Quantifying Uncertainty of Damage in Composites Using a Quasi Monte Carlo Technique

A quasi-Monte Carlo solver for thermal radiation in participating media

<abstract> A bidirectional Monte Carlo (BDMC) method based on reversibility of bundle trajectory and reciprocity of thermal radiative energy exchange was developed to solve radiative heat transfer in absorbing and scattering medium. Two types of sampling models were introduced into the Monte Carlo (MC) simulation, namely the equivalent sampling and the weight sampling, respectively. Mathematical formula for the sampling models and the statistical calculation of sampling bundles were derived. Furthermore, the reciprocity error correlation of radiative exchange factors between the BDMC method and the traditional Monte Carlo (TMC) method were demonstrated and analyzed. Radiative heat transfer in a two-dimensional rectangular domain with absorbing and scattering media was solved by using both the BDMC method and the TMC method. Radiative exchange factors and radiative equilibrium temperature profiles predicted by the BDMC method were compared with those predicted by the TMC method. The performance parameter P, defined to evaluate the performance of MC methods, was computed and compared between the BDMC and the TMC methods. The results showed the superiority of BDMC method compared with the TMC method for radiative heat transfer, in addition, the weight sampling was proved to be more flexible than the equivalent sampling in the BDMC method. </abstract>

The way of measuring diameter by use of measuring bow height and chord length is commonly adopted for the large diameter work piece. In the process of computing the diameter of large work piece, measurement uncertainty is an important parameter and is always employed to evaluate the reliability of the measurement results. Therefore, it is essential to present reliable methods to evaluate the measurement uncertainty, especially in precise measurement. Because of the limitations of low convergence and unstable results of the Monte-Carlo (MC) method, the quasi-Monte-Carlo (QMC) method is used to estimate the measurement uncertainty. The QMC method is an improvement of the ordinary MC method which employs highly uniform quasi random numbers to replace MC's pseudo random numbers. In the process of evaluation, first, more homogeneous random numbers (quasi random numbers) are generated based on Halton's sequence. Then these random numbers are transformed into the desired distribution random numbers. The desired distribution random numbers are used to simulate the measurement errors. By computing the simulation results, measurement uncertainty can be obtained. An experiment of cylinder diameter measurement and its uncertainty evaluation are given. In the experiment, the guide to the expression of uncertainty in measurement method, MC method, and QMC method are validated. The result shows that the QMC method has a higher convergence rate and more stable evaluation results than that of the MC method. Therefore, the QMC method can be applied effectively to evaluate the measurement uncertainty.

Increment variance reduction techniques are add-ons to Monte Carlo (MC) simulations. They make MC simulations converging faster by repeating the number of simulations with an incremental rate derived from mathematical functions. Besides speeding up MC simulations, the major advantage of increment techniques is their ability to handle large numbers of simulations avoiding memory saturation and overflow which occur when a plain MC simulation is involved in the pricing of multi-name credit derivatives. A trend among authors pricing financial securities with MC simulation has been to choose Quasi-Monte Carlo (QMC) methods using deterministic sequences instead of MC methods involving pseudorandom generators such as congruential generator and Mersenne twister. The Increment family models circumvent the constraint of identifying the optimal QMC sequence to price a given security by using a common generator such as Matlab-LCG-Xor RNG and determining the optimal mathematical function of incrementation of MC simulations that will make the pricing of the security adequate. Market participants in need of selecting a reliable numerical method for pricing complex financial securities such as multi-name credit derivatives will find our paper appealing.

Absence of closed form solutions for many financial models has given rise to numerical and simulation techniques in the recent past. In the current study direct simulation, which is one of the popular approaches in aerospace engineering is used for studying applications in finance, especially option pricing.Monte Carlo (MC) is one of the popular simulation approaches for approximating the value of the quantity under question. However, the slow convergence rate, O(N−1/2) for N number of samples of the MC method has motivated research in Quasi Monte-Carlo (QMC) techniques. QMC methods use low discrepancy (LD) sequences that provide faster, more accurate results than MC methods. In this paper, we focus on the parallelization of the QMC method on a heterogeneous network of workstations (HNOWs) for option pricing. HNOWs are machines with different processing capabilities and have distinct execution time for the same task. It is, therefore, vital to allocate and schedule the tasks depending on the performance a...

The Monte-Carlo (MC) technique is very efficient solution for statistical problem. Various MC methods can easily be applied to statistical circuit performance analysis. Recently, as the number of process parameters and their impact, has increasingly affected circuit performance, a sufficient sample size is required in order to consider high dimensionality, profound nonlinearity, and stringent accuracy requirements. Also, it is important to identify the performance of circuit as soon as possible. In this paper, Fast MC method is proposed for efficient analysis of circuit performance. The proposed method analyzes performance using enhanced-precision Latin Hypercube Sampling Monte Carlo (LHSMC). To increase the accuracy of the analysis, we calculate the effective dimension for the low discrepancy value on critical parameters. This will guarantee a robust input vector for the critical parameters. Using a 90㎚ process parameter and OP-AMP, we verified the accuracy and reliability of the proposed method in comparison with the standard MC, LHS and Quasi Monte Carlo (QMC).

The splitting of Quasi-Monte Carlo (QMC) point sequences into interleaved substreams has been suggested to raise the speed of distributed numerical integration and to lower the traffic on the network. The usefulness of this approach in GRID environments is discussed. After specifying requirements for using QMC techniques in GRID environments in general we review and evaluate the proposals made in literature so far. In numerical integration experiments we investigate the quality of single leaped QMC point sequence substreams, comparing the respective properties of Sobol', Halton, Faure, Niederreiter-Xing, and Zinterhof sequences in detail. Numerical integration results obtained on a distributed system show that leaping sensitivity varies tremendously among the different sequences and we provide examples of deteriorated results caused by leaping effects, especially in heterogeneous settings which would be expected in GRID environments.

In this chapter we deal with numerical computation of integrals over the domain ℝ s (s >1) with respect to a positive weight function. For one-dimensional integrals Gauss Hermite formulas compute integrals with respect to a Gaussian weight with quite high accuracy but for high-dimensional integrals the effort increases exponentially. For integrals over the s-dimensional unit cube probabilistic methods like Monte Carlo (MC) are not affected by this so-called “curse of dimensions,” but the convergence rate is rather poor. If the integration nodes are not pure random points but special deterministic point sequences (the method is called Quasi Monte Carlo (QMC) due to this fact) this rate can be significantly improved. These lowdiscrepancy sequences appear also in the computation of integrals over ℝ s .

A high-order solution of electromagnetic (EM) scattering from electrically large targets with uncertain shapes is proposed in this letter. At first, the uncertain geometry can be described with several independent variables by using the nonuniform rational B-spline surfaces. Then, the second-order Taylor series expansion is used to present the geometrical uncertainty in terms of the combined field integral equation. As a result, higher accuracy can be obtained by using this high-order solver. Finally, the mean value and variance of the radar cross section (RCS) are derived and can be compared with the traditional Monte Carlo (MC) method. Numerical results show the proposed method can provide higher accuracy than the traditional one-order solver and higher efficiency than the traditional MC method.

This paper presents an application of the Global Sensitivity Analysis (GSA) method in evaluating performance of impedance measurement algorithms implemented in modern Intelligent Electronic Devices (IEDs) when affected by uncertainty. To be specific, the GSA is applied in testing an algorithm for distance protection of a single transmission line in a system with two lines on the same towers. Factors contributing to uncertainty are discussed in the paper. The GSA method decomposes variance of the estimation error according to the factors and their interactions, and in this way identifies the factors that have the highest impact on the estimation error. Computation of the GSA sensitivity indices relies on the Quasi-Monte Carlo (QMC) technique with Sobol's quasi-random sampling. This is time-consuming for large number of factors therefore, fast pre-screening process based on the Morris method is implemented to remove non-influential parameters and reduce the factor space before running the QMC. The methodology is implemented using fault simulation software DIgSILENT PowerFactory to calculate fault impedance as a function of uncertain parameters (i.e. factors) as well as voltage and current signals that are inputs to an IED under test. In this simulation, factor space is sampled using Sobol's quasi-random sequence generated with the standard GSA software tool called SIMLAB. All simulations of the transmission line faults for various parameter values and calculation of the performance indices (i.e. sensitivities) is automated using the program developed in DIgSILENT Programming Language (DPL). This testing technique is applied to study the fault impedance measurement implemented in SEL-421. (6 pages)

Capturing stochastic variabilities in pedestrian flows: A dynamic continuum modeling approach

Accurate and efficient estimation of Sobol’ sensitivity indices is a cornerstone of variance-based global sensitivity analysis, providing critical insights into how uncertainties in input parameters affect model outputs. This is particularly important for large-scale environmental, engineering, and financial models, where understanding parameter influence is essential for improving model reliability, guiding calibration, and supporting informed decision-making. However, computing Sobol’ indices requires evaluating high-dimensional integrals, presenting significant numerical and computational challenges. In this study, we present a comparative analysis of two of the best available Quasi-Monte Carlo (QMC) techniques: polynomial lattice rules (PLRs) and modified Sobol’ sequences. The performance of both approaches is systematically assessed in terms of performance and accuracy. Extensive numerical experiments demonstrate that the proposed PLR-based framework achieves superior precision for several sensitivity measures, while modified Sobol’ sequences remain competitive for lower-dimensional indices. Our results show that IPLR-α3 outperforms traditional QMC methods in estimating both dominant and weak sensitivity indices, offering a robust framework for high-dimensional models. These findings provide practical guidelines for selecting optimal QMC strategies, contributing to more reliable sensitivity analysis and enhancing the predictive power of complex computational models.

Predicting the long-term performance of pavement structures is crucial for efficient infrastructure management, but uncertainties in material properties, traffic loads, and environmental conditions pose significant challenges. This research introduces a novel reliability model for pavement performance prediction using Quasi-Monte Carlo (QMC) simulation. Unlike traditional Monte Carlo methods, QMC employs deterministic sequences for enhanced sampling efficiency and accuracy in estimating probabilistic quantities like reliability indices and probabilities of failure. The study investigates the impact of QMC on pavement performance predictions, considering uncertainties in traffic load repetitions, fatigue life, and rutting life. The QMC approach demonstrates superior performance by producing stable mean estimates and significantly narrowing confidence intervals as sample size increases. This research highlights the advantages of QMC simulation in probabilistic pavement design, offering a computationally efficient framework for robust and reliable performance predictions. The findings support QMC as a valuable tool for achieving durable, safe, and cost-effective pavement infrastructure.

For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA’s Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed.

For the approximation of multidimensional integrals, two types of methods are widely used. Monte Carlo (MC) methods are the best known and require the use of a pseudorandom generator. Quasi-Monte Carlo (QMC) methods use low discrepancy point sets and are deterministic. The idea is to use points that are more regularly distributed over the integration space than random points. The best known methods to achieve this are the lattice rules and (t,s) sequences (or (t,m,s) nets) (A.B. Owen, 1998; H. Niederreiter; 1992; I.H. Sloan and S. Joe, 1994). The paper compares Monte Carlo methods, lattice rules, and other low discrepancy point sets on the problem of evaluating Asian options. The combination of these methods with variance reduction techniques is also explored.