quasi-Monte Carlo Methods Research Articles

Uncertainty analysis and reliability improvement of planetary roller screw mechanism using active learning Kriging model

The uncertainties in the geometry, material and operation conditions may cause structural failure of the planetary roller screw mechanism (PRSM). The uncertainty analysis model is the key to the reliability assessment of the PRSM, however, the relevant studies have been rarely reported in the past. This paper focuses on establishing a preliminary mathematical model of the PRSM considering uncertain factors. The quasi-Monte Carlo (QMC) method is introduced to improve the solving efficiency of the multidimensional and nonlinear implicit limit state function (LSF). Then, the parameter sensitivities of the uncertain factors to the load distribution and contact characteristics are comprehensively ranked by the design of experiment (DoE). The computational cost for constructing the active learning Kriging (ALK) model of PRSM is decreased by only selecting the most sensitive variables. Moreover, the ALK model and QMC method (ALK-QMC) are combined to explore how the main factors affect the structural reliability of PRSM, which further guides the implementation of multi-objective optimization to improve the reliability by the developed NSGA-II-Downhill algorithm. Finally, the theoretical model and optimization results are verified by the finite element method.

Adaptive Two-Stage Monte Carlo Algorithm for Accuracy Estimation of Total Least Squares

The existing theory of the total least squares (TLS) accuracy estimation contains the following problems: (1) the approximation function method based on the Taylor series expansion cannot avoid the derivative operation; (2) the selection of the number of simulations for the Monte Carlo method is subjective; and (3) the adaptive Monte Carlo algorithm is complex. Aiming at these problems, we introduce the adaptive two-stage Monte Carlo (ATMC) algorithm into the theory of accuracy estimation of the TLS. In order to consider the biases of estimation for parameters, residuals, and the estimation of the variance of unit weight, the computing process for accuracy estimation of the TLS contains bias estimation and accuracy estimation, and the algorithm is provided. In addition, based on the quasi-Monte Carlo method, an adaptive two-stage quasi-Monte Carlo (ATQMC) algorithm is proposed for the calculation of the expected bias in parameter estimates, and the algorithm flow is given. The experimental results verified the effectiveness of the ATMC and ATQMC algorithms. Compared with the ATMC algorithm, the ATQMC algorithm proposed in this paper is more efficient in the calculation of the expected bias in parameter estimates.

Numerical Regularization for 4-loop Self-Energy Feynman Diagrams

In recent work we computed 4-loop integrals for self-energy diagrams with 11 massive internal lines. Presently we perform numerical integration and regularization for diagrams with 8 to 11 lines, while considering massive and massless cases. For dimensional regularization, a sequence of integrals is computed depending on a parameter (ε) that is incorporated via the space-time dimension, and approaches zero. We consider diagrams where the leading term in the expansion is in 1/ε 2 or in 1/ε or finite. The numerical integration methods include non-adaptive – double exponential, and Quasi-Monte Carlo methods with composite lattice rules implemented in CUDA C for acceleration on GPUs. The leading term coefficients in the integral expansion are obtained via linear or nonlinear extrapolation as ε tends to zero.

Pricing of Arithmetic Average Asian Option by Combining Variance Reduction and Quasi-Monte Carlo Method

Financial derivatives have developed rapidly over the past few decades due to their risk-averse nature, with options being the preferred financial derivatives due to their flexible contractual mechanisms, particularly Asian options. The Black–Scholes stock option pricing model is often used in conjunction with Monte Carlo simulations for option pricing. However, the Black–Scholes model assumes that the volatility of asset returns is constant, which does not square with practical financial markets. Additionally, Monte Carlo simulation suffers from slow error convergence. To address these issues, we first correct the asset volatility in the Black–Scholes model using a GARCH model. Then, the low error convergence rate of the Monte Carlo method is improved using variance reduction techniques. Meanwhile, the quasi-Monte Carlo approach based on low discrepancy sequences is used to refine the error convergence rate. We also provide a simulation experiment and result analysis to validate the effectiveness of our proposed method.

The efficient calculation methods for stochastic nonlinear transient heat conduction problems

This paper proposes an efficient model reduction strategy in conjunction with non-intrusive statistical moment methods for the stochastic nonlinear transient heat conduction problem. In the proposed strategy, the deterministic reduced basis vectors (RBVs) are adopted to approximate the random temperature response, and the thermal conductivity of the random nonlinear heat conduction matrix in the RBVs space is decoupled from the geometric characteristic matrix (GCM). The computational efficiency of solving the reduced random nonlinear heat conduction models can be enhanced by implementing these two technologies. The large coefficient of variation and the small coefficient of variation stochastic heat conduction problems, respectively, are addressed by the two non-intrusive statistical methods, i.e., the quasi-Monte Carlo method (QMCM) based on number theory and the modified stochastic perturbation method (MSPM). These two methods necessitate deterministic analysis and are easily integrated into the model reduction method. In four numerical instances, the selection of RBVs, the influence of the nonlinear term on the random temperature response, and the accuracy and efficiency of the two non-intrusive statistical moment methods are discussed. The numerical results highlight that the model reduction strategy based on the decoupling of thermal conductivity and GCM, in conjunction with the QMCM and the MSPM, enables the analysis of stochastic nonlinear transient heat conduction problems with a large coefficient of variation and a small coefficient of variation, respectively.

Quasi-Monte Carlo Methods for Binary Event Models with Complex Family Data

The generalized linear mixed model for binary outcomes with the probit link function is used in many fields but has a computationally challenging likelihood when there are many random effects. We extend a previously used importance sampler, making it much faster in the context of estimating heritability and related effects from family data by adding a gradient and a Hessian approximation and making a faster implementation. Additionally, a graph-based method is suggested to simplify the likelihood when there are thousands of individuals in each family. Simulation studies show that the resulting method is orders of magnitude faster, has a negligible efficiency loss, and confidence intervals with nominal coverage. We also analyze data from a large study of obsessive-compulsive disorder based on Swedish multi-generational data. In this analysis, the proposed method yielded similar results to a previous analysis, but was much faster. Supplementary materials for this article are available online.

A high-performance calculation scheme for stochastic dynamic problems

The goal of stochastic dynamic problems is to efficiently and accurately calculate statistical moments and random dynamic responses. Thus, a high-performance calculation scheme is proposed to achieve the goal. First, to enhance the accuracy of the statistical moments, we proposed an adaptive weights quasi-Monte Carlo (AWQMC) method. Compared to the integration weights of the traditional quasi-Monte Carlo method, the adaptive weights can better reflect the discrepancy of the sample points to obtain more accurate statistical moments. Additionally, the adaptive weights only depend on the sample set, which can be applied to other stochastic problems. Second, the reduced-order model technology is introduced to improve the calculation efficiency of random dynamic responses. Combining of the Galerkin method, a novel reduced-order model based on deterministic reduced bases is proposed for random dynamic system. Then, an efficient dynamic equation solver is constructed based on the deterministic reduced bases and the Newmark method. Finally, according to the three numerical examples, the high-performance calculation scheme combined the AWQMC method and the efficient dynamic equation solver can significantly improve the accuracy and efficiency of statistical moments for stochastic dynamic problems.

A note on the CBC-DBD construction of lattice rules with general positive weights

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule for an s-dimensional integral is specified by its generating vectorz∈Zs and its number of points N. While there are many results on the existence of “good” rank-1 lattice rules, there are no explicit constructions of good generating vectors for dimensions s≥3. Therefore one resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors for integration of functions in weighted Korobov classes equipped with product weights. Here, we generalize this result to arbitrary positive weights, answering an open question from the paper of Ebert et al. We include a section on how the algorithm can be implemented in the case of POD weights, implying that the CBC-DBD construction is competitive with the classical CBC construction.

The Use of Quasi Monte Carlo Method with Halton Random Number Sequence in Determining the Price of European Type Options: in PT Telekomunikasi Indonesia Stock’s

An investor must be wise in managing the funds he has to carry out investment activities. Investors can use options as an alternative to investing because they can increase profits and avoid investment risks. Options are one of the most widely used derivative products. The main problem when entering into an option contract is determining the right price to be paid by the option buyer to the option seller. This research was made to determine the price of European-type stock options. Case studies on PT Telekomunikasi Indonesia, Tbk shares in the 2021-2022 period. The analysis was performed using the Quasi-Monte Carlo method with Halton's random number sequence. Based on the results of this study, it is expected to be a consideration in deciding to buy European-type stock options at PT Telekomunikasi Indonesia, Tbk

Biomineral Flocculation of Kaolinite and Microalgae: Laboratory Experiments and Stochastic Modeling

AbstractIn estuaries and coastal waters, the flocculation of cohesive sediments is significantly affected by extracellular polymeric substances secreted by microalgae. In this study, laboratory experiments are carried out to explore the role of two typical microalgae (Skeletonema costatum and Cyclotella meneghiniana) on suspended fine‐grained sediments under various turbulent shear and environmental conditions. The results show that the shear rate is a dominant factor in controlling the biomineral flocculation processes, and the existence of microalgae generally enhances the aggregation of mineral sediments with microalgae. It is also found that floc size distributions (FSDs) and mean sizes are clearly influenced by different algal species and concentrations, especially under intensive turbulent environments. Moreover, a quasi‐Monte Carlo (QMC) based bivariate population balance model is developed to simulate the FSDs of sediment‐algae flocs. This model is reasonably validated by two analytical solutions and laboratory observations, and indicates a potential extension for tracking the properties of microalgae‐associated fine sediment flocs due to the discrete nature of the QMC method.

A random walk algorithm to estimate a lower bound of the star discrepancy

Abstract In many Monte Carlo applications, one can substitute the use of pseudorandom numbers with quasirandom numbers and achieve improved convergence. This is because quasirandom numbers are more uniform that pseudorandom numbers. The most common measure of that uniformity is the star discrepancy. Moreover, the main error bound in quasi-Monte Carlo methods, called the Koksma–Hlawka inequality, has the star discrepancy in the formulation. A difficulty with this bound is that computing the star discrepancy is very costly. The star discrepancy can be computed by evaluating a function called the local discrepancy at a number of points. The supremum of these local discrepancy values is the star discrepancy. If we have a point set in [ 0 , 1 ] s {[0,1]^{s}} with N members, we need to compute the local discrepancy at N s {N^{s}} points. In fact, computing star discrepancy is NP-hard. In this paper, we will consider an approximate algorithm for a lower bound on the star discrepancy based on using a random walk through some of the N s {N^{s}} points. This approximation is much less expensive that computing the star discrepancy, but still accurate enough to provide information on convergence. Our numerical results show that the random walk algorithm has the same convergence rate as the Monte Carlo method, which is O ( N - 1 2 {O(N^{-\frac{1}{2}}} ).

Convergence of Uniformity Criteria and the Application in Numerical Integration

Quasi-Monte Carlo (QMC) methods have been successfully used for the estimation of numerical integrations arising in many applications. In most QMC methods, low-discrepancy sequences have been used, such as digital nets and lattice rules. In this paper, we derive the convergence rates of order of some improved discrepancies, such as centered L2-discrepancy, wrap-around L2-discrepancy, and mixture discrepancy, and propose a randomized QMC method based on a uniform design constructed by the mixture discrepancy and Baker’s transformation. Moreover, the numerical results show that the proposed method has better approximation than the Monte Carlo method and many other QMC methods, especially when the number of dimensions is less than 10.

Super-polynomial accuracy of one dimensional randomized nets using the median of means

Letffbe analytic on[0,1][0,1]with|f(k)(1/2)|⩽Aαkk!|f^{(k)}(1/2)|\leqslant A\alpha ^kk!for some constantsAAandα>2\alpha >2and allk⩾1k\geqslant 1. We show that the median estimate ofμ=∫01f(x)dx\mu =\int _0^1f(x)\,\mathrm {d} xunder random linear scrambling withn=2mn=2^mpoints converges at the rateO(n−clog⁡(n))O(n^{-c\log (n)})for anyc>3log⁡(2)/π2≈0.21c> 3\log (2)/\pi ^2\approx 0.21. We also get a super-polynomial convergence rate for the sample median of2k−12k-1random linearly scrambled estimates, whenk/mk/mis bounded away from zero. Whenffhas app’th derivative that satisfies aλ\lambda-Hölder condition then the median of means has errorO(n−(p+λ)+ϵ)O( n^{-(p+\lambda )+\epsilon })for anyϵ>0\epsilon >0, ifk→∞k\to \inftyasm→∞m\to \infty. The proof techniques use methods from analytic combinatorics that have not previously been applied to quasi-Monte Carlo methods, most notably an asymptotic expression from Hardy and Ramanujan on the number of partitions of a natural number.

Quasi Monte Carlo time-frequency analysis

We study signal processing tasks in which the signal is mapped via some generalized time-frequency transform to a higher dimensional time-frequency space, processed there, and synthesized to an output signal. We show how to approximate such methods using a quasi-Monte Carlo (QMC) approach. We consider cases where the time-frequency representation is redundant, having feature axes in addition to the time and frequency axes. The proposed QMC method allows sampling both efficiently and evenly such redundant time-frequency representations. Indeed, 1) the number of samples required for a certain accuracy is log-linear in the resolution of the signal space, and depends only weakly on the dimension of the redundant time-frequency space, and 2) the quasi-random samples have low discrepancy, so they are spread evenly in the redundant time-frequency space. One example of such redundant representation is the localizing time-frequency transform (LTFT), where the time-frequency plane is enhanced by a third axis. This higher dimensional time-frequency space improves the quality of some time-frequency signal processing tasks, like the phase vocoder (an audio signal processing effect). Since the computational complexity of the QMC is log-linear in the resolution of the signal space, this higher dimensional time-frequency space does not degrade the computation complexity of the proposed QMC method. The proposed QMC method is more efficient than standard Monte Carlo methods, since the deterministic QMC sample points are optimally spread in the time-frequency space, while random samples are not.

Evaluating View Factors Using a Hybrid Monte-Carlo Method

AbstractThis paper demonstrates that the well-known method for calculating view factors, the Monte Carlo method, combined with ray tracing is not necessarily the most efficient strategy. The Monte Carlo method and quasi-Monte Carlo method combined with numerical integration, provided the surfaces in a configuration are not too close together, are more accurate for the same run-time than a ray tracing-based Monte Carlo method. The Monte Carlo method based on numerical integration is complementary to the Monte Carlo method based on ray tracing. When many rays are required to calculate an accurate view factor, few function evaluations in a numerical integration approach are necessary to achieve the same accuracy. Where the surfaces in a configuration are touching, the Monte Carlo method with numerical integration converges to the exact view factor very slowly due to a singularity in the view factor multi-integral. For these configurations, a hybrid Monte Carlo method and quasi-Monte Carlo method are demonstrated to be the stochastic methods of choice.

Normal-transformation-based probabilistic load flow with correlated wind and load forecast errors

With aggressive government policies driving the electric power systems around the globe towards high renewable energy penetration, methodologies for considering forecast uncertainties in power system analysis and planning are gaining relevance. Probabilistic load flow (PLF) is a useful tool to carry out risk-based power system analysis and planning. This article discusses the need and philosophy of Normal-transformations, elaborating on Nataf transformation and polynomial normal transformation (PNT), to solve PLF and presents a unified approach for representation. Through detailed statistical modeling, the Gaussian mixture (GM) model is found to fit the historical wind and load power forecast error data better compared to widely used parametric models. Sampling of correlated marginal distributions, for scenario generation in PLF, requires the quantile function of the GM distribution, which is not available in a closed expression. A strategy, using quantile functions of Gaussian component sampling and interpolation, to approximate the GM’s quantile function is presented. The presented approximation enables the use of GM for solving PLF with correlated marginals, which is not addressed in the literature. This work also contributes to prevailing research questions by two new performance comparisons: between various higher-order PNT embedded to point estimate methods (PEMs) for PLF, and between PEM and Quasi-Monte Carlo (QMC) methods for PLF analysis. When PEMs are employed for PLF, there is no improvement observed in going for higher than third order PNT (3PNT). Results show that estimating the first four moments of the output variables through QMC-PLF is more accurate compared to PEM-PLF. Formulae useful for implementation of various higher-order PNT, for QMC and PEM, are derived and presented for direct use. Finally, the impact of correlated non-normal modeling of inputs on PLF output variables’ distribution is demonstrated.

Thermal performance and heat transfer reliability analysis in helically corrugated helical tube

In this study, the characteristics of heat transfer and flow in helically corrugated helical tubes are investigated. Numerical calculations are carried out to study the influence of different corrugation depths and corrugation pitches on the Nusselt number and friction coefficient, considering a mass flow rate range of 0.75–1 kg/s. The results indicate that the sudden change in the flow velocity generated by the corrugated structure and the secondary flow generated by the helical structure improve both heat transfer efficiency and flow resistance. Furthermore, the Quasi-Monte Carlo method is employed to calculate the heat transfer reliability when the structural parameters and mass flow rates are regarded as random variables in practical engineering. To improve the calculation efficiency, the Kriging model is proposed to replace the numerical calculation process for predicting the Nusselt number. The results demonstrated that the Kriging model has high accuracy and efficiency in predicting the Nusselt number and calculating the heat transfer reliability. When the mean values of corrugation depth, corrugation pitch, inlet diameter, and mass flow rate are 1.5 mm, 32 mm, 16 mm, and 0.85 kg/s, respectively, the heat transfer reliability of the helically corrugated helical tube was found to be 99.28% under the condition that the error limit is 7.5% of the mean value.

Seismic reliability evaluation of spatially correlated pipeline networks by quasi-Monte Carlo simulation

The reliability assessment of pipeline networks under nature disasters helps to improve the disaster prevention capability and overall resilience of pipeline networks. For the earthquake hazard, the seismic damage of pipelines in the network is usually correlated due to the spatial correlation of critical parameters, including characteristics of pipeline, soil properties and ground motions. This paper proposes a framework to evaluate the seismic reliability of pipeline network under spatially correlated parameters based on the quasi-Monte Carlo (QMC) simulation. The influence of spatial correlated random variables on the seismic reliability of pipeline network is investigated with explicit consideration of the correlated random variables including the peak ground velocity, the wall thickness and the yield stress of pipe segment. The framework has been implemented in seismic reliability evaluation of two pipeline networks. The result shows that the QMC method has better simulation accuracy than the standard MC method under same sampling numbers. The spatial correlations have different influences on the seismic connectivity reliability of pipeline networks with different topological redundancy and parallelism of connective paths from sources to user nodes. For the pipeline network with greater redundancy and parallelism, the model without considering the spatial correlations overestimates the network connectivity reliability.

Stochastic Static Analysis of Planar Elastic Structures with Multiple Spatially Uncertain Material Parameters

Engineering structures are often assembled from parts with different materials. When uncertainty quantification techniques are applied, the curse of dimensionality increases the computational complexity. Here, a stochastic Galerkin method for planar elasticity allowing for multiple regions with independent uncertain materials is introduced. The method allows for efficient solution of linear systems both in fully assembled and matrix-free formulations. The selection of the stochastic basis polynomials is performed using a priori knowledge of the decay of the random fields. The statistical quantities of interest are the expected solution and variance, both of which can be computed efficiently after the Galerkin system has been solved. Analysis of the results indicates that the proposed method is highly efficient in terms of both computational resource requirements and discretization of the stochastic dimensions. The results were verified with Monte Carlo and quasi-Monte Carlo methods.

Modeling ultrasonic wave fields using a Quasi-Monte Carlo method: Wave transmission through complicated interfaces.

The sound fields generated by ultrasonic transducers can be modeled using the Quasi-Monte Carlo (QMC) method with a high level of accuracy and efficiency from Zhang [J. Acoust. Soc. Am. 149(1), 7-15 (2021)]. In this work, this method is extended to simulate transmitted wave fields through complicated interfaces. When a wave propagates in two-layer media, the vibrating waves over the interface radiated by the transducer can be treated as the source for generating waves in the second medium, thus, a nested-form Rayleigh integral expression can be used as a model equation for the transmitted wave calculation. When the QMC method is used to solve the nested integral, pseudo-random samples for constructing the transducer and the interface are sampled separately and the transmitted wave fields are obtained using the final sample mean. Numerical examples and results are presented when the wave transmits normally or obliquely through planar or curved interfaces. The results indicate that the high level of accuracy and efficiency remains when the QMC method is used to model the transmitted wave fields. One important advantage is that wave fields can be well simulated using the QMC method when the wave transmits through a complicated interface as long as the interface can be constructed using pseudo-random samples.