Monte Carlo Numerical Integration Research Articles

Research on three-dimensional wake model of horizontal axis wind turbine based on Weibull function

In wind turbine wake models, Gaussian models depend on multidimensional integration to ascertain the distribution of wake velocity deficits. These integrations, which often involve complex boundary conditions, significantly enhance the complexity of mathematical computations. Due to the difficulty of obtaining analytical solutions, numerical integration methods such as Monte Carlo or other numerical integration techniques are commonly employed. This study presents a three-dimensional wake model (3DJW) for horizontal axis wind turbines, utilizing the Weibull function to simplify wake deficit characterization instead of traditional Gaussian distribution methods. The 3DJW model considers wind shear effects and mass conservation laws to enhance predictions of vertical wake velocities. By integrating incoming wind conditions and turbine parameters, the model efficiently computes downstream wake velocities, improving computational efficiency. To enhance predictions in the ultra-far wake region, an improved three-dimensional Weibull wake model is proposed using the exponential fitting method. Validation through wind tunnel experiments and wind farm data demonstrates the model's accuracy in predicting wake deficits at the hub height, with relative errors in horizontal and vertical profiles mostly within 5% and 3%, respectively. The proposed model enables accurate and rapid calculation of wake velocities at any spatial location downstream, facilitating enhanced energy utilization and reduced costs.

Structural Safety of the Steel Hall under Dynamic Excitation Using the Relative Probabilistic Entropy Concept.

This work aimed to analyze the reliability of a steel hall that was recently erected in central Poland subjected to dynamic wind excitation using the stochastic finite element method. Reliability analysis was completed using the relative entropy concept delivered by Bhattacharyya and contrasted with the first-order reliability method recommended by the engineering design codes. Bhattacharyya probabilistic relative entropy was additionally rescaled in this study to fit the demands and recommended admissibility intervals given in Eurocode 0. The finite element method study was carried out thanks to a discrete model created in the system ABAQUS 2019, while all further statistical and probabilistic computations were programmed and completed in the symbolic environment of MAPLE 2019. Contrary to most engineering analyses in steel structure areas, this study included the important warping effect while designing the hall ridges and the purlins. Dynamic structural responses were determined via the Hilber-Hughes-Taylor algorithm and their series were numerically obtained for a series of input uncertainty parameters representing several mechanical and environmental quantities. The generalized 10th order iterative stochastic perturbation technique was contrasted in this context with statistical estimators from the Monte Carlo simulations and numerical integration resulting from the semi-analytical approach. The key research finding of this study was an extremely good coincidence between the FORM indices and the rescaled relative probabilistic entropies for the given stochastic excitations, which additionally did not depend on a choice of one of the three proposed numerical approaches.

Statistical Modeling and Performance Analysis of Cooperative Communication in Frequency-Selective Underwater Acoustic Channel Using Vector Sensor

In this work, an angle of arrival (AoA) based statistical model for the path gain in an underwater (UW) channel is proposed considering the randomness of the line of sight and specular paths. The channel is assumed to be wide sense stationary (WSS), time-variant, and frequency-selective. The proposed model is used to obtain a novel characterization of a multi-hop multi-branch (MHMB) cooperative framework for a frequency-selective UW channel. In this general system architecture, multipath exists between any pair of consecutive hops. It is observed that when AoA is Gaussian distributed, then the path gain has a log-normal distribution. We use the Gauss Quadrature Rule (GQR) representation of the Moment Generating Function (MGF) of a log-normal distribution to derive the expressions for average bit error rate (BER) and ergodic capacity (EC) for a MHMB cooperative framework. Results for i) Pressure sensor (PS) as receiver and ii) Vector sensor (VS) as receiver for the dual-hop cooperative model are obtained and compared. Excellent matching between results acquired from Monte Carlo (MC) simulation and numerical integration (NI) expressions validate the accuracy of the derived expressions. Expectedly VS based communication systems perform better.

Analytical investigations of high-energy electromagnetic showers in strong magnetic fields

Abstract The diffusion equation of high-energy electromagnetic showers in strong magnetic fields is solved analytically, under the condition that the product of particle energy and field strength considerably exceeds $mc^2H_{\rm c}\simeq 2.3 \times 10^{7}$ TeV G, by applying Mellin and Laplace transforms. Differential and integral energy spectra of shower electrons/positrons and photons are evaluated by applying the saddle point method. Both spectra expressed by asymptotic expansions are also derived based on singularities of the Laplace–Mellin transform of the spectrum. The results are compared with those derived by a Monte Carlo method and numerical integration methods. Energy flows, peak positions, and peak values of transition curves, as well as track lengths of shower particles, are predicted and discussed, together with other characteristic properties of showers in strong magnetic fields, wherein good agreement between the low-energy limit of the power-law index for our differential energy spectra and the low-energy photon index of $\Gamma$ observed in Fermi LAT is pointed out and discussed.

Outage Probability Bounds of EGC Over Dual-Branch Non-Identically Distributed Independent Lognormal Fading Channels With Optimized Parameters

Outage probability analysis of diversity reception over lognormal fading channels is challenging because the cumulative distribution function (CDF) of a sum of lognormal random variables (RVs) does not yield a closed-form expression. Furthermore, traditional asymptotic analysis techniques fail for lognormal fading channels. The well-applied approximation techniques, such as Fenton, Wilkinson, and Schwartz and Yeh approximations, become unreliable when the signal-to-noise ratio (SNR) is high. In this paper, we derive closed-form bounds of outage probability for dual-branch equal-gain combining system over non-identically distributed lognormal fading channels. The bounds are shown tight from low-to-high SNR regimes, which provides an efficient way to evaluate the EGC system without resorting to expensive Monte Carlo simulation or numerical integration. Furthermore, the new analytical expressions can be used to test at what SNR the existing approximation techniques become inaccurate, and can also be used as criteria for system design.

Generic Consistency for Approximate Stochastic Programming and Statistical Problems

In stochastic programming, statistics, or econometrics, the aim is in general the optimization of a criterion function that depends on a decision variable $\theta$ and reads as an expectation with respect to a probability $\mathbb{P}$. When this function cannot be computed in closed form, it is customary to approximate it through an empirical mean function based on a random sample. On the other hand, several other methods have been proposed, such as quasi--Monte Carlo integration and numerical integration rules. In this paper, we propose a general approach for approximating such a function, in the sense of epigraphical convergence, using a sequence of functions of simpler type which can be expressed as expectations with respect to probability measures $\mathbb{P}_{n}$ that, in some sense, approximate $\mathbb{P}$. The main difference with the existing results lies in the fact that our main theorem does not impose conditions directly on the approximating probabilities but only on some integrals with respect to them. In addition, the $\mathbb{P}_{n}$'s can be transition probabilities, i.e., are allowed to depend on a further parameter, $\xi$, whose value results from deterministic or stochastic operations, depending on the underlying model. This framework allows us to deal with a large variety of approximation procedures such as Monte Carlo, quasi--Monte Carlo, numerical integration, quantization, several variations on Monte Carlo sampling, and some density approximation algorithms. As by-products, we discuss convergence results for stochastic programming and statistical inference based on dependent data, for programming with estimated parameters, and for robust optimization; we also provide a general result about the consistency of the bootstrap for $M$-estimators.

A New Asymptotic Analysis Technique for Diversity Receptions Over Correlated Lognormal Fading Channels

Prior asymptotic performance analyses are based on the series expansion of the moment-generating function (MGF) or the probability density function (PDF) of channel coefficients. However, these techniques fail for lognormal fading channels because the Taylor series of the PDF of a lognormal random variable is zero at the origin and the MGF does not have an explicit form. Although lognormal fading model has been widely applied in wireless communications and free-space optical communications, few analytical tools are available to provide elegant performance expressions for correlated lognormal channels. In this paper, we propose a novel framework to analyze the asymptotic outage probabilities of selection combining (SC), equal-gain combining (EGC), and maximum-ratio combining (MRC) over equally correlated lognormal fading channels. Based on these closed-form results, we show: 1) the outage probability of EGC or MRC becomes an infinitely small quantity compared to that of SC at high signal-to-noise ratio (SNR); 2) channel correlation can cause an infinite performance loss at high SNR; and 3) negatively correlated lognormal channels can outperform the independent lognormal channels. The analyses reveal insights into the long-standing problem of asymptotic performance analyses over correlated lognormal channels, and circumvent the time-consuming Monte Carlo simulation and numerical integration.

Asymptotic Analysis and Tight Performance Bounds of Diversity Receptions Over Beckmann Fading Channels With Arbitrary Correlation

Prior results on exact error rates and outage probabilities of diversity receptions over arbitrarily correlated fading channels involve intractable nested integrals, which makes the performance analyses challenging to carry out. In this paper, we derive asymptotic expressions and asymptotically tight bounds for error rates and outage probabilities of diversity receptions (including maximal-ratio combining, selection combining, and equal-gain combining) over arbitrarily correlated Beckmann fading channels. These new asymptotic performance expressions reveal insights into the diversity systems over arbitrarily correlated Beckmann fading channels, and the accuracy of the asymptotic performance expressions is explicitly quantified by the performance bounds. Using these analytical tools, one can assess the performance of the diversity reception systems over arbitrarily correlated Beckmann fading channels without resorting to time-consuming Monte Carlo simulation or numerical multifold integration.

On the efficient numerical solution of lattice systems with low-order couplings

We apply the Quasi Monte Carlo (QMC) and recursive numerical integration methods to evaluate the Euclidean, discretized time path-integral for the quantum mechanical anharmonic oscillator and a topological quantum mechanical rotor model. For the anharmonic oscillator both methods outperform standard Markov Chain Monte Carlo methods and show a significantly improved error scaling. For the quantum mechanical rotor we could, however, not find a successful way employing QMC. On the other hand, the recursive numerical integration method works extremely well for this model and shows an at least exponentially fast error scaling.

Virial coefficients and the equation of state of the hard tetrahedron fluid

Virial coefficients of the fluid of hard regular tetrahedra are calculated up to the ninth, using computer-algebra-generated Ree–Hoover diagrams and Monte Carlo numerical integration. The virial expansion is compared to the results of isobaric Monte Carlo simulations. Convergence of the virial series is studied and higher virial coefficients are estimated from a combination of the known virial coefficients and simulation data. Convergence of the virial series is compared to that of the fluid of hard spheres.

Playdoh: A lightweight Python library for distributed computing and optimisation

Parallel computing is now an essential paradigm for high performance scientific computing. Most existing hardware and software solutions are expensive or difficult to use. We developed Playdoh, a Python library for distributing computations across the free computing units available in a small network of multicore computers. Playdoh supports independent and loosely coupled parallel problems such as global optimisations, Monte Carlo simulations and numerical integration of partial differential equations. It is designed to be lightweight and easy to use and should be of interest to scientists wanting to turn their lab computers into a small cluster at no cost.

Stochastic development in biologically structured population models

Variation in organismal development is ubiquitous in nature but omitted from most age- and stage-structured population models. I give a general approach for formulating and analyzing its role in density-independent population models using the framework of integral projection models. The approach allows flexible assumptions, including correlated development times among multiple life stages. I give a new Monte Carlo numerical integration approach to calculate long-term growth rate, its sensitivities, stable age-stage distributions and reproductive value. This method requires only simulations of individual life schedules, rather than iteration of full population dynamics, and has practical and theoretical appeal because it ties easily implemented simulations to numerical solution of demographic equations. I show that stochastic development is demographically important using two examples. For a desert cactus, many stochastic development models, with independent or correlated stage durations, can generate the same stable stage distribution (SSD) as the real data, but stable age-within-stage distributions and sensitivities of growth rate to demographic rates differ greatly among stochastic development scenarios. For Mediterranean fruit flies, empirical variation in maturation time has a large impact on population growth. The systematic model formulation and analysis approach given here should make consideration of variable development models widely accessible and readily extendible.

On rate and BER analysis for finite-dimensional lattice coding for the dirty paper channel

Costa's dirty paper coding (DPC) offers a good framework for precoding for transmission over additive Gaussian channels with additive states non-causally known at the transmitter but not at the receiver. In this paper, rate calculation, error probability analysis and code design for DPC are investigated from a practical point-of-view. Based on Monte-Carlo numerical integration and simulations, we first show that the gap to the full AWGN capacity can be partially bridged using some finite-dimensional lattices with good packing and quantizing properties. Then the difficult problem of codebook selection is addressed through some illustrative examples. Analysis sheds light on the dual roles of “packing” and “shaping” as well as on their inter-connection at finite-dimensional coding, by opposition to the asymptotic case (i.e., infinite dimension) where the two coding components are decoupled.

Direct numerical integration of one-loop Feynman diagrams forN-photon amplitudes

One approach to the calculation of cross sections for infrared-safe observables in high energy collisions at next-to-leading order is to perform all of the integrations, including the virtual loop integration, by Monte Carlo numerical integration. In a previous paper, two of us have shown how one can perform such a virtual loop integration numerically after first introducing a Feynman parameter representation. In this paper, we perform the integration directly, without introducing Feynman parameters, after suitably deforming the integration contour. Our example is the N-photon scattering amplitude with a massless electron loop. We report results for N = 6 and N = 8.

Semiparametric Bayesian modeling of random genetic effects in family‐based association studies

We consider the inference problem of estimating covariate and genetic effects in a family-based case-control study where families are ascertained on the basis of the number of cases within the family. However, our interest lies not only in estimating the fixed covariate effects but also in estimating the random effects parameters that account for varying correlations among family members. These random effects parameters, though weakly identifiable in a strict theoretical sense, are often hard to estimate due to the small number of observations per family. A hierarchical Bayesian paradigm is a very natural route in this context with multiple advantages compared with a classical mixed effects estimation strategy based on the integrated likelihood. We propose a fully flexible Bayesian approach allowing nonparametric modeling of the random effects distribution using a Dirichlet process prior and provide estimation of both fixed effect and random effects parameters using a Markov chain Monte Carlo numerical integration scheme. The nonparametric Bayesian approach not only provides inference that is less sensitive to parametric specification of the random effects distribution but also allows possible uncertainty around a specific genetic correlation structure. The Bayesian approach has certain computational advantages over its mixed-model counterparts. Data from the Prostate Cancer Genetics Project, a family-based study at the University of Michigan Comprehensive Cancer Center including families having one or more members with prostate cancer, are used to illustrate the proposed methods. A small-scale simulation study is carried out to compare the proposed nonparametric Bayes methodology with a parametric Bayesian alternative.

Estimating genealogies from unlinked marker data: A Bayesian approach

An issue often encountered in statistical genetics is whether, or to what extent, it is possible to estimate the degree to which individuals sampled from a background population are related to each other, on the basis of the available genotype data and some information on the demography of the population. In this article, we consider this question using explicit modelling of the pedigrees and gene flows at unlinked marker loci, but then restricting ourselves to a relatively recent history of the population, that is, considering the genealogy at most some tens of generations backwards in time. As a computational tool we use a Markov chain Monte Carlo numerical integration on the state space of genealogies of the sampled individuals. As illustrations of the method, we consider the question of relatedness at the level of genes/genomes (IBD estimation), using both simulated and real data.

Bayesian modeling for genetic association in case-control studies: accounting for unknown population substructure

A two-stage parametric Bayesian method is proposed to examine the association between a candidate gene and the occurrence of a disease after accounting for population substructure. This procedure, implemented via a Markov chain Monte Carlo numerical integration technique, first estimates the posterior probability of different unknown population substructures and then integrates this information into a disease-gene association model through the technique of Bayesian model averaging. The model relaxes certain assumptions of previous analyses and provides a unified computational framework to obtain an estimate of the log odds ratio parameter corresponding to the genetic factor after allowing for the allele frequencies to vary across subpopulations. The uncertainty in estimating the population substructure is taken into account while providing credible intervals for parameters in the disease-gene association model. Simulations on unmatched case-control studies that mimic an admixed Argentinean population are performed to demonstrate the statistical properties of our model. The method is also applied to a real data set coming from a genetic association study on obesity.

Solution of the time-dependent, multi-particle Schrödinger equation using Monte Carlo and numerical integration

We study the solution of the multi-particle, time-dependent Schrödinger equation using quantum Monte Carlo methods and numerical integration. The Monte Carlo method is based on a mixed scheme, combining classical dynamics for the nuclei and quantum mechanics for the electrons. The numerical solution uses a discretization of the Schrödinger equation in real space and time. The two methods have been applied to light elements and silicon, respectively, and the dynamical events considered are the dissociation of H 3 and the fragmentation of small silicon clusters. Benchmark calculations, performed for the ground state of H, He and small clusters of H and Si, are compared with Hartree–Fock calculations, also carried out in the course of this study. This comparison shows that both methods regain the exact stationary limit.

Bayesian neural networks for bivariate binary data: an application to prostate cancer study

Prostate cancer is one of the most common cancers in American men. The cancer could either be locally confined, or it could spread outside the organ. When locally confined, there are several options for treating and curing this disease. Otherwise, surgery is the only option, and in extreme cases of outside spread, it could very easily recur within a short time even after surgery and subsequent radiation therapy. Hence, it is important to know, based on pre-surgery biopsy results how likely the cancer is organ-confined or not. The paper considers a hierarchical Bayesian neural network approach for posterior prediction probabilities of certain features indicative of non-organ confined prostate cancer. In particular, we find such probabilities for margin positivity (MP) and seminal vesicle (SV) positivity jointly. The available training set consists of bivariate binary outcomes indicating the presence or absence of the two. In addition, we have certain covariates such as prostate specific antigen (PSA), gleason score and the indicator for the cancer to be unilateral or bilateral (i.e. spread on one or both sides) in one data set and gene expression microarrays in another data set. We take a hierarchical Bayesian neural network approach to find the posterior prediction probabilities for a test and validation set, and compare these with the actual outcomes for the first data set. In case of the microarray data we use leave one out cross-validation to access the accuracy of our method. We also demonstrate the superiority of our method to the other competing methods through a simulation study. The Bayesian procedure is implemented by an application of the Markov chain Monte Carlo numerical integration technique. For the problem at hand, our Bayesian bivariate neural network procedure is shown to be superior to the classical neural network, Radford Neal's Bayesian neural network as well as bivariate logistic models to predict jointly the MP and SV in a patient in both the data sets as well as in the simulation study.

A practical treatment for the three-body interactions in the transcorrelated variational Monte Carlo method: Application to atoms from lithium to neon

We suggest a practical solution to dealing with the three-body interactions in the transcorrelated variational Monte Carlo method (TC-VMC). In the TC-VMC method, which was suggested in our previous paper [N. Umezawa and S. Tsuneyuki, J. Chem. Phys. 119, 10015 (2003)], the Jastrow-Slater-type wave function is efficiently optimized through a self-consistent procedure by minimizing the variance of the local energy. The three-body terms in the transcorrelated self-consistent-field equation, which have been simply ignored in our previous works, are efficiently calculated by the Monte Carlo numerical integration. We found that our treatment for the three-body interactions is successful for atoms from Li to Ne.