quasi-Monte Carlo Methods Research Articles

Influence of manufacturing errors on the aerodynamic working stability of axial flow compressors

In turbomachinery, manufacturing errors in blades have been found to impact aerodynamic performance. This paper aims to investigate the effect of manufacturing errors on the stability of compressor operation. To this end, a geometric uncertainty reduced-order model is developed to generate the normal profile error of rotor blades based on specific tolerances, considering the simultaneous variation of blade parameters during manufacturing. The stability margin improvement of the compressor is then calculated using computational fluid dynamics (CFD), and a neural network is employed to predict the relationship between the uncertainty input variables and the stability margin improvement. Finally, a large number of samples are generated using the Quasi-Monte Carlo method, and Sobol sensitivity analysis is performed. According to the results, manufacturing errors can reduce the stability margin by an average of 0.9% and up to 3% in the limiting case. The parameters that have the greatest impact on the stability of the compressor are the blade chord length of the hub section and the maximum thickness of the tip section. These two parameters affect blade tip blockage by influencing spanwise secondary flow on the suction surface. Additionally, the decrease in maximum thickness at the tip section affects the separation of the boundary layers on the suction surface, resulting in additional effects.

Stabilized weighted reduced order methods for parametrized advection-dominated optimal control problems governed by partial differential equations with random inputs

Abstract In this work, we analyze Parametrized Advection-Dominated distributed Optimal Control Problems with random inputs in a Reduced Order Model (ROM) context. All the simulations are initially based on a finite element method (FEM) discretization; moreover, a space-time approach is considered when dealing with unsteady cases. To overcome numerical instabilities that can occur in the optimality system for high values of the Péclet number, we consider a Streamline Upwind Petrov–Galerkin technique applied in an optimize-then-discretize approach. We combine this method with the ROM framework in order to consider two possibilities of stabilization: Offline-Only stabilization and Offline-Online stabilization. Moreover we consider random parameters and we use a weighted Proper Orthogonal Decomposition algorithm in a partitioned approach to deal with the issue of uncertainty quantification. Several quadrature techniques are used to derive weighted ROMs: tensor rules, isotropic sparse grids, Monte-Carlo and quasi Monte-Carlo methods. We compare all the approaches analyzing relative errors between the FEM and ROM solutions and the computational efficiency based on the speedup-index.

Dimension reduction for Quasi-Monte Carlo methods via quadratic regression

Quasi-Monte Carlo (QMC) methods have been gaining popularity in computational finance as they are competitive alternatives to Monte Carlo methods that can accelerate numerical accuracy. This paper develops a new approach for reducing the effective dimension combined with a randomized QMC method. A distinctive feature of the proposed approach is its sample-based transformation that enables us to choose a flexible manipulation via regression. In the proposed approach, the first step is to perform a regression using the samples to estimate the parameters of the regression model. An optimal transformation is proposed based on the regression result to minimize the effective dimension. An advantage of this approach is that adopting a statistical approach allows greater flexibility in selecting the regression model. In addition to a linear model, this paper proposes a dimension reduction method based on a linear-quadratic model for regression. In numerical experiments, we focus on pricing different types of exotic options to test the effectiveness of the proposed approach. The numerical results show that different regression models are chosen depending on the underlying risk process and the type of derivative securities. In particular, we show several examples where the proposed method works while existing dimension reductions are ineffective.

Groundwater LNAPL Contamination Source Identification Based on Stacking Ensemble Surrogate Model

Groundwater LNAPL (Light Non-Aqueous Phase Liquid) contamination source identification (GLCSI) is essential for effective remediation and risk assessment. Addressing the GLCSI problem often involves numerous repetitive forward simulations, which are computationally expensive and time-consuming. Establishing a surrogate model for the simulation model is an effective way to overcome this challenge. However, how to obtain high-quality samples for training the surrogate model and which method should be used to develop the surrogate model with higher accuracy remain important questions to explore. To this end, this paper innovatively adopted the quasi-Monte Carlo (QMC) method to sample from the prior space of unknown variables. Then, this paper established a variety of individual machine learning surrogate models, respectively, and screened three with higher training accuracy among them as the base-learning models (BLMs). The Stacking ensemble framework was utilized to integrate the three BLMs to establish the ensemble surrogate model for the groundwater LNAPL multiphase flow numerical simulation model. Finally, a hypothetical case of groundwater LNAPL contamination was designed. After evaluating the accuracy of the Stacking ensemble surrogate model, the differential evolution Markov chain (DE-MC) algorithm was applied to jointly identify information on groundwater LNAPL contamination source and key hydrogeological parameters. The results of this study demonstrated the following: (1) Employing the QMC method to sample from the prior space resulted in more uniformly distributed and representative samples, which improved the quality of the training data. (2) The developed Stacking ensemble surrogate model had a higher accuracy than any individual surrogate model, with an average R2 of 0.995, and reduced the computational burden by 99.56% compared to the inversion process based on the simulation model. (3) The application of the DE-MC algorithm effectively solved the GLCSI problem, and the mean relative error of the identification results of unknown variables was less than 5%.

Statistical sampling method to verify the homogeneity of full-scale cement-solidified radioactive waste

Homogeneity is an important factor for ensuring the structural stability of solidified radioactive waste, and the most effective approach for assessing its homogeneity is by performing compressive strength measurements using the minimum amount of coring specimens. The efficiency of detecting inhomogeneous waste is affected by the coring position and number of coring positions. However, no guidelines exist for coring solidified waste for compressive-strength tests. Therefore, this study compared uniform, random, and quasi-Monte Carlo sampling methods to determine the most effective core position. Further, the effects of different sampling amounts on the detection rate of inhomogeneous solidified waste were observed, and the detection rate of the inhomogeneous waste was obtained by modeling the coring procedure of solidified radioactive waste using MATLAB. Thus, a sampling method and a method for increasing the specimen amount, both of which can efficiently detect inhomogeneous waste during compressive strength tests, were presented in this paper. The results of this study can be applied as background data for developing homogeneity assessment guidelines for solidified radioactive waste.

Markov chain quasi-Monte Carlo method for forecasting fire hotspots in Sarawak, Malaysia.

Stochastic modeling approaches have attracted many researchers to the field. However, fire hotspot detection suffers from not using a Markov chain quasi-Monte Carlo (MCQMC) as a forecasting methodology. This paper proposes improvements to the computational time by combining the strengths of the Markov chain Monte Carlo (MCMC) and quasi-Monte Carlo (QMC) methods. The proposed method can lead to more precise and stable results, particularly in problems with high-dimensional integration or complex probability distributions. The proposed method is applied to a case study of fire hotspot detection in Sarawak, Malaysia. The outcome of this study reveals that the MCQMC method is more computationally efficient, taking only 0.0746seconds compared to MCMC's 0.0914seconds and QMC's 0.0994seconds. It is shown that the best option derived by the proposed method is effective in predicting fire hotspots and providing quick solutions to protect the environment and communities from forest fires.

Deep learning based on randomized quasi-Monte Carlo method for solving linear Kolmogorov partial differential equation

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations (PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization (ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is O(n−1+ϵ) for arbitrarily small ϵ>0, while for the MC method it is O(n−1/2+ϵ) for arbitrarily small ϵ>0. Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as n increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative L2 error than that based on the MC method.

Dominant patterns in small directed bipartite networks: ubiquitous generalized tripod gait

The synchronization patterns exhibited by small networks of neurons that regulate biological processes (CPGs) have aroused growing scientific interest. In many of these networks there is a main behavioral pattern within the parameter space. In particular, in the context of insect locomotion, tripod walking stands out as a predominant pattern, both in natural observations (where insects walk on tripod gait) and in mathematical models. This predominance appears to be stable under parameter variations within the network, suggesting a possible correlation with the underlying network topology. Tripod walking can be naturally extended to all CPGs with a bipartite connectivity. Then a natural question arises: Are “generalized tripod gaits” equally dominant among synchronization patterns within those networks? To investigate this, we carried out a comprehensive study covering all bipartite networks of up to nine neurons. For each of those networks we numerically explore the phase space using a quasi-MonteCarlo method to see what are the main synchronization patterns that the network can achieve. Then, all those patterns are grouped according to their dynamics. Generalized tripod gait was observed in all cases examined as the dominant pattern again. However, certain cases revealed additional stable patterns, mainly associated with the 3-colorings of the respective graph structures.

Exploring spatiotemporal patterns of algal cell density in lake Dianchi with explainable machine learning

The escalating global occurrence of algal blooms poses a growing threat to ecosystem services. In this study, the spatiotemporal heterogeneity of water quality parameters was leveraged to partition Lake Dianchi into three clusters. Considering water quality parameters and both the delayed and instantaneous effects of meteorological factors, ensemble learning, and quasi-Monte Carlo methods were employed to predict daily algal cell density (AD) between January 2021 and January 2024. Consistently, superior predictive accuracy across all three clusters was exhibited by the Stacking-Elastic-Net regularization model. Furthermore, the minimum combination of drivers that achieved near-optimal accuracy for each cluster was identified, striking a balance between accuracy and cost. The ranking of the effect of drivers on AD varied by cluster, while the delayed effect of meteorological factors on AD generally outweighed their instantaneous effect for all clusters. Additionally, the heterogeneous or homogeneous thresholds and responses between drivers and AD were explored. These findings could serve as a scientific and cost-effective basis for government agencies to develop regional sustainable strategies for managing water quality.

Sufficient Conditions for Central Limit Theorems and Confidence Intervals for Randomized Quasi-Monte Carlo Methods

Randomized quasi-Monte Carlo methods have been introduced with the main purpose of yielding a computable measure of error for quasi-Monte Carlo approximations through the implicit application of a central limit theorem over independent randomizations. But to increase precision for a given computational budget, the number of independent randomizations is usually set to a small value so that a large number of points are used from each randomized low-discrepancy sequence to benefit from the fast convergence rate of quasi-Monte Carlo. While a central limit theorem has been previously established for a specific but computationally expensive type of randomization, it is also known in general that fixing the number of randomizations and increasing the length of the sequence used for quasi-Monte Carlo can lead to a non-Gaussian limiting distribution. This paper presents sufficient conditions on the relative growth rates of the number of randomizations and the quasi-Monte Carlo sequence length to ensure a central limit theorem and also an asymptotically valid confidence interval. We obtain several results based on the Lindeberg condition for triangular arrays and expressed in terms of the regularity of the integrand and the convergence speed of the quasi-Monte Carlo method. We also analyze the resulting estimator’s convergence rate.

Multilevel Monte Carlo Methods for Stochastic Convection–Diffusion Eigenvalue Problems

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection–diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov–Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

Randomized Quasi-Monte Carlo Methods on Triangles: Extensible Lattices and Sequences

Randomized Quasi-Monte Carlo Methods on Triangles: Extensible Lattices and Sequences

Performance analysis of electrical signal output of multi-state flexoelectric structures with parameter uncertainties through quasi-Monte Carlo method

Flexoelectric effect is a more universal electromechanical coupling effect than piezoelectric effect. Flexoelectric beams as the main structural component of flexoelectric power signal output have broad application prospects in the next generation of micro–nano electromechanical systems. However, the electrical signal output of flexoelectric structures in macro-scale is far less than the output of the piezoelectric signal. Therefore, it is urgent to explore the influence of the parameter uncertainties on the electrical signal output of the flexoelectric structures, in order to improve the electrical signal output of flexoelectric materials with excellent design performance. Based on the quasi-static theory, the output voltage model and the output charge model of flexoelectric structures as well as the effective piezoelectric coefficient model are constructed. Then the influences of the flexoelectric parameters on the output voltage and the output charge are researched as well as the influence of the effective piezoelectric coefficient. Finally, the influences of uncertain parameters under different electrical states (e.g. the electrical open circuit and short circuit states) on the output performance of flexoelectric signal are studied by the quasi-Monte Carlo method, in order to further provide a reference for the reliability analysis and optimization design of the flexoelectric structures.

Efficient Methods for Reliability Sensitivity Analysis of Distribution Parameters and Their Application

To efficiently evaluate the influence of the distribution parameters of the input variables on the failure probability of engineering structures and improve the reliability and safety of engineering structures in a targeted manner, new methods for the global reliability sensitivity analysis (RSA) of distribution parameters are proposed in this study based on the cubature formula (CF), surrogate sampling probability density function (SSPDF), and quasi-Monte Carlo (QMC) method. By introducing CF, the proposed methods can effectively improve the computational efficiency of the nested expectation and variance operators in the reliability sensitivity indices of the distribution parameters. Based on the concept of SSPDF, a surrogate importance sampling probability density function was developed. This not only overcomes the problem of the computational effort of propagating parameter uncertainty to the failure probability function (FPF), which depends on the dimensionality of the parameters; it also further improves the efficiency of the RSA of the parameters in the case of a small failure probability. Finally, by incorporating the idea of the QMC method, the process of calculating the reliability sensitivity indices of the parameters is reduced from a double-loop to a single-loop one. Three engineering examples are used in this study to demonstrate the efficiency and accuracy of the new algorithms.

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

Probabilistic analysis of bolted joint anti-loosening under cyclic transverse load

The loosening of bolted joints under cyclic transverse load is a crucial failure mode, and the uncertainty of the relevant parameters of the bolted joint will affect the loosening state. However, there are few bolt-loosening probability analysis and evaluation methods based on parameter uncertainty in previous studies. In this paper, different loosening states are analyzed based on the relationship between pitch torque, thread friction torque, and bearing friction torque. Considering the uncertainty of process parameters, a probabilistic analysis method of the bolted joint anti-loosening under cyclic transverse load is proposed. In the proposed method, the finite element model verified by the fastening experiment is used to verify the accuracy of the bolted joints’ loosening criterion. Then, the loosening criterion is used as the Limit State Function (LSF). According to the LSF, an Adaptive Kriging-Kernel Density Importance Sampling (AK-KDIS) method is proposed for solving the anti-loosening reliability. This method combines adaptive Kriging model with the importance sampling method, effectively reducing the number of sampling samples and the number of LSF calls. The classical and robust Quasi-Monte Carlo (QMC) method is also used to evaluate the reliability of anti-loosening to verify the accuracy of the proposed AK-KDIS method. Finally, the practical application of probabilistic analysis of anti-loosening is presented, and the effectiveness of the proposed method is analyzed in detail. The results indicate that the AK-KDIS method has higher efficiency while ensuring calculation accuracy compared with the QMC method.

Probabilistic evaluation of dynamic positioning operability with a Quasi-Monte Carlo approach

During the design phase of an offshore unit, estimating the station-keeping capabilities of the dynamic positioning (DP) system is mandatory. This means, in conventional offshore applications, to determine the maximum sustainable wind speed as a function of the encounter heading, which the unit may counteract by employing the onboard actuators or mooring lines only. Besides the deterministic estimation of DP capability, it is possible to assess the operability of the DP system following a non-deterministic probabilistic process by employing the site-specific joint wind-wave distributions to model the environment. In such a case, the operability results from a Monte Carlo integration process. Here it is proposed to enhance the applicability of the probabilistic analysis of DP operability, investigating the application of a Quasi-Monte Carlo method. In this sense, the procedure uses quasi-random samplings following a Sobol sequence instead of employing random samples of the joint distributions. In this paper, the Quasi-Monte Carlo process is tested and compared on a reference ship, highlighting the improvements to the established probabilistic DP prediction concerning the number of calculations needed to estimate operability. The significant reduction of computational time makes the newly implemented method suitable for the early design stage applications.

PRICING OF CALL OPTIONS USING THE QUASI MONTE CARLO METHOD

A call option is a type of option that grants the option holder the right to buy an asset at a specified price within a specified period of time. Determining the option price period of time within a certain period of time is the most important part of determining an investment strategy. Various methods can be employed to determine the prices of options, such as Quasi-Monte Carlo and Monte Carlo simulations. The purpose of this research is to determine the price of European-type call options using the Quasi-Monte Carlo method. The data used is daily stock closing price data on the Apple Inc. for the period October 1, 2021, to September 30, 2022. Apple Inc. stock options in this study were chosen because it is the largest technology company in the world in 2022. The steps taken in this study are to determine the parameters obtained from historical data such as the initial risk-free interest rate (r), stock price (, volatility , maturity time (T), and strike price (K). Next is to generate Halton’s quasi-randomized sequence and simulate the stock price by substituting the parameters by substituting the parameters. Then proceed to calculate the call option payoff and estimate the call option price by averaging the call option payoff values. The results showed that the call option price of the company Apple Inc. using the Quasi-Monte Carlo with Halton’s quasi-randomized sequence on the 1000000th simulation with a standard error of 0,045 is $90,163. The call option price obtained can be used as a reference for investors in purchasing options to minimize losses from call option investments in that period.

Truncated Cone – A Universal Primitive Shape for Axisymmetric View Factor Systems

Abstract It is demonstrated that axisymmetric view factor systems can be modeled as a composite of truncated cones. For the hemisphere primitive shape, it is represented as a composite of truncated cones. One conclusion of the investigation is eight truncated cones are required to give a reasonable approximation to a hemisphere. The sensitivity of the run-time for the Monte Carlo methods to the number of surfaces is investigated and the run-time of the Monte Carlo method combined with ray tracing scales as the square of the number of surfaces, whereas the run-time of the hybrid Monte Carlo method scales in a weakly linear way with the number of surfaces. Representing a hemisphere with eight surfaces, for the view factor system considered and an root mean squared error (RMS) threshold of 0.001 the hybrid Monte Carlo method and quasi-Monte Carlo method have a speed-up of 8.3 and 55 compared to the Monte Carlo method with ray tracing.

Functional quantization of rough volatility and applications to volatility derivatives

We develop a product functional quantization of rough volatility. Since the optimal quantizers can be computed offline, this new technique, built on the insightful works by [Luschgy, H. and Pagès, G., Functional quantization of Gaussian processes. J. Funct. Anal., 2002, 196(2), 486–531; Luschgy, H. and Pagès, G., High-resolution product quantization for Gaussian processes under sup-norm distortion. Bernoulli, 2007, 13(3), 653–671; Pagès, G., Quadratic optimal functional quantization of stochastic processes and numerical applications. In Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 101–142, 2007 (Springer: Berlin Heidelberg)], becomes a strong competitor in the new arena of numerical tools for rough volatility. We concentrate our numerical analysis on the pricing of options on the VIX and realized variance in the rough Bergomi model [Bayer, C., Friz, P.K. and Gatheral, J., Pricing under rough volatility. Quant. Finance, 2016, 16(6), 887–904] and compare our results to other benchmarks recently suggested.