today. We present a new analysis discussing whether these effects are detectable, considering both currently available data and simulated results from an optimal CMB experiment. We find that the WMAP (Wilkinson Microwave Anisotropy Probe) data show no evidence for the particular signature considered in this work but give an upper bound on the parameters of the model. However, a hypothetical experiment shows that with proper data, the trans-Planckian effects should be detectable through alternate sampling methods. This fuzzy conclusion is a result of the nature of the oscillations, since they give rise to a likelihood hypersurface riddled with local maxima. A simple Bayesian analysis shows no significant evidence for the simulated data to prefer a trans-Planckian model. Conventional Markov chain Monte Carlo (MCMC) methods are not suitable for exploring this complicated landscape, but alternative methods are required to solve the problem. This, however, requires extremely high-precision data.

On population-based simulation for static inference

In this paper we present a review of population-based simulation for static inference problems. Such methods can be described as generating a collection of random variables {X n } n=1,?,N in parallel in order to simulate from some target density ? (or potentially sequence of target densities). Population-based simulation is important as many challenging sampling problems in applied statistics cannot be dealt with successfully by conventional Markov chain Monte Carlo (MCMC) methods. We summarize population-based MCMC (Geyer, Computing Science and Statistics: The 23rd Symposium on the Interface, pp. 156---163, 1991; Liang and Wong, J. Am. Stat. Assoc. 96, 653---666, 2001) and sequential Monte Carlo samplers (SMC) (Del Moral, Doucet and Jasra, J. Roy. Stat. Soc. Ser. B 68, 411---436, 2006a), providing a comparison of the approaches. We give numerical examples from Bayesian mixture modelling (Richardson and Green, J. Roy. Stat. Soc. Ser. B 59, 731---792, 1997).

Analysis of time‐dependent cylindrical problems using Monte Carlo

AbstractMonte Carlo method is well known for solving static problems such as Laplace's or Poisson's equation. In this article, we extend the applicability of the conventional Monte Carlo method to solve time‐dependent cylindrical problems. We present results in one‐dimensional and two‐dimensional regions that agree with the exact solutions. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2571–2573, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22790

TU‐B‐BRB‐01: Monte Carlo — II: Clinical Impact

This presentation is aimed at facilitating participants with the clinical application of Monte Carlo dose calculation algorithms in radiotherapy treatment planning and dosimetry verification. Following a brief introduction to the clinical implementation and commissioning of the Monte Carlo dose calculation software detailed discussions will be given on the actual and potential clinical impact of Monte Carlo dose calculation algorithms on conventional electron beam therapy and advanced mixed beam treatments.Patient treatment plans generated using conventional dose calculation algorithms and Monte Carlo methods will be compared with the causes of the dose discrepancies discussed. Further discussions will be conducted on the use of Monte Carlo dose calculation as a radiotherapy treatment QA tool to validate individual patient plans and as an investigation tool to improve target dose conformity and normal tissue sparing using novel treatment techniques.Educational Objectives:1. To introduce important aspects of clinical implementation and commissioning of Monte Carlo dose calculation algorithms for radiotherapy treatment planning.2. To describe the role of Monte Carlo dose calculation in electron therapy and advanced mixed beam treatment planning.3. To describe the applications of Monte Carlo dose calculations in treatment planning and beam delivery QA for advanced radiotherapy treatments.

An efficient sampling method for stochastic inverse problems

A general framework is developed to treat inverse problems with parameters that are random fields. It involves a sampling method that exploits the sensitivity derivatives of the control variable with respect to the random parameters. As the sensitivity derivatives are computed only at the mean values of the relevant parameters, the related extra cost of the present method is a fraction of the total cost of the Monte Carlo method. The effectiveness of the method is demonstrated on an example problem governed by the Burgers equation with random viscosity. It is specifically shown that this method is two orders of magnitude more efficient compared to the conventional Monte Carlo method. In other words, for a given number of samples, the present method yields two orders of magnitude higher accuracy than its conventional counterpart.

A new fast velocity-diffusion modelling for impurity transport in integrated edge plasma simulation

A fast velocity-diffusion model, which applies to an impurity Monte–Carlo (MC) code, such as IMPMC, is newly developed. The algorithm significantly reduces the computational time, especially in detached plasma, its speed attains more than 100 times faster than the conventional MC method. The performance of IMPMC is further optimized on the massive parallel computer. Thereby the IMPMC is combined with a so-called divertor code, and a self-consistent modelling of divertor plasma and impurity transport is successfully achieved. The integrated code, SONIC (SOLDOR/NEUT2D/IMPMC) is characterized by treatment for kinetic effects and complicated dissociation process of methane. As initial simulations with SONIC, the dynamic evolution of X-point MARFE in JT-60U is investigated. Neutral carbons chemically sputtered from the dome in the private region can deeply penetrate into the main plasma and are subsequently ionized/recombined near the X-point. The resultant carbon ions cause large radiation during MARFE phase.

Revaluation of the first-order upwind difference scheme to solve coarse-grained master equations

This study addresses the initial-boundary value problem of coarse-grained probability measure on the state space in which a differentiable vector field v is given and, as a consequence, the differenced continuity equation using the first-order upwind difference scheme (UDS) based on the finite volume method appears as the physical substance on the coarse-grained dynamics. At first, the UDS is theoretically shown to be equivalent to a class of coarse-grained master equations (CGME), brought by a principle that we cannot distinguish state points in the same partition with each other. The principle is based on the formulation of non-equilibrium statistical mechanics to resolve the macroscopic irreversibility. Moreover the entropy production evaluated by the UDS is also shown to be in accord with the average volume contraction rate in the steady state. This is essential for the non-equilibrium statistical dynamics and was numerically confirmed. Under the principle of coarse graining the UDS is very superior to the conventional Monte-Carlo method in computer time and storage and is very useful to solve the CGME.

Efficient molecular simulations for environmentally benign processes

An efficient sampling technique namely the Hammersley sequence sampling (HSS) technique based on quasi-random numbers is applied to molecular simulations to increase computational efficiency and reduce the error in property calculations for efficient and environmentally friendly product and process design. Due to its k-dimensional uniformity properties, this sampling technique reduces the computational intensity and converges faster as compared to conventional Monte Carlo (MC) method based on pseudo-random numbers. Two case studies are presented in order to show the effect of the sampling technique on molecular simulations. The first case study involves the computation of equation of state (EOS) for a Lennard Jones fluid and the second case study concerns the prediction of Gibbs free energies of transfer for the calculation of octanol–water partition coefficients. The results show that the HSS technique provides about 3-fold gain in efficiency and reduces the number of cycles required to reach equilibration. Furthermore, the errors in property calculations are reduced for both of the case studies.

A Table-Based Random Sampling Simulation for Bioluminescence Tomography

As a popular simulation of photon propagation in turbid media, the main problem of Monte Carlo (MC) method is its cumbersome computation. In this work a table-based random sampling simulation (TBRS) is proposed. The key idea of TBRS is to simplify multisteps of scattering to a single-step process, through randomly table querying, thus greatly reducing the computing complexity of the conventional MC algorithm and expediting the computation. The TBRS simulation is a fast algorithm of the conventional MC simulation of photon propagation. It retained the merits of flexibility and accuracy of conventional MC method and adapted well to complex geometric media and various source shapes. Both MC simulations were conducted in a homogeneous medium in our work. Also, we present a reconstructing approach to estimate the position of the fluorescent source based on the trial-and-error theory as a validation of the TBRS algorithm. Good agreement is found between the conventional MC simulation and the TBRS simulation.

Fast LROC analysis of Bayesian reconstructed emission tomographic images using model observers

Lesion detection and localization is an important task in emission computed tomography. Detection and localization performance with signal location uncertainty may be summarized by a scalar figure of merit, the area under the localization receiver operating characteristic (LROC) curve, ALROC. We consider model observers to compute ALROC for two-dimensional maximum a posteriori (MAP) reconstructions. Model observers may be used to rapidly prototype studies that use human observers. We address the case background-known-exactly (BKE) and signal known except for location. Our ALROC calculation makes use of theoretical expressions for the mean and covariance of the reconstruction and, unlike conventional methods that also use model observers, does not require computation of a large number of sample reconstructions. We validate the results of the procedure by comparison to ALROC obtained using a gold-standard Monte Carlo method employing a large set of reconstructed noise samples. Under reasonable simulation conditions, our theoretical calculation is about one to two orders of magnitude faster than the conventional Monte Carlo method.

Estimation of the four-wave mixing noise probability-density function by the multicanonical Monte Carlo method

The performance of high-powered wavelength-division multiplexed (WDM) optical networks can be severely degraded by four-wave-mixing- (FWM-) induced distortion. The multicanonical Monte Carlo method (MCMC) is used to calculate the probability-density function (PDF) of the decision variable of a receiver, limited by FWM noise. Compared with the conventional Monte Carlo method previously used to estimate this PDF, the MCMC method is much faster and can accurately estimate smaller error probabilities. The method takes into account the correlation between the components of the FWM noise, unlike the Gaussian model, which is shown not to provide accurate results.

A comparative assessment of efficient uncertainty analysis techniques for environmental fate and transport models: application to the FACT model

This work presents a comparative assessment of efficient uncertainty modeling techniques, including Stochastic Response Surface Method (SRSM) and High Dimensional Model Representation (HDMR). This assessment considers improvement achieved with respect to conventional techniques of modeling uncertainty (Monte Carlo). Given that traditional methods for characterizing uncertainty are very computationally demanding, when they are applied in conjunction with complex environmental fate and transport models, this study aims to assess how accurately these efficient (and hence viable) techniques for uncertainty propagation can capture complex model output uncertainty. As a part of this effort, the efficacy of HDMR, which has primarily been used in the past as a model reduction tool, is also demonstrated for uncertainty analysis. The application chosen to highlight the accuracy of these new techniques is the steady state analysis of the groundwater flow in the Savannah River Site General Separations Area (GSA) using the subsurface Flow And Contaminant Transport (FACT) code. Uncertain inputs included three-dimensional hydraulic conductivity fields, and a two-dimensional recharge rate field. The output variables under consideration were the simulated stream baseflows and hydraulic head values. Results show that the uncertainty analysis outcomes obtained using SRSM and HDMR are practically indistinguishable from those obtained using the conventional Monte Carlo method, while requiring orders of magnitude fewer model simulations.

Reverse Monte Carlo Method for Transient Radiative Transfer in Participating Media

Abstract The Monte Carlo (MC) method has been widely used to solve radiative transfer problems due to its flexibility and simplicity in simulating the energy transport process in arbitrary geometries with complex boundary conditions. However, the major drawback of the conventional (or forward) Monte Carlo method is the long computational time for converged solution. Reverse or backward Monte Carlo (RMC) is considered as an alternative approach when solutions are only needed at certain locations and time. The reverse algorithm is similar to the conventional method, except that the energy bundle (photon ensemble) is tracked in a time-reversal manner. Its migration is recorded from the detector into the participating medium, rather than from the source to the detector as in the conventional MC. There is no need to keep track of the bundles that do not reach a particular detector. Thus, RMC method takes up much less computation time than the conventional MC method. On the other hand, RMC will generate less information about the transport process as only the information at the specified locations, e.g., detectors, is obtained. In the situation where detailed information of radiative transport across the media is needed the RMC may not be appropriate. RMC algorithm is most suitable for diagnostic applications where inverse analysis is required, e.g., optical imaging and remote sensing. In this study, the development of a reverse Monte Carlo method for transient radiative transfer is presented. The results of non-emitting, absorbing, and anisotropically scattering media subjected to an ultra short light pulse irradiation are compared with the forward Monte Carlo and discrete ordinates methods results.

DENSITY OF STATES METHOD AT FINITE ISOSPIN DENSITY

The density of states method is applied for lattice QCD at a finite isospin density. The advantage of this method is that one can easily obtain results for various values of parameters (quark mass, coupling constant and the number of flavors). We compare results for the chiral condensate and the quark number density with those from the R-algorithm and find that they are in good agreement. By calculating the chiral condensate we obtain information on the phase structure for various quark flavors and isospin chemical potentials. We also show results for the chiral condensate at two different quark masses and at two different isospin densities which are not easily obtainable in the conventional Monte Carlo method.

Microscopic modelling of semiconductor-based infrared photodetectors: a weighted Monte Carlo approach

A microscopic investigation of hot-carrier vertical transport phenomena in semiconductor-based infrared photodetectors is presented. In order to overcome the intrinsic limitations of the conventional Monte Carlo method in describing the electro-optical response of such quantum devices—i.e. huge photocurrent fluctuations—a novel weighted Monte Carlo scheme is proposed. As shown by our simulated experiments, this new Monte Carlo strategy, particularly suited for the analysis of steady-state conditions, allows for a fully three-dimensional analysis of the basic microscopic processes governing new-generation infrared photodetectors.

Weighted simulation of steady-state transport within the standard Monte Carlo paradigm

In this paper we propose a novel Weighted Monte Carlo scheme which overcomes the intrinsic limitations of the conventional Monte Carlo method in describing the electro-optical response of some semiconductor-based quantum devices—e.g., huge photocurrent fluctuations in photodetectors. More specifically, to avoid potential numerical instabilities of existing weighted Monte Carlo approaches at long simulation times, we derive a version of the method specifically designed for the study of the steady-state regime. The latter, based again on the particle-counting paradigm, comes out to be statistically stable also in the long-time limit.

Constant-number Monte Carlo simulation of aggregating and fragmenting particles

The constant-number Monte Carlo method introduced by Matsoukas and co-workers for simulating particulate systems is applied to the kinetics of aggregating and fragmenting particles. The efficiency of this approach is increased by incorporating a modified version of Gillespie’s full-conditioning algorithm for selecting an aggregation or fragmentation event. After the steps comprising the algorithm are outlined, it is validated by simulations for several aggregation and fragmentation kernels for which the population balance equations can be solved exactly. The results agree very well with the analytical expressions except for those kernels that give rise to a gelation transition, such as the product kernel kij=ij. In this case, the simulation data are accurate below the transition time tg, but deviate significantly above tg. The accuracy of the simulation method in describing gelling kernels, including those of the form kij=(ij)ω, is also investigated. For a strongly gelling kernel, tg is accurately predicted by maxima in the time derivative of the second moment of the particle mass and the time dependence of the number of size classes in the simulation. Gel formation is simulated by setting a threshold size g above which particles have properties of the gel in the Stockmayer or Flory models. The Stockmayer model can be accurately simulated for a value of g that depends on the number of particles in the simulation. Simulation of the Flory model is less successful; results are obtained more efficiently by using the conventional constant-volume Monte Carlo method.