Traditional Monte Carlo Method Research Articles

Estimating Tail Probability in MMPP/D/1 Queue with Importance Sampling by Service Rate Adjustments

The Asynchronous Transfer Mode (ATM) is an efficient technology for call relays, and it transmits information from multiple services including data, video, or voice. This information is conveyed at ATM multiplexers in small fixed-size packets called cells. The acceptable cell loss probability at ATM multiplexers is about 10−12. Important Sampling (IS) is an efficient method for estimating tiny probabilities that cannot be achieved by traditional Monte Carlo (MC) methods. This research presents a novel approach for evaluating the tail probability in the MMPP/D/1 queue system utilizing importance sampling simulation in the ATM network. To generate more rare events, a virtual queue is implemented in the dequeue process by decreasing the processing rate in the queue. In this way, the tail probability can be estimated on a real-time network.

A comparative lattice analysis of SU(2) dark glueballs* *Supported in part by the Natural Science Foundation of China (12125503, 12335003), and the Natural Science Foundation of Shandong Province, China (ZR2022ZD26)

We study the mass and scattering cross section of glueballs as dark matter candidates using lattice simulations. We employ both naive and improved gauge actions in dimensions with several β values, and we adopt both the traditional Monte Carlo method and flow-based model based on machine learning techniques to generate lattice configurations. The mass of a dark scalar glueball with and the Nambu-Bethe-Salpeter wave function are calculated. Using a coupling constant of as an illustration, we compare the dark glueball mass calculated from the configurations generated from the two methods. While consistent results can be achieved, the two methods demonstrate distinct advantages. Using the Runge-Kutta method, we extract the glueball interaction potential and two-body scattering cross section. From the observational constraints, we obtain the lower bound of the mass of scalar glueballs as candidates of dark matter.

A Markov Chain Genetic Algorithm Approach for Non-Parametric Posterior Distribution Sampling of Regression Parameters

This paper proposes a genetic algorithm-based Markov Chain approach that can be used for non-parametric estimation of regression coefficients and their statistical confidence bounds. The proposed approach can generate samples from an unknown probability density function if a formal functional form of its likelihood is known. The approach is tested in the non-parametric estimation of regression coefficients, where the least-square minimizing function is considered the maximum likelihood of a multivariate distribution. This approach has an advantage over traditional Markov Chain Monte Carlo methods because it is proven to converge and generate unbiased samples computationally efficiently.

Stochastic simulation of magnetically controlled convection in porous media with random porosity via Karhunen-Loève expansion and intrusive polynomial chaos expansion

Understanding and quantifying uncertainty factors are crucial for accurately predicting thermomagnetic convection phenomena. This study presented a mathematical model and algorithm framework for uncertainty analysis of thermomagnetic convection in porous media with random porosity. The proposed method combined Karhunen-Loève expansion and intrusive polynomial chaos expansion to represent the input random parameters and output response, respectively. The Galerkin projection method was employed to decouple the stochastic governing equations of thermomagnetic convection into deterministic governing equations that can be efficiently solved using the finite volume method. By solving the polynomial chaos expansion of the output response and employing the stochastic projection method to solve the associated decoupled governing equations, the temporal evolution and statistical characteristics of the output response were obtained. The study revealed that porosity uncertainty in the porous medium affects the thermomagnetic convection of paramagnetic fluid, exhibiting significant chaos effects. Monte Carlo simulations were performed to validate the accuracy of the proposed approach and demonstrate its computational efficiency compared to traditional Monte Carlo methods. This research provides valuable insights into the uncertainty analysis of thermomagnetic convection and offers a promising methodology for analyzing heat transfer in various porous media applications.

Current State of Advances in Quantification and Modeling of Hydrological Droughts

Hydrological droughts may be referred to as sustained and regionally extensive water shortages as reflected in streamflows that are noticeable and gauged worldwide. Hydrological droughts are largely analyzed using the truncation level approach to represent the desired flow condition such as the median, mean, or any other flow quantile of an annual, monthly, or weekly flow sequence. The quantification of hydrologic droughts is accomplished through indices, such as the standardized streamflow index (SSI) in tandem with the standardized precipitation index (SPI) commonly used in meteorological droughts. The runs of deficits in the SSI sequence below the truncation level are treated as drought episodes, and thus, the theory of runs forms an essential tool for analysis. The parameters of significance from the modeling perspective of hydrological droughts (or tantamount to streamflow droughts in this paper) are the longest duration and the largest magnitude over a desired return period of T-year (or month or week) of the streamflow sequences. It is to be stressed that the magnitude component of the hydrological drought is of paramount importance for the design and operation of water resource storage systems such as reservoirs. The time scales chosen for the hydrologic drought analysis range from daily to annual, but for most applications, a monthly scale is deemed appropriate. For modeling the aforesaid parameters, several methodologies are in vogue, i.e., the empirical fitting of the historical drought sequences through a known probability density function (pdf), extreme number theorem, Markov chain analysis, log-linear, copulas, entropy-based analyses, and machine learning (ML)-based methods such as artificial neural networks (ANN), wavelet transform (WT), support vector machines (SVM), adaptive neuro-fuzzy inference systems (ANFIS), and hybrid methods involving entropy, copulas, and machine learning-based methods. The forecasting of the hydrologic drought is rigorously conducted through machine learning-based methodologies. However, the traditional stochastic methods such as autoregressive integrated moving average (ARIMA), seasonal autoregressive integrated moving average (SARIMA), copulas, and entropy-based methods are still popular. New techniques for flow simulation are based on copula and entropy-based concepts and machine learning methodologies such as ANN, WT, SVM, etc. The simulated flows could be used for deriving drought parameters in consonance with traditional Monte Carlo methods of data generation. Efforts are underway to use hydrologic drought models for reservoir sizing across rivers. The ML methods whilst combined in the hybrid form hold promise in drought forecasting for better management of existing water resources during the drought periods. Data mining and pre-processing techniques are expected to play a significant role in hydrologic drought modeling and forecasting in future.

Fatigue Reliability Assessment for Orthotropic Steel Decks: Considering Multicrack Coupling Effects

Multiple fatigue cracks are generally present in practical engineering due to the existence of welding; the size and number of cracks of orthotropic steel bridge decks are greatly uncertain. The component failure conditions caused by these cracks may have correlations. Currently, it is still a challenging issue to develop a physical model of multiple fatigue crack propagation in bridge decks and perform a fatigue reliability assessment, which is also the motivation that drives the innovation of this study. A fatigue reliability evaluation method is presented for orthotropic steel bridge decks, considering the coupling effect of multiple cracks and the randomness of vehicle loading. A numerical simulation method for multifatigue crack growth is developed by combining the ABAQUS and FRANC3D programs. The equivalent crack depth under different spacing and depths of collinear cracks is calculated by using numerical simulation and the multicrack equivalent characterization method. The critical damage accumulation function of multiple fatigue cracks is established using linear elastic fracture mechanics. Subsequently, the critical damage accumulation function of multiple fatigue cracks is established based on linear elastic fracture mechanics. In order to solve the time-consuming problem of traditional Monte Carlo method, the iHL-RF method and AK-MCS method are developed for fatigue reliability analysis. The results show that compared with the single-crack model, the fatigue reliability of orthotropic steel deck will be crucially reduced considering the coupling effect of double cracks. The MCS, iHL-RF and AK-MCS methods can effectively solve the fatigue reliability analysis problem. Compared with the MCS method, the reliability calculation time based on AK-MCS method is significantly reduced. The AK-MCS method-based method reduces the time for calculating the reliability of orthotropic steel decks by 50% compared with the iHL-RF method. The reliability analysis of orthotropic steel deck bridge based on AK-MCS method is proved to be efficient and accurate.

Cosmological Parameter Estimation with Genetic Algorithms

Genetic algorithms are a powerful tool in optimization for single and multimodal functions. This paper provides an overview of their fundamentals with some analytical examples. In addition, we explore how they can be used as a parameter estimation tool in cosmological models to maximize the likelihood function, complementing the analysis with the traditional Markov chain Monte Carlo methods. We analyze that genetic algorithms provide fast estimates by focusing on maximizing the likelihood function, although they cannot provide confidence regions with the same statistical meaning as Bayesian approaches. Moreover, we show that implementing sharing and niching techniques ensures an effective exploration of the parameter space, even in the presence of local optima, always helping to find the global optima. This approach is invaluable in the cosmological context, where an exhaustive space exploration of parameters is essential. We use dark energy models to exemplify the use of genetic algorithms in cosmological parameter estimation, including a multimodal problem, and we also show how to use the output of a genetic algorithm to obtain derived cosmological functions. This paper concludes that genetic algorithms are a handy tool within cosmological data analysis, without replacing the traditional Bayesian methods but providing different advantages.

Binary decision diagram-based reliability modeling of phased-mission manufacturing system processing multi-type products

ABSTRACT The development of science and technology drives phased-mission manufacturing systems to be increasingly complex, which poses significant challenges to system reliability analysis. Although it has been considered that machine degradation is the key factor affecting system reliability, very little has been done to explore the effect of multi-type products on machine degradation, where the dependency of phases processing multi-type products further increases the difficulty in reliability modeling. Focusing on the phased-mission manufacturing system processing multi-type products, this paper presents a novel machine reliability model to investigate the mixed degradation process caused by multi-type products, which characterizes the dynamics of machine reliability. Further, given the proposed dynamic machine reliability model dealing with phase dependency, a binary-decision-diagram-model-based algorithm is proposed to analyze the reliability of phased-mission manufacturing systems when processing multi-type products. Finally, the experimental result verifies the effectiveness of the proposed method, and the accuracy is validated by the traditional Monte Carlo method. Also, the proposed method shows great potential to provide decision support for the operation management of phased-mission manufacturing systems.

Fast and robust Bayesian inference using Gaussian processes with GPry

We present the GPry algorithm for fast Bayesian inference of general (non-Gaussian) posteriors with a moderate number of parameters. GPry does not need any pre-training, special hardware such as GPUs, and is intended as a drop-in replacement for traditional Monte Carlo methods for Bayesian inference. Our algorithm is based on generating a Gaussian Process surrogate model of the log-posterior, aided by a Support Vector Machine classifier that excludes extreme or non-finite values. An active learning scheme allows us to reduce the number of required posterior evaluations by two orders of magnitude compared to traditional Monte Carlo inference. Our algorithm allows for parallel evaluations of the posterior at optimal locations, further reducing wall-clock times. We significantly improve performance using properties of the posterior in our active learning scheme and for the definition of the GP prior. In particular we account for the expected dynamical range of the posterior in different dimensionalities. We test our model against a number of synthetic and cosmological examples. GPry outperforms traditional Monte Carlo methods when the evaluation time of the likelihood (or the calculation of theoretical observables) is of the order of seconds; for evaluation times of over a minute it can perform inference in days that would take months using traditional methods. GPry is distributed as an open source Python package (pip install gpry) and can also be found at https://github.com/jonaselgammal/GPry.

Reliability evaluation of steel beam resting on elastic foundation subjected to moving load using monte carlo simulation and latin hypercube sampling

Steel beams resting on the elastic foundation are an applied technique to minimize the impact of thermal expansion and vibrations on civil engineering structures such as bridges and buildings, due to their flexibility and ability to withstand large loads. In practice, the performance of steel beams resting on elastic foundations is affected by random factors such as exceeding load-bearing capacity, undesirable deformations, loss of soil resilience, or environmental corrosion. This study proposes a technique to evaluate the reliability of steel beams resting on the elastic foundation subjected to moving loads using discrete random input parameters through the Latin hypercube sampling and Monte Carlo simulation method. The reliability of steel beams obtained from the proposed method is then compared to that of the traditional Monte Carlo simulation. The results reveal that the proposed technique provides a high accuracy and computational resource savings compared to the traditional Monte Carlo method. Finally, the influence of the random input parameters is also assessed based on the Sobol' sensitivity index.

Uncertainty Quantification for Thermodynamic Simulations with High-Dimensional Input Spaces Using Sparse Polynomial Chaos Expansion: Retrofit of a Large Thermal Power Plant

The assessment of the future thermodynamics performance of a retrofitted heat and power production unit is prone to many uncertainties due to the large number of parameters involved in the modeling of all its components. To carry out uncertainty quantification analysis, alternatives to the traditional Monte Carlo method must be used due to the large stochastic dimension of the problem. In this paper, sparse polynomial chaos expansion (SPCE) is applied to the retrofit of a large coal-fired power plant into a biomass-fired combined heat and power unit to quantify the main drivers and the overall uncertainty on the plant’s performance. The thermodynamic model encompasses over 180 components and 1500 parameters. A methodology combining the use of SPCE and expert judgment is proposed to narrow down the sources of uncertainty and deliver reliable probability distributions for the main key performance indicators (KPIs). The impact of the uncertainties on each input parameter vary with the considered KPI and its assessment through the computation of Sobol’ indices. For both coal and biomass operations, the most impactful input parameters are the composition of the fuel and its heating value. The uncertainty on the performance and steam quality parameters is not much affected by the retrofit. Key furnace parameters exhibit a skewed probability distribution with large uncertainties, which is a strong attention point in terms of boiler operation and maintenance.

Pointing probability Driven semi-analytic Monte Carlo Method (PDMC) – Part I: Global variance reduction for large-scale radiation transport analysis

We propose a Pointing Probability Driven Semi-Analytic Monte Carlo Method (PDMC) to replace the traditional Monte Carlo method for reactor physics analysis and radiation transport analysis. This paper describes the calculation principle and acceleration effect of the PDMC method in radiation transport analysis, highlighting its advantages in dealing with the deep-penetration problem. The PDMC method no longer requires particles to be transported to the detector. During the particle transport process, the global response of each particle state is calculated with a pointing probability to construct global information quickly. The statistical unbiasedness of the global response of a single particle state is demonstrated through rigorous mathematical derivation. The PDMC method does not rely on a deterministic program or iterative calculation, ensuring its universality and efficiency of global variance reduction (GVR). The PDMC method is tested in the China Fusion Engineering Test Reactor (CFETR) and the HBR2 benchmark. The results show that the PDMC method can significantly improve the efficiency of Monte Carlo deep-penetration simulation and is helpful for the GVR for large-scale radiation analysis.

Dose calculation in nuclear medicine for internal organs using Gamos NMGUI software

Accurately calculating the dose for patients in cancer treatment is a crucial task. Currently, there are many methods as well as simulation software for dose calculation, most of which have certain limitations in terms of flexibility, ease of use, and error in organ’s borders when calculating the dose at voxel level using the traditional Monte Carlo method. However, GAMOS developers have implemented an application specifically for nuclear medicine dosimetry - Gamos NMGUI. This software simplifies the dose calculation process with just a few buttons. The addition of the “parallel geometry” utility eliminates error at the boundaries, resulting in more optimal dosesfor patients. The results of the S values for the ICRP phantoms with Gamos NMGUI show a good agreement with the results calculated by Stephanie Lamart et al. The doses for the boundaries of some organs (Gallbladder Wall, Stomach Wall, Heart Wall, Urinary Bladder Wall) calculated by the “parallel geometry” method has deviations of up to 26% (heart wall) when compared with the traditional MC method. This shows the need to improve the dose calculation by using the parallel geometry technique. In summary, Gamos NMGUI is a convenient, simple but highly effective dose software.

A Monte Carlo Thermal Radiative Transfer Solver with Nonlinear Elimination

In this paper, we present a new Monte Carlo method for solving the thermal radiative transfer (TRT) equations via the method of nonlinear elimination (NLEM). This method is inspired by the previous application of NLEM to thermal radiation diffusion. Our approach, called diffusion accelerated Implicit Monte Carlo (DAIMC), is a hybrid technique which combines a Monte Carlo method for solving a purely-absorbing transport equation and a diffusion solution that accounts for effective scattering, or absorption–reemission. The method aims to improve the implicitness of the traditional implicit Monte Carlo (IMC) method. We derive DAIMC generally for 3D Cartesian geometries, but in this paper, we present results and analysis in 1D slab geometry. These preliminary results indicate that DAIMC implementations may provide more accurate and robust TRT solutions than IMC in certain test problems.

Evaluation of probabilistic model solving methods for modern power electronic distribution networks with wind power integration

The emergence of more and more power electronic loads and Distributed Generation (DG) presents many challenges to the reliable operation of Distribution Networks (DN). The uncertainties of power electronic loads and DGs have increasingly profound impacts on distribution networks. Currently, probabilistic model solving methods for distribution networks often ignore the power quality and the unbalanced operation of the distribution network. Besides, the traditional Monte Carlo method is accurate, but the heavy calculation burden limits its application in practice. Therefore, the fast and accurate solving of probabilistic models of modern power electronic distribution networks urgently needs to be studied. Therefore, in this paper, a probability solving model for a three-phase unbalanced modern power electronic distribution network with DG integration is developed, and the probability model is solved using Point Estimation Method (PEM) combined with Gram–Charlier expansion and Monte Carlo Simulation (MCS). Besides, this paper presents a detailed analysis comparing the results of PEM and MCS solutions from the perspective of voltage, THD, and line loss. The results prove that the Probability density functions (PDFs) of the parameters obtained by the two methods are almost identical, and the relative errors of the indicators are less than 1%. Moreover, the PEM uses only 1.9% of the time taken by the traditional MCS method in solving modern probabilistic models of distribution networks with power electronic devices integration (wind power, EVC, .. etc.). It is shown that the PEM used in this paper, combined with Gram–Charlier expansion, is an accurate and efficient solving method for modern distribution networks, providing a reference for subsequent research.

Interval Harmonic Power Flow Algorithm Based on Relative Distance Measure

In the harmonic power flow calculation of the power grid, the indeterminacy of new energy generation, the volatility and intermittency of nonlinear load will lead to the uncertainty of the injected harmonic current. In order to deal with the uncertainty information in the power grid, relative distance measurement (RDM) is used instead of point values to describe the uncertainty of distributed generations(DGs) output and state variables. In this paper, the RDM harmonic source model is built to solve the interval harmonic power flow. Through a typical 33-node distribution network calculation example, it is testified that the RDM Power flow algorithm has sufficient completeness and low conservatism compared with the traditional interval algorithm and Monte Carlo method.

Lightning Protection With a Differentiated Configuration of Arresters in a Distribution Network

When concerning the installation and operation costs, it is always desirable to protect an overhead distribution network with a limited quantity of surge arresters (SAs). This paper presents a novel procedure for designing differentiated protection against lightning, by identifying appropriate positions for those SAs on the overhead line. Rather than the traditional flashover rate of a whole line, indices in terms of flashover numbers at individual poles and associated with possible stroke locations are introduced in this paper. Based on these flashover indices, a differentiated configuration of SAs can be identified using a selection strategy. After several rounds of selection, a differentiated SA protection scheme can be obtained, given by a pre-set quantity of SAs. In this procedure, a simplified probability-based approach is proposed to determine the flashover indices. By comparing the traditional Monte Carlo method, this approach can achieve a 10-fold increase in computational efficiency with reasonable accuracy. Two different selection strategies are introduced in this paper. Compared with the selection strategy using the flashover indices at individual poles, the selection strategy based on a group of poles is recommended because it yields a configuration with better lightning performance with the same quantity of SAs. This design procedure is applied to practical distribution networks with a shield wire and nearby structures. The influence of grounding impedance, a shield wire, and nearby structures is addressed. Finally, a generalized step-by-step procedure is provided to assist in a differentiated SA protection design.

Uncertainty EM Scattering Prediction for Inhomogeneous Dielectric Bodies of Revolution

A perturbed volumetric integral equation is fabricated to predict the uncertain scattering characteristics for the in-homogeneous dielectric bodies of revolution (BoRs) targets. The transverse surface of the body of revolution is discretized in terms of the rectangular and triangular basis functions. In this way, all the unknowns are assigned to the electric flux density and the governing equation can be solved mode by mode. Both the geometrical and dielectric uncertainties are taken into consideration for EM scattering characteristics. On one hand, the geometrical shape uncertainty is described with the nonuniform rational B-spline surface by using several independent random variables. On the other hand, the dielectric uncertainty is represented by the perturbed permittivity. Thus, the volume integral equation is reconstructed by the perturbed electric flux density. Both the mean value and variance of the radar cross Section can be derived to account for the geometrical and dielectric uncertainties. Numerical results are given to show that the calculation efficiency of the proposed method is higher than that of the traditional Monte Carlo method.

Publisher’s Note

The emergence of more and more power electronic loads and Distributed Generation (DG) presents many challenges to the reliable operation of Distribution Networks (DN). The uncertainties of power electronic loads and DGs have increasingly profound impacts on distribution networks. Currently, probabilistic model solving methods for distribution networks often ignore the power quality and the unbalanced operation of the distribution network. Besides, the traditional Monte Carlo method is accurate, but the heavy calculation burden limits its application in practice. Therefore, the fast and accurate solving of probabilistic models of modern power electronic distribution networks urgently needs to be studied. Therefore, in this paper, a probability solving model for a three-phase unbalanced modern power electronic distribution network with DG integration is developed, and the probability model is solved using Point Estimation Method (PEM) combined with Gram–Charlier expansion and Monte Carlo Simulation (MCS). Besides, this paper presents a detailed analysis comparing the results of PEM and MCS solutions from the perspective of voltage, THD, and line loss. The results prove that the Probability density functions (PDFs) of the parameters obtained by the two methods are almost identical, and the relative errors of the indicators are less than 1%. Moreover, the PEM uses only 1.9% of the time taken by the traditional MCS method in solving modern probabilistic models of distribution networks with power electronic devices integration (wind power, EVC, .. etc.). It is shown that the PEM used in this paper, combined with Gram–Charlier expansion, is an accurate and efficient solving method for modern distribution networks, providing a reference for subsequent research.

Hamiltonian sequential Monte Carlo with application to consumer choice behavior

The practical use of nonparametric Bayesian methods requires the availability of efficient algorithms for posterior inference. The inherently serial nature of traditional Markov chain Monte Carlo (MCMC) methods imposes limitations on their efficiency and scalability. In recent years, there has been a surge of research activity devoted to developing alternative implementation methods that target parallel computing environments. Sequential Monte Carlo (SMC), also known as a particle filter, has been gaining popularity due to its desirable properties. SMC uses a genetic mutation-selection sampling approach with a set of particles representing the posterior distribution of a stochastic process. We propose to enhance the performance of SMC by utilizing Hamiltonian transition dynamics in the particle transition phase, in place of random walk used in the previous literature. We call the resulting procedure Hamiltonian Sequential Monte Carlo (HSMC). Hamiltonian transition dynamics have been shown to yield superior mixing and convergence properties relative to random walk transition dynamics in the context of MCMC procedures. The rationale behind HSMC is to translate such gains to the SMC environment. HSMC will facilitate practical estimation of models with complicated latent structures, such as nonparametric individual unobserved heterogeneity, that are otherwise difficult to implement. We demonstrate the behavior of HSMC in a challenging simulation study and contrast its favorable performance with SMC and other alternative approaches. We then apply HSMC to a panel discrete choice model with nonparametric consumer heterogeneity, allowing for multiple modes, asymmetries, and data-driven clustering, providing insights for consumer segmentation, individual level marketing, and price micromanagement.