Traditional Monte Carlo Method Research Articles

On the parametric uncertainty quantification of the Rothermel's rate of spread model

Parametric uncertainty quantification of the Rothermel's fire spread model is presented using the Polynomial Chaos expansion method under a Non-Intrusive Spectral Projection (NISP) approach. Several Rothermel's model input parameters have been considered random with an associated prescribed probability density function. Two different vegetation fire scenarios are considered and NISP method results and performance are compared with four other stochastic methodologies: Sensitivity Derivative Enhance Sampling; two Monte Carlo techniques; and Global Sensitivity Analysis. The stochastic analysis includes a sensitivity analysis study to quantify the direct influence of each random parameter on the solution. The NISP approach achieved performance three orders of magnitude faster than the traditional Monte Carlo method. The NISP capability to perform uncertainty quantification associated with fast convergence makes it well suited to be applied for stochastic prediction of fire spread.

Hamiltonian Monte Carlo Sampling in Bayesian Empirical Likelihood Computation

SummaryWe consider Bayesian empirical likelihood estimation and develop an efficient Hamiltonian Monte Carlo method for sampling from the posterior distribution of the parameters of interest. The method proposed uses hitherto unknown properties of the gradient of the underlying log-empirical-likelihood function. We use results from convex analysis to show that these properties hold under minimal assumptions on the parameter space, prior density and the functions used in the estimating equations determining the empirical likelihood. Our method employs a finite number of estimating equations and observations but produces valid semiparametric inference for a large class of statistical models including mixed effects models, generalized linear models and hierarchical Bayes models. We overcome major challenges posed by complex, non-convex boundaries of the support routinely observed for empirical likelihood which prevent efficient implementation of traditional Markov chain Monte Carlo methods like random-walk Metropolis–Hastings sampling etc. with or without parallel tempering. A simulation study confirms that our method converges quickly and draws samples from the posterior support efficiently. We further illustrate its utility through an analysis of a discrete data set in small area estimation.

Using the Reliability Theory for Assessing the Decision Confidence Probability for Comparative Life Cycle Assessments.

Comparative decision making process is widely used to identify which option (system, product, service, etc.) has smaller environmental footprints and for providing recommendations that help stakeholders take future decisions. However, the uncertainty problem complicates the comparison and the decision making. Probability-based decision support in LCA is a way to help stakeholders in their decision-making process. It calculates the decision confidence probability which expresses the probability of a option to have a smaller environmental impact than the one of another option. Here we apply the reliability theory to approximate the decision confidence probability. We compare the traditional Monte Carlo method with a reliability method called FORM method. The Monte Carlo method needs high computational time to calculate the decision confidence probability. The FORM method enables us to approximate the decision confidence probability with fewer simulations than the Monte Carlo method by approximating the response surface. Moreover, the FORM method calculates the associated importance factors that correspond to a sensitivity analysis in relation to the probability. The importance factors allow stakeholders to determine which factors influence their decision. Our results clearly show that the reliability method provides additional useful information to stakeholders as well as it reduces the computational time.

Efficient Sequential Monte-Carlo Samplers for Bayesian Inference

In many problems, complex non-Gaussian and/or nonlinear models are required to accurately describe a physical system of interest. In such cases, Monte-Carlo algorithms are remarkably flexible and extremely powerful approaches to solve such inference problems. However, in the presence of a high-dimensional and/or multimodal posterior distribution, it is widely documented that standard Monte-Carlo techniques could lead to poor performance. In this paper, the study is focused on a Sequential Monte-Carlo (SMC) sampler framework, a more robust and efficient Monte-Carlo algorithm. Although this approach presents many advantages over traditional Monte-Carlo methods, the potential of this emergent technique is, however, largely underexploited in signal processing. In this paper, we aim at proposing some novel strategies to improve the efficiency and facilitate practical implementation of the SMC sampler. First, we propose an automatic and adaptive strategy that selects the sequence of distributions within the SMC sampler that minimizes the asymptotic variance of the estimator of the posterior normalization constant. The second original contribution we present improves the global efficiency of the SMC sampler by introducing a novel correction mechanism that allows the use of the particles generated through all of the iterations of the algorithm (instead of only particles from the last iteration). This is a significant contribution as it removes the need to discard a large portion of the samples obtained, as is standard in standard SMC methods. This will improve estimation performance in practical settings where the computational budget is important to consider.

Reduced cost mission design using surrogate models

This paper uses surrogate models to reduce the computational cost associated with spacecraft mission design in three-body dynamical systems. Sampling-based least squares regression is used to project the system response onto a set of orthogonal bases, providing a representation of the ΔV required for rendezvous as a reduced-order surrogate model. Models are presented for mid-field rendezvous of spacecraft in orbits in the Earth–Moon circular restricted three-body problem, including a halo orbit about the Earth–Moon L2 libration point (EML-2) and a distant retrograde orbit (DRO) about the Moon. In each case, the initial position of the spacecraft, the time of flight, and the separation between the chaser and the target vehicles are all considered as design inputs. The results show that sample sizes on the order of 102 are sufficient to produce accurate surrogates, with RMS errors reaching 0.2m/s for the halo orbit and falling below 0.01m/s for the DRO. A single function call to the resulting surrogate is up to two orders of magnitude faster than computing the same solution using full fidelity propagators. The expansion coefficients solved for in the surrogates are then used to conduct a global sensitivity analysis of the ΔV on each of the input parameters, which identifies the separation between the spacecraft as the primary contributor to the ΔV cost. Finally, the models are demonstrated to be useful for cheap evaluation of the cost function in constrained optimization problems seeking to minimize the ΔV required for rendezvous. These surrogate models show significant advantages for mission design in three-body systems, in terms of both computational cost and capabilities, over traditional Monte Carlo methods.

Fast Unscented Transformation-Based Transient Stability Margin Estimation Incorporating Uncertainty of Wind Generation

Intermittent generation from wind farms leads to fluctuating power system operating conditions pushing the stability margin to its limits. The traditional way of determining the worst case generation dispatch for a system with several semi-scheduled wind generators yields a conservative solution. This paper proposes a fast estimation of the transient stability margin (TSM) incorporating the uncertainty of wind generation. First, the Kalman filter (KF) is used to provide linear estimation of system angle and then unscented transformation (UT) is used to estimate the distribution of the TSM. The proposed method is compared with the traditional Monte Carlo (MC) method and the effectiveness of the proposed approach is verified using Single Machine Infinite Bus (SMIB) and IEEE 14 generator Australian dynamic system. This method will aid grid operators to perform fast online calculations to estimate TSM distribution of a power system with high levels of intermittent wind generation.

A coupled ensemble filtering and probabilistic collocation approach for uncertainty quantification of hydrological models

In this study, a coupled ensemble filtering and probabilistic collocation (EFPC) approach is proposed for uncertainty quantification of hydrologic models. This approach combines the capabilities of the ensemble Kalman filter (EnKF) and the probabilistic collocation method (PCM) to provide a better treatment of uncertainties in hydrologic models. The EnKF method would be employed to approximate the posterior probabilities of model parameters and improve the forecasting accuracy based on the observed measurements; the PCM approach is proposed to construct a model response surface in terms of the posterior probabilities of model parameters to reveal uncertainty propagation from model parameters to model outputs. The proposed method is applied to the Xiangxi River, located in the Three Gorges Reservoir area of China. The results indicate that the proposed EFPC approach can effectively quantify the uncertainty of hydrologic models. Even for a simple conceptual hydrological model, the efficiency of EFPC approach is about 10 times faster than traditional Monte Carlo method without obvious decrease in prediction accuracy. Finally, the results can explicitly reveal the contributions of model parameters to the total variance of model predictions during the simulation period.

An intelligent method for contact fatigue reliability analysis of spur gear under EHL

To complete the contact fatigue reliability analysis of spur gear under elastohydrodynamic lubrication (EHL) efficiently and accurately, an intelligent method is proposed. Oil film pressure is approximated using quadratic polynomial with intercrossing term and then mapped into the Hertz contact zone. Considering the randomness of the EHL, material properties and fatigue strength correction factors, the probabilistic reliability analysis model is established using artificial neural network (ANN). Genetic algorithm (GA) is employed to search the minimum reliability index and the design point by introducing an adjusting factor in penalty function. Reliability sensitivity analysis is completed based on the advanced first order second moment (AFOSM). Numerical example shows that the established probabilistic reliability analysis model could correctly reflect the effect of EHL on contact fatigue of spur gear, and the proposed intelligent method has an excellent global search capability as well as a highly efficient computing performance compared with the traditional Monte Carlo method (MCM).

Application of response surface method for contact fatigue reliability analysis of spur gear with consideration of EHL

In order to consider the effects of elastohydrodynamic lubrication (EHL) on contact fatigue reliability of spur gear, an accurate and efficient method that combines with response surface method (RSM) and first order second moment method (FOSM) was developed for estimating the contact fatigue reliability of spur gear under EHL. The mechanical model of contact stress analysis of spur gear under EHL was established, in which the oil film pressure was mapped into hertz contact zone. Considering the randomness of EHL, material properties and fatigue strength correction factors, the proposed method was used to analyze the contact fatigue reliability of spur gear under EHL. Compared with the results of 1.5×105 by traditional Monte-Carlo, the difference between the two failure probability results calculated by the above mentioned methods is 2.2×10−4, the relative error of the failure probability results is 26.8%, and time-consuming only accounts for 0.14% of the traditional Monte-Carlo method (MCM). Sensitivity analysis results are in very good agreement with practical cognition. Analysis results show that the proposed method is precise and efficient, and could correctly reflect the influence of EHL on contact fatigue reliability of spur gear.

A consistent hierarchy of generalized kinetic equation approximations to the master equation applied to surface catalysis.

We develop a hierarchy of approximations to the master equation for systems that exhibit translational invariance and finite-range spatial correlation. Each approximation within the hierarchy is a set of ordinary differential equations that considers spatial correlations of varying lattice distance; the assumption is that the full system will have finite spatial correlations and thus the behavior of the models within the hierarchy will approach that of the full system. We provide evidence of this convergence in the context of one- and two-dimensional numerical examples. Lower levels within the hierarchy that consider shorter spatial correlations are shown to be up to three orders of magnitude faster than traditional kinetic Monte Carlo methods (KMC) for one-dimensional systems, while predicting similar system dynamics and steady states as KMC methods. We then test the hierarchy on a two-dimensional model for the oxidation of CO on RuO2(110), showing that low-order truncations of the hierarchy efficiently capture the essential system dynamics. By considering sequences of models in the hierarchy that account for longer spatial correlations, successive model predictions may be used to establish empirical approximation of error estimates. The hierarchy may be thought of as a class of generalized phenomenological kinetic models since each element of the hierarchy approximates the master equation and the lowest level in the hierarchy is identical to a simple existing phenomenological kinetic models.

A stochastic Galerkin approach to uncertainty quantification in poroelastic media

Recent concerns over the safety of oil and natural gas extraction, fracking, and carbon sequestration have driven the need to develop methods for uncertainty quantification for coupled subsurface flow and deformation processes. Traditional Monte Carlo methods are versatile but exhibit prohibitively slow convergence. In this work, we develop an intrusive polynomial chaos expansion method for Biot’s poroelasticity equations based on the Galerkin projection with uniform and log-normally distributed material parameters. We analyze accuracy and efficiency of our method and compare it to the Monte Carlo method. We verify exponential convergence of the stochastic Galerkin approach.

A novel algorithm for solving the true coincident counting issues in Monte Carlo simulations for radiation spectroscopy.

Coincident counts can be observed in experimental radiation spectroscopy. Accurate quantification of the radiation source requires the detection efficiency of the spectrometer, which is often experimentally determined. However, Monte Carlo analysis can be used to supplement experimental approaches to determine the detection efficiency a priori. The traditional Monte Carlo method overestimates the detection efficiency as a result of omitting coincident counts caused mainly by multiple cascade source particles. In this study, a novel "multi-primary coincident counting" algorithm was developed using the Geant4 Monte Carlo simulation toolkit. A high-purity Germanium detector for ⁶⁰Co gamma-ray spectroscopy problems was accurately modeled to validate the developed algorithm. The simulated pulse height spectrum agreed well qualitatively with the measured spectrum obtained using the high-purity Germanium detector. The developed algorithm can be extended to other applications, with a particular emphasis on challenging radiation fields, such as counting multiple types of coincident radiations released from nuclear fission or used nuclear fuel.

Efficient estimation of the probability density function of transmission loss in uncertain ocean environments

Predictions of acoustic transmission loss (TL) in the ocean are useful in a variety of naval and ocean engineering applications. However, real ocean environments are imperfectly known and have many uncertainties. Consequently, TL calculations performed for these environments are also uncertain. Traditional Monte-Carlo methods used to obtain probability density functions (PDF) of TL in such circumstances require many field calculations, and are impractical for real-time applications. This presentation describes how the PDF of TL at a point of interest may be estimated using a single TL-field calculation and the statistics of TL within a range-depth area containing that point. To document the performance of this technique, RAM was used to perform TL field calculations for 100 Hz and 300 Hz sound sources in realistic ocean sound channels with eight uncertain environmental parameters describing the bathymetry, sound speed profile, and bottom properties. Preliminary results indicate that TL PDFs generated with area statistics have an L1-error ≤ 0.5 when compared with 1000-calculation Monte-Carlo PDF results throughout most of the sound channel. However, greater errors were observed in the near field as well as in areas where the sound channel depth was less than 100 m, for 100 Hz sound. [Sponsored by ONR.]

Adaptive Workflow Scheduling on Cloud Computing Platforms with IterativeOrdinal Optimization

The scheduling of multitask jobs on clouds is an NP-hard problem. The problem becomes even worse when complex workflows are executed on elastic clouds, such as Amazon EC2 or IBM RC2. The main difficulty lies in the large search space and high overhead of generating optimal schedules, especially for real-time applications with dynamic workloads. In this work, a new iterative ordinal optimization (IOO) method is proposed. The ordinal optimization method is applied in each iteration to achieve sub-optimal schedules. IOO aims at generating more efficient schedules from a global perspective over a long period. We prove through overhead analysis the advantages in time and space efficiency in using the IOO method. The IOO method is designed to adapt to system dynamism to yield suboptimal performance. In cloud experiments on IBM RC2 cloud, we execute 20,000 tasks in LIGO ( Laser Interferometer Gravitational-wave Observatory ) verification workflow on 128 virtual machines. The IOO schedule is generated in less than 1,000 seconds, while using the Monte Carlo simulation takes 27.6 hours, 100 times longer to yield an optimal schedule. The IOO-optimized schedule results in a throughput of 1,100 tasks/sec with 7 GB memory demand, compared with 60 percent decrease in throughput and 70 percent increase in memory demand in using the Monte Carlo method. Our LIGO experimental results clearly demonstrate the advantage of using the IOO-based workflow scheduling over the traditional blind-pick, ordinal optimization, or Monte Carlo methods. These numerical results are also validated by the theoretical complexity and overhead analysis provided.

Wavelet Neural Modeling for Hydrologic Time Series Forecasting with Uncertainty Evaluation

An approach, with the basic idea of resampling wavelet neural parameters, was proposed for probabilistic forecasting of hydrologic time series by the wavelet neural model. Parameters in wavelet neural model are assumed as following uniform distribution, and both proper convergence criterion and likelihood function are used to train the wavelet neural structure and judge the acceptance of parameter set. By training and learning wavelet neural structure as many times (i.e., resampling neural parameters) until becoming stable, all sets of wavelet neural parameters are composed as the resampling results, based on which probabilistic forecasting of hydrologic time series is attained. Optimal forecasting result can be gained by computing mathematical mean of the resampling results, and uncertainty can be described by proper confidence interval. Results of one runoff example indicated the identical performance of the proposed approach and wavelet regression model, but both perform better than conventional neural model. The proposed approach has similar efficiency as the Bayesian method for uncertainty evaluation, and both show higher efficiency than traditional Monte-Carlo method. Choice of proper convergence criterion is an important task when using the proposed approach, because it directly determines the convergence rate, accuracy and uncertainty level of probabilistic forecasting result. Overall, several key issues should be carefully considered for obtaining more reasonable probabilistic forecasting results by the proposed approach, including choice of proper likelihood function, accurate wavelet decomposition of series, and determination of proper wavelet neural structure.

Radiative Heat Transfer in Participating Medium and Dynamic Region Monte Carlo Method by Region Adaption

A dynamic region Monte Carlo method (DRMC) is proposed to simulate radiative heat transfer in participating medium. The basic principle and solution procedure of this method is described; radiative heat transfer in a two-dimensional rectangular region of absorbing, emitting, and/or scattering gray medium is analyzed. A comparison between DRMC and the traditional Monte Carlo method (TMC) is investigated by analyzing the simulated temperature distribution, the computing time, and the number of the sampling bundles. The investigation results show that, to compare with TMC, the DRMC can obviously reduce the computing time and storage capacity under the same solution precision for radiative transfer in optically thick medium; the DRMC allows bypassing the difficulties encountered by TMC in the limit of optically thick extinction.

Optimal design of fractional order linear system with stochastic inputs/parametric uncertainties by hybrid spectral method

This paper reports the design of a fractional linear system under stochastic inputs/uncertainties. The design methods were based on the hybrid spectral method for expanding the system signals over orthogonal functions. The use of the hybrid spectral method led to algebraic relationships between the first and second order stochastic moments of the input and output of a system. The spectral method could obtain a highly accurate solution with less computational demand than the traditional Monte Carlo method. Based on the hybrid spectral framework, the optimal design was elaborated by minimizing the suitably defined constrained-optimization problem.

Uncertainty Quantification in Transcranial Magnetic Stimulation via High-Dimensional Model Representation

A computational framework for uncertainty quantification in transcranial magnetic stimulation (TMS) is presented. The framework leverages high-dimensional model representations (HDMRs), which approximate observables (i.e., quantities of interest such as electric (E) fields induced inside targeted cortical regions) via series of iteratively constructed component functions involving only the most significant random variables (i.e., parameters that characterize the uncertainty in a TMS setup such as the position and orientation of TMS coils, as well as the size, shape, and conductivity of the head tissue). The component functions of HDMR expansions are approximated via a multielement probabilistic collocation (ME-PC) method. While approximating each component function, a quasi-static finite-difference simulator is used to compute observables at integration/collocation points dictated by the ME-PC method. The proposed framework requires far fewer simulations than traditional Monte Carlo methods for providing highly accurate statistical information (e.g., the mean and standard deviation) about the observables. The efficiency and accuracy of the proposed framework are demonstrated via its application to the statistical characterization of E-fields generated by TMS inside cortical regions of an MRI-derived realistic head model. Numerical results show that while uncertainties in tissue conductivities have negligible effects on TMS operation, variations in coil position/orientation and brain size significantly affect the induced E-fields. Our numerical results have several implications for the use of TMS during depression therapy: 1) uncertainty in the coil position and orientation may reduce the response rates of patients; 2) practitioners should favor targets on the crest of a gyrus to obtain maximal stimulation; and 3) an increasing scalp-to-cortex distance reduces the magnitude of E-fields on the surface and inside the cortex.

Robust Optimization of Static Reserve Planning With Large-Scale Integration of Wind Power: A Game Theoretic Approach

This paper presents a methodology based on game theory for power system static reserve planning with large-scale integration of wind power to ensure the generating capacity adequacy. First, a min–max game model is proposed for decision-making problems with uncertainties. Then, it is applied to the static reserve capacity planning problem. In the proposed model, the system planner (as one player) aims to find the minimum static reserve capacity to meet the total load demand while keeping the system reliability index within a desired value. Nature (as the other player), which determines the wind power output, is modeled as an attacker who wants to worsen the system reliability level due to its uncertainty and inaccurate prediction. Then, a two-stage relaxation algorithm is introduced to solve the min–max game. Finally, the proposed model for static reserve planning is applied to the IEEE Reliability Test System (RTS), and its robustness and high efficiency are demonstrated by comparing it to the traditional expectation method and Monte Carlo method.

A fast single particle Monte-Carlo method of computing the breakdown threshold of multipactor in microwave device

To compute the breakdown thresholds of multipactor in microwave devices, a fast single particle Monte-Carlo (SP-MC) method is presented, which considers the random nature of secondary electrons and their initial energies, phases and angles. With Runge-Kutta method and Furman model, the motion of the electron and the secondary electron yield (SEY) of the wall of the device are computed. An effective SEY is regarded as a criterion to estimate whether multipactor occurs, which is computed by averaging the SEYs for all impacts. As an example, the multipactor in a transmission line composed of parallel plates is investigated with the presented SP-MC method, traditional Monte-Carlo method, statistical theory method and particle-in-cell method separately. The results obtained from the SP-MC method accord well with those from the statistical theory method and particle-in-cell method, including the results of the susceptibility zones, break thresholds on specific products of frequency and gap space. Moreover, the SP-MC method is more adaptive than the statistical theory method, more stable than the traditional Monte-Carlo method and much more efficient than the particle-in-cell method.