Applications and Parallel Implementation of QMC Integration

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.

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

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.

In this article, we evaluate a quasi-Monte Carlo (QMC) method with various low-discrepancy sequences (LDS) in illumination optical systems which are adopted in some commercial products, and clarify the method's effectiveness quantitatively. We assumed the evaluated systems were an illumination optical system with a perfectly diffusing surface, and we compared them against the theoretical irradiance distribution. The evaluation results indicate that the QMC method delivers higher asymptotic convergence rate than the MC method does, and there is little difference between each LDS. In evaluation of simple optical systems that can be boiled down to low-dimensional numerical integration problems, the QMC method was found to be extremely effective.

Property variations in a structure strongly impact the macroscopic mechanical performance as regions with lower strength will be prone to damage initiation or acceleration. Consideration of the variability in material property is critical for high-resolution simulations of damage initiation and propagation. While the recent progressive damage analyses consider randomness in property fields, accurately quantifying the uncertainty in damage measures remains computationally expensive. Stochastic damage analyses require extensive sampling of random property fields and numerous replications of the underlying nonlinear deterministic simulations. This paper demonstrates that a Quasi-Monte Carlo (QMC) method, which uses a multidimensional low discrepancy sobol sequence, is a computationally economical way to obtain the mean and standard deviations in cracks evolving in composites. An extended finite element method (XFEM) method with spatially random strength fields simulates the damage initiation and evolution in a model composite. We compared the number of simulations required for Monte Carlo (MC) and QMC techniques to measure the influence of input variability on the mean crack-length in an open-hole angle-ply tensile test. We conclude that the low discrepancy sampling and QMC technique converges substantially faster than traditional MC methods.

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...

Low discrepancy sequences in high dimensions: How well are their projections distributed?

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

Digital net properties of a polynomial analogue of Frolov's construction

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.

This paper proposes an uncertainty quantification (UQ)-incorporated design optimization technique that integrates UQ considering the statistical characteristics of sensing margin with machine learning (ML) for the design of bitline sense amplifiers (BLSAs) in dynamic random access memory (DRAM), including offset compensation operation. The increased offset due to process variation exacerbated by device scaling degrades the reliability of DRAM read operations, making quantitative analysis of process variation essential. Conventional Monte Carlo (MC)-based UQ analysis provides high accuracy but requires extensive simulation costs. Accordingly, this study systematically defines key input and output variables for conventional BLSA and two representative offset compensation BLSA structures, introduces Sobol sequence-based quasi-Monte Carlo (QMC) sampling to improve simulation efficiency, and generalizes the Standard UQ methodology to predict statistical distributions of circuit performance parameters under process variation. Furthermore, a 3-Stage UQ algorithm that sequentially learns nominal, mean, and standard deviation values is proposed to enhance the efficiency of ML model training. HSPICE simulation results based on the TSMC 65 nm process demonstrated that the proposed 3-Stage UQ method reduced the simulation time required for ML training data generation by an average of 43% compared to the Standard UQ method and achieved excellent prediction performance even in data-limited environments. Additionally, design optimization combined with Bayesian Optimization (BO) confirmed performance improvements across all circuit structures. The methodology presented in this study enables effective process variation-aware optimization for DRAM designs requiring high reliability and yield, and is expected to contribute to next-generation memory circuit design automation.

We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space H s ( S d ) \mathbb {H}^s( \mathbb {S}^d) with smoothness parameter s > d / 2 s > d/2 defined over the unit sphere S d \mathbb {S}^d in R d + 1 \mathbb {R}^{d+1} . Focusing on N N -point configurations that achieve optimal order QMC error bounds (as is the case for efficient spherical designs), we are led to introduce the concept of QMC designs: these are sequences of N N -point configurations X N X_N on S d \mathbb {S}^d such that the worst-case error satisfies sup f ∈ H s ( S d ) , ‖ f ‖ H s ≤ 1 | 1 N ∑ x ∈ X N f ( x ) − ∫ S d f ( x ) d σ d ( x ) | = O ( N − s / d ) , N → ∞ , \begin{equation*} \sup _{\substack {f \in \mathbb {H}^s( \mathbb {S}^d ), \\ \| f \|_{\mathbb {H}^s} \leq 1}} \Bigg | \frac {1}{N} \sum _{\mathbf {x} \in X_N} f( \mathbf {x} ) - \int _{\mathbb {S}^d} f( \mathbf {x} ) \, \mathrm {d} \sigma _d( \mathbf {x} ) \Bigg | = \mathcal {O}\big ( N^{-s/d} \big ), \qquad N \to \infty , \end{equation*} with an implied constant that depends on the H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) -norm, but is independent of N N . Here σ d \sigma _d is the normalized surface measure on S d \mathbb {S}^d . We provide methods for generation and numerical testing of QMC designs. An essential tool is an expression for the worst-case error in terms of a reproducing kernel for the space H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) with s > d / 2 s > d/2 . As a consequence of this and a recent result of Bondarenko et al. on the existence of spherical designs with appropriate number of points, we show that minimizers of the N N -point energy for this kernel form a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) . Furthermore, without appealing to the Bondarenko et al. result, we prove that point sets that maximize the sum of suitable powers of the Euclidean distance between pairs of points form a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d ) with s s in the interval ( d / 2 , d / 2 + 1 ) {(d/2,d/2+1)} . For such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order Quasi Monte Carlo integration. Numerical experiments suggest that many familiar sequences of point sets on the sphere (equal area points, spiral points, minimal [Coulomb or logarithmic] energy points, and Fekete points) are QMC designs for appropriate values of s s . For comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute QMC designs for any s > d / 2 s>d/2 . If ( X N ) (X_N) is a sequence of QMC designs for H s ( S d ) \mathbb {H}^s( \mathbb {S}^d) , we prove that it is also a sequence of QMC designs for H s ′ ( S d ) \mathbb {H}^{s’}( \mathbb {S}^d) for all s ′ ∈ ( d / 2 , s ) s’\in (d/2,s) . This leads to the question of determining the supremum of such s s (here called the QMC strength of the sequence), for which we provide estimates based on computations for the aforementioned sequences.

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).

Many important practical problems connected to energy efficiency in buildings, ecology, metallurgy, the development of wireless communication systems, the optimization of radar technology, quantum computing, pharmacology, and seismology are described by large-scale mathematical models that are typically represented by systems of partial differential equations. Such systems often involve numerous input parameters. It is crucial to understand how susceptible the solutions are to uncontrolled variations or uncertainties within these input parameters. This knowledge helps in identifying critical factors that significantly influence the model’s outcomes and can guide efforts to improve the accuracy and reliability of predictions. Sensitivity analysis (SA) is a method used efficiently to assess the sensitivity of the output results from large-scale mathematical models to uncertainties in their input data. By performing SA, we can better manage risks associated with uncertain inputs and make more informed decisions based on the model’s outputs. In recent years, researchers have developed advanced algorithms based on the analysis of variance (ANOVA) technique for computing numerical sensitivity indicators. These methods have also incorporated computationally efficient Monte Carlo integration techniques. This paper presents a comprehensive theoretical and experimental investigation of Monte Carlo algorithms based on “symmetrized shaking” of Sobol’s quasi-random sequences. The theoretical proof demonstrates that these algorithms exhibit an optimal rate of convergence for functions with continuous and bounded first derivatives and for functions with continuous and bounded second derivatives, respectively, both in terms of probability and mean square error. For the purposes of numerical study, these approaches were successfully applied to a particular problem. A specialized software tool for the global sensitivity analysis of an air pollution mathematical model was developed. Sensitivity analyses were conducted regarding some important air pollutant levels, calculated using a large-scale mathematical model describing the long-distance transport of air pollutants—the Unified Danish Eulerian Model (UNI-DEM). The sensitivity of the model was explored focusing on two distinct categories of key input parameters: chemical reaction rates and input emissions. To validate the theoretical findings and study the applicability of the algorithms across diverse problem classes, extensive numerical experiments were conducted to calculate the main sensitivity indicators—Sobol’ global sensitivity indices. Various numerical integration algorithms were employed to meet this goal—Monte Carlo, quasi-Monte Carlo (QMC), scrambled quasi-Monte Carlo methods based on Sobol’s sequences, and a sensitivity analysis approach implemented in the SIMLAB software for sensitivity analysis. During the study, an essential task arose that is small in value sensitivity measures. It required numerical integration approaches with higher accuracy to ensure reliable predictions based on a specific mathematical model, defining a vital role for small sensitivity measures. Both the analysis and numerical results highlight the advantages of one of the proposed approaches in terms of accuracy and efficiency, particularly for relatively small sensitivity indices.

We are interested in Monte Carlo (MC) methods for solving the diffusion equation: in the case of a constant diffusion coefficient, the solution is approximated by using particles and in every time step, a constant stepsize is added to or subtracted from the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable stepsize introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of stepsizes and stepping probabilities has recently been proposed and the numerical tests have given satisfactory results. In this paper, we describe a quasi-Monte Carlo (QMC) method for solving the diffusion equation in a spatially nonhomogeneous medium: we replace the random samples in the corrected MC scheme by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles are reordered according to their successive coordinates at each time step. We illustrate the method with numerical examples: in dimensions 1 and 2, we show that the QMC approach leads to improved accuracy when compared with the original MC method using the same number of particles.