quasi-Monte Carlo Methods Research Articles

Risk assessment of voltage overrun probability of photovoltaic access distribution system voltage based on quasi-Monte Carlo method

With the rapid development of renewable energy, distributed PV access to the distribution grid has become a common form. However, distributed PV access to the distribution network (DN) may cause voltage overrun problems, which brings challenges to the stability and reliability of the DN. In order to reasonably assess the impact of PV grid connection, the paper proposes a probabilistic trend calculation based on the quasi-Monte Carlo mode method to obtain the node voltage overrun probability under the premise of considering the stochastic nature of PV power generation and load fluctuation, and combines with the utility preference exponential function to characterize the overrun severity, to construct a comprehensive assessment model of voltage overrun probability risk. Taking the IEEE33 node system as an example, the simulation results show that the rational selection of access points effectively reduces the risk of overrun of the whole DN, and improves the safety of the distribution network.

Air Pollution Modeling of UNI-DEM Model by using Innovative Stochastic Approaches

In this paper we develop advanced multidimensional sensitivity analysis, based on some innovative stochastic approaches for performing air pollution modelling on a large-scale model of long-range transport of air pollutants. The Unified Danish Eulerian Model (UNI-DEM) is a very important mathematical model, and can be applied in many different studies related to damaging effects caused by high air pollution levels. We shall use it in this paper to get a reliable answer to a some very important questions regrading environmental protection. We develop some advanced Monte Carlo and quasi-Monte Carlo methods, based on special lattice and digital sequences. In this paper we will improve the digital ecosystem modeling by improving the existence stochastic approaches. The computational efficiency (in terms of relative error and computational time) of the advanced Monte Carlo algorithms for multidimensional numerical integration has been studied to analyze the sensitivity of UNI-DEM model output to variation of input emissions of the anthropogenic pollutants and of rates of several chemical reactions. The algorithms will be applied to compute global Sobol sensitivity measures corresponding to the influence of several input parameters on the concentrations of important air pollutants. The study will be done for the areas of several European cities with different geographical locations.

Application of high-credible statistical results calculation scheme based on least squares Quasi-Monte Carlo method in multimodal stochastic problems

Solving stochastic problems based on multimodal distributions with high accuracy and efficiency is the focus of current research in the stochastic field. However, since exact statistical results in practical engineering problems are unknown, how to efficiently obtain statistical results with high credibility is of great importance. Therefore, a high-credible statistical results calculation scheme is proposed in this paper. First, the multimodal model is transformed into a weighted sum form of multiple unimodal models by Gaussian mixture model. Then, a high-credible statistical results calculation scheme is designed based on the reusability of low-discrepancy sequences in the quasi-Monte Carlo (QMC) method. The scheme calculates the coefficient of variation for the random response statistical results of the unimodal model across various sample ranges. When the coefficient of variation is less than the tolerance error, which indicates low fluctuation, the statistical results are deemed to have converged to the exact solution. In order to further accelerate the calculation efficiency of the proposed scheme, this paper proposes a least squares quasi-Monte Carlo (LSQMC) method to yield high accuracy statistical moments. Compared with the averageness weights in the QMC method, the non-averageness weights obtained by LSQMC method can more effectively reflect the numerical Characteristics of the random variables and the location distribution of the sampling points in the probability space. Finally, numerical examples have demonstrated that the high-credible statistical results calculation scheme based on the LSQMC method can obtain the high accuracy statistical moments with high efficiency in multimodal stochastic problems.

Reliability-oriented global sensitivity analysis using subset simulation and space partition

This paper presents a novel reliability-oriented global sensitivity analysis method using the Subset Simulation (SS) method and the Space Partition (SP) scheme. It can exploit the inherent information of uncertainty within all the conditional samples in each simulation level of SS along the spirit of the SP scheme. The reasoning of the first order indices is firstly provided, followed by a heuristic study on the influence of parametric settings upon the statistical properties of the indices. By extending the calculation process to the higher order indices and formally optimizing the partition scheme considering the coefficient of variation of all the indices, the framework of the proposed reliability-oriented global sensitivity for arbitrary order of indices can be formed. The performance of the direct Monte Carlo method, the Quasi-Monte Carlo method, SP and the proposed method is compared through three numerical examples and two engineering applications, to demonstrate the merits of the proposed one.

Variability Analysis of the Hysteretic Behavior of Fiber-Reinforced Polymer (FRP)-Confined Concrete Columns Based on a Secondary Development Model

A reasonable material hysteretic constitutive model has a significant influence on the seismic simulation results of structures. To better describe the hysteresis seismic performance of fiber-reinforced polymer (FRP)-constrained concrete, a new modified hysteresis constitutive model is proposed based on the existing model and with sufficient consideration of the drop section of the skeleton curve. The validity of the proposed model is verified by comparing with quasi-static experimental data of FRP-confined reinforced concrete (FRP-C RC) columns in the literature. Subsequently, the compressive strength of concrete is selected as a major variable, and a quasi-Monte Carlo method is utilized to generate random samples, which are substituted into the proposed modified model and some comparison models. Finally, the hysteretic behavior of FRP-C RC columns is analyzed from the perspective of the material strength variability. The results demonstrate that (1) The proposed hysteretic constitutive model is able to provide rational predictions of the hysteretic behavior of FRP-C RC columns, and the mean relative error of each specimen is less than 6%. It can be applied to carbon FRPs (CFRPs) and glass FRPs (GFRPs), as well as different cross-sectional forms such as cylindrical and square columns. (2) A large number of hysteretic behavior cases of FRP-C RC columns can be successfully analyzed from the perspective of concrete material variability combined with finite element software. The average and variation coefficient of the maximum horizontal force of FRP-C reinforced C30 concrete columns are 76.77 kN and 0.0488, respectively, while the average and variation coefficient of the maximum horizontal force of FRP-C reinforced C50 concrete columns are 91.14 kN and 0.0454, respectively. (3) The average value and variation coefficient of the maximum horizontal force and equivalent damping ratio of FRP-C RC columns are affected by the compressive strength, axial compression ratio and reinforcement ratio, which show a certain regularity.

An efficient quasi-Monte Carlo method with forced fixed detection for photon scatter simulation in CT.

Detected scattered photons can cause cupping and streak artifacts, significantly degrading the quality of CT images. For fast and accurate estimation of scatter intensities resulting from photon interactions with a phantom, we first transform the path probability of photons interacting with the phantom into a high-dimensional integral. Secondly, we develope a new efficient algorithm called gQMCFFD, which combines graphics processing unit(GPU)-based quasi-Monte Carlo (QMC) with forced fixed detection to approximate this integral. QMC uses low discrepancy sequences for simulation and is deterministic versions of Monte Carlo. Numerical experiments show that the results are in excellent agreement and the efficiency improvement factors are 4 ∼ 46 times in all simulations by gQMCFFD with comparison to GPU-based Monte Carlo methods. And by combining gQMCFFD with sparse matrix method, the simulation time is reduced to 2 seconds in a single projection angle and the relative difference is 3.53%.

Efficient machine learning algorithm with enhanced cat swarm optimization for prediction of compressive strength of GGBS-based geopolymer concrete at elevated temperature

In order to assess building damage and develop fire safety applications, it is crucial to examine the mechanical behavior of concrete after exposure to high temperatures. Compressive strength is the most crucial mechanical characteristic of concrete which is vital for the quality assurance of engineering structures. But it is difficult to estimate the compressive strength with accuracy after geopolymer concrete (GPC) is exposed to extreme temperatures. The residual compressive strength of GGBS-based GPC is predicted in this study using a novel enhanced cat swarm-optimized extreme learning machine (ELM-ECSO) model. The ELM-ECSO compared executing statistical parameters such as the determination coefficient (R2), adjusted determination coefficient (Adj. R2), mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE) with simple ELM and ELM with cat swarm optimized to check its generalizability. To create a dataset for training and testing of the model, several experiments using various mix designs were carried out at high temperatures. The quasi-Monte Carlo method was implemented for the sensitivity analysis of the model. Assessment results showed that the ELM-ECSO model transcends the other two models owing to the highest prediction accuracy and showing the least error. Shapiro-Wilk statistical test was carried out to compare all the models. The sensitivity analysis indicated temperature exposure and curing age as the most impactful parameter while predicting the residual compressive strength of GPC. GPC with a Na2SiO3/NaOH ratio of 2.5 and alkaline solution/GGBS ratio of 0.40 shows better resilience against higher temperatures.

An Active Learning Strategy for Joint Surrogate Models Construction with Compatibility Conditions: Application to VPP

Static Velocity Prediction Programs (VPP) are standard tools in sailing yachts’ design and performance assessment. Predicting the maximal steady velocity of a yacht involves resolving constrained optimization problems. These problems have a prohibitive computational cost when using high-fidelity global modeling of the yacht. This difficulty has motivated the introduction of modular approaches, decomposing the global model into subsystems modeled independently and approximated by surrogate models (response surfaces). The maximum boat speed for prescribed conditions solves an optimization problem for the trimming parameters of the model constrained by compatibility conditions between the subsystems’ surrogate solution (e.g., the yacht equilibrium). The accuracy of the surrogates is then critical for the quality of the resulting VPP. This paper relies on Gaussian Process (GP) models of the subsystems and introduces an original sequential Active Learning Method (ALM) for their joint construction. Our ALM exploits the probabilistic nature of the GP models to decide the enrichment of the training sets using an infilling criterion that combines the predictive uncertainty of the surrogate models and the likelihood of equilibrium at every input point. The resulting strategy enables the concentration of the computational effort around the manifolds where equilibrium is satisfied. The results presented compare ALM with a standard (uninformed) Quasi-Monte Carlo method, which samples the input space of the subsystems uniformly. ALM surrogates have higher accuracy in the equilibrium regions for equal construction cost, with improved mean prediction and reduced prediction uncertainty. We further investigate the effect of the prediction uncertainty on the numerical VPP and in a routing problem.

Difference sets in pseudocyclic association schemes

Wilson (1972) proved that for fixed positive integers d and r such that 2d divides r(r−1), there exists a subset Y of size r of the finite field of order q such that the r(r−1) nonzero differences occurring from Y are evenly distributed to the dth cyclotomic classes provided that q is sufficiently large relative to d and r. On the other hand, Kajiura, Matsumoto and Okuda (2019) introduced the concept of difference sets in commutative association schemes in their study on a quasi-Monte Carlo method to approximate integrations over the point sets. Then, Wilson's result implies the existence of difference sets in cyclotomic schemes. In this paper, we generalize Wilson's result into pseudocyclic commutative association schemes and show the existence of difference sets in pseudocyclic association schemes with sufficiently large numbers of points.

A Review of Representative Points of Statistical Distributions and Their Applications

Statistical modeling relies on a diverse range of statistical distributions, encompassing both univariate and multivariate distributions and/or discrete and continuous distributions. In the literature, numerous statistical methods have been proposed to approximate continuous distributions. The most commonly used approach is the use of the empirical distribution which is obtained from a random sample drawn from the distribution. However, it is very likely that the empirical distribution suffers from an accuracy problem when used to approximate the underlying distribution, especially if the sample size is not sufficient. In order to improve statistical inferences, various alternative forms of discrete approximation to the distribution were proposed in the literature. The choice of support points for the discrete approximation, known as Representative Points (RPs), becomes extremely important in terms of distribution approximations. In this paper we give a review of the three main methods for constructing RPs, namely based on the Monte Carlo method, the number-theoretic method (or quasi-Monte Carlo method), and the mean square error method, aiming to introduce such important methods to the statistical or mathematical community. Additional approaches for forming RPs are also briefly discussed. The review focuses on certain critical aspects such as theoretical properties and computational algorithms for constructing RPs. We also address the issue of the application of RPs through studying practical problems and provide evidence of RPs’ advantages over random samples in approximating the distribution.

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: regularity and error analysis

Abstract Stochastic partial differential equation (PDE) eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration analysis, the criticality of a nuclear reactor or photonic crystal structures. In this paper we present a multilevel quasi-Monte Carlo (MLQMC) method for approximating the expectation of the minimal eigenvalue of an elliptic eigenvalue problem with coefficients that are given as a series expansion of countably-many stochastic parameters. The MLQMC algorithm is based on a hierarchy of discretizations of the spatial domain and truncations of the dimension of the stochastic parameter domain. To approximate the expectations, randomly shifted lattice rules are employed. This paper is primarily dedicated to giving a rigorous analysis of the error of this algorithm. A key step in the error analysis requires bounds on the mixed derivatives of the eigenfunction with respect to both the stochastic and spatial variables simultaneously. Under stronger smoothness assumptions on the parametric dependence, our analysis also extends to multilevel higher-order quasi-Monte Carlo rules. An accompanying paper (Gilbert, A. D. & Scheichl, R. (2023) Multilevel quasi-Monte Carlo methods for random elliptic eigenvalue problems II: efficient algorithms and numerical results. IMA J. Numer. Anal.) focusses on practical extensions of the MLQMC algorithm to improve efficiency and presents numerical results.

A fitting program for structural determination of molecular clusters from rotational spectroscopy

The characterization of the structures of molecular clusters, which serve as building blocks for bulk substances, provides crucial insight into the interactions between constituent units. Chirped-pulse Fourier transform microwave (CP-FTMW) spectroscopy, combined with state-of-the-art quantum chemical calculations, is a powerful tool for characterizing the structures of molecular clusters, as the rotational spectra are directly related to the mass distribution of a molecule or cluster. However, determining the structures of large or complex clusters from experimental rotational spectra remains challenging due to their structural flexibility. Ab initio and density functional theory calculations for searching their stable structures could be significantly time-consuming and method-dependent. To address these challenges, we have developed an approach that relies on the experimental rotational constants to search for potential molecular structures without quantum chemical optimization. Our approach involves creating an initial set of conformers through either a semi-empirical sampling program or the quasi-Monte Carlo method. After-ward, the trust region reflective algorithm is utilized for structure fitting. This procedure enables us to quickly generate potential conformers and gain access to precise structural information. We apply our fitting program to water hexamer and benzaldehyde-water clusters, and the resulting topological structures align extremely well with the experimental results.

Research on Seismic Connectivity Reliability Analysis of Water Distribution System Based on CUDA

To improve the seismic connectivity reliability (SCR) analysis efficiency of water distribution systems (WDS) based on Monte Carlo (MC) simulation, the quasi-Monte Carlo (QMC) method sampled by a low-discrepancy sequence is applied. Furthermore, a parallel algorithm combined with the breadth-first search algorithm for SCR analysis of WDS based on the QMC method and Compute Unified Device Architecture (CUDA) platform was proposed. A city WDS was taken as a computational example, the accuracy and efficiency of the traditional MC algorithm and parallel algorithm were compared, and the influence of the Sobol sequence and pseudo-random number sequence was analysed. The analysis results show that when 1,000,000 simulations are performed, the maximum error of the calculation results of the two methods is 0.2%, and the parallel method can obtain a six-fold speedup ratio compared with the serial method, indicating that the proposed parallel method is correct, meets the accuracy requirements, and helps to improve the SCR analysis efficiency. When the number of simulations is the same, the simulation results based on the Sobol sequence are more accurate than those based on the pseudo-random number sequence. The proposed parallel method also achieves a good acceleration effect in the SCR analysis of large-scale WDS.

Wavenumber-Explicit Parametric Holomorphy of Helmholtz Solutions in the Context of Uncertainty Quantification

A crucial role in the theory of uncertainty quantification (UQ) of PDEs is played by the regularity of the solution with respect to the stochastic parameters; indeed, a key property one seeks to establish is that the solution is holomorphic with respect to (the complex extensions of) the parameters. In the context of UQ for the high-frequency Helmholtz equation, a natural question is therefore: how does this parametric holomorphy depend on the wavenumber ? The recent paper [35] showed for a particular nontrapping variable-coefficient Helmholtz problem with affine dependence of the coefficients on the stochastic parameters that the solution operator can be analytically continued a distance into the complex plane. In this paper, we generalize the result in [35] about -explicit parametric holomorphy to a much wider class of Helmholtz problems with arbitrary (holomorphic) dependence on the stochastic parameters; we show that in all cases the region of parametric holomorphy decreases with and show how the rate of decrease with is dictated by whether the unperturbed Helmholtz problem is trapping or nontrapping. We then give examples of both trapping and nontrapping problems where these bounds on the rate of decrease with of the region of parametric holomorphy are sharp, with the trapping examples coming from the recent results of [31]. An immediate implication of these results is that the -dependent restrictions imposed on the randomness in the analysis of quasi-Monte Carlo methods in [35] arise from a genuine feature of the Helmholtz equation with large (and not, for example, a suboptimal bound).

Implementation of the self-consistent phonons method with abinitio potentials (AI-SCP).

The self-consistent phonon (SCP) method allows one to include anharmonic effects when treating a many-body quantum system at thermal equilibrium. The system is then described by an effective temperature-dependent harmonic Hamiltonian, which can be used to estimate its various dynamic and static properties. In this paper, we combine SCP with abinitio (AI) potential energy evaluation in which case the numerical bottleneck of AI-SCP is the evaluation of Gaussian averages of the AI potential energy and its derivatives. These averages are computed efficiently by the quasi-Monte Carlo method utilizing low-discrepancy sequences leading to a fast convergence with respect to the number, S, of the AI energy evaluations. Moreover, a further substantial (an-order-of-magnitude) improvement in efficiency is achieved once a numerically cheap approximation of the AI potential is available. This is based on using a perturbation theory-like (the two-grid) approach in which it is the average of the difference between the AI and the approximate potential that is computed. The corresponding codes and scripts are provided.

Analyticity of parametric elliptic eigenvalue problems and applications to quasi-Monte Carlo methods

In the present paper, we study the analyticity of the leftmost eigenvalue of the linear elliptic partial differential operators with random coefficient and analyse the convergence rate of the quasi-Monte Carlo method for approximation of the expectation of this quantity. The random coefficient is assumed to be represented by an affine expansion a 0 ( x ) + ∑ j ∈ N y j a j ( x ) , where elements of the parameter vector y = ( y j ) j ∈ N ∈ U ∞ are independent and identically uniformly distributed on U := [ − 1 2 , 1 2 ] . Under the assumption ‖ ∑ j ∈ N ρ j | a j | ‖ L ∞ ( D ) < ∞ with some positive sequence ( ρ j ) j ∈ N ∈ ℓ p ( N ) for p ∈ ( 0 , 1 ] we show that for any y ∈ U ∞ , the elliptic partial differential operator has a countably infinite number of eigenvalues ( λ j ( y ) ) j ∈ N which can be ordered non-decreasingly. Moreover, the spectral gap λ 2 ( y ) − λ 1 ( y ) is uniformly positive in U ∞ . From this, we prove the holomorphic extension property of λ 1 ( y ) to a complex domain in C ∞ and estimate partial derivatives of λ 1 ( y ) with respect to the parameter y by using Cauchy's formula for analytic functions. Based on these bounds we prove the dimension-independent convergence rate of the quasi-Monte Carlo method to approximate the expectation of λ 1 ( y ) .

An adaptive divided-difference perturbation method for solving stochastic problems

Most stochastic problems are based on the independent and symmetrically distributed random variables. Using this principle, in this study, we propose an efficient calculation scheme to obtain the statistical moments based on high-order stochastic perturbation (SP) expansion. This solves the long-existing problem of the SP method, namely, the low calculation efficiency of statistical moments using high-order perturbation expansion, and the complexity of the calculation format. The scheme involves adopting the divided-difference method to approximate the partial derivative items of the SP method. The proposed method, which is similar to the collocation point method, only requires the deterministic calculation result for each node in the divided-difference method. In addition, as the adoption of high-order perturbation expansion is limited in multidimensional random problems due to the curse of dimensionality, an adaptive partial derivative items selection method is proposed in this paper to efficiently and reasonably ignore the expansion items that have an insignificant influence on statistical moments. This considerably improves the calculation efficiency of the statistical moments. Finally, the adaptive divided-difference perturbation (ADDP) method proposed in this paper is compared with the quasi-Monte Carlo method and the adaptive sparse grid methods using three numerical examples. The results prove that the ADDP method not only has high efficiency but also leads to significant improvements in the accuracy of statistical moments.

Modeling ultrasonic wave fields scattered by flaws using a quasi-Monte Carlo method: Theoretical method and experimental verification

The modeling and visualization of wave fields scattered by flaws can be helpful in terms of guiding the testing and evaluation of flaws using an ultrasonic nondestructive method. In this work, the ultrasonic scattering of wave fields from flaws with different shapes is modeled using a quasi-Monte Carlo (QMC) method and measured through experiments for verification. The incident wave fields generated by a transducer can be modeled using the Rayleigh integral expression and calculated using the QMC method. When the size of the flaw is much larger than the wavelength, the incident wave over the lit portion of flaw can be treated as the source for the scattering of wave fields, and these wave fields can also be modeled using the proposed QMC method. In this paper, water is treated as the material and an embedded solid component is considered as the flaw. Numerical examples and results are presented for flaws with different shapes and sizes, and the properties of these scattering wave fields are analyzed and discussed. Experiments are performed to measure the scattering wave fields using a needle transducer, and it is shown that the results agree with the simulations, thus verifying the proposed modeling method. The work presented here can assist in understanding the wave-flaw interaction and can help in optimizing ultrasonic nondestructive testing.

On the Error Rate of Importance Sampling with Randomized Quasi-Monte Carlo

Importance sampling (IS) is valuable in reducing the variance of Monte Carlo sampling for many areas, including finance, rare event simulation, and Bayesian inference. It is natural and obvious to combine quasi-Monte Carlo (QMC) methods with IS to achieve a faster rate of convergence. However, a naive replacement of Monte Carlo with QMC may not work well. This paper investigates the convergence rates of randomized QMC-based IS for estimating integrals with respect to a Gaussian measure, in which the IS measure is a Gaussian or distribution. We prove that if the target function satisfies the so-called boundary growth condition and the covariance matrix of the IS density has eigenvalues no smaller than 1, then randomized QMC with the Gaussian proposal has a root mean squared error of for arbitrarily small . Similar results with a distribution as the proposal are also established. These sufficient conditions help to assess the effectiveness of IS in QMC. For some particular applications, we find that the Laplace IS, a very general approach to approximate the target function by a quadratic Taylor approximation around its mode, has eigenvalues smaller than 1, making the resulting integrand less favorable for QMC. From this point of view, when using Gaussian distributions as the IS proposal, a change of measure via Laplace IS may transform a favorable integrand into an unfavorable one for QMC although the variance of Monte Carlo sampling is reduced. We also study the effect of positivization tricks on the error rate when the integrand has mixed sign. If the smooth positivization proposed by Owen and Zhou (J. Amer. Statist. Assoc., 95 (2000), pp. 135–143) is used, the rate is retained. This is not the case if taking the positive and negative parts of the integrand. We also give some examples to verify our propositions and warn against naive replacement of Monte Carlo with QMC under IS proposals. Numerical results suggest that using Laplace IS with distributions is more robust than that with Gaussian distributions.

Deterministic and stochastic phase-field modeling of anisotropic brittle fracture

We investigate variational phase-field formulations of anisotropic brittle fracture to model zigzag crack patterns in cubic materials. Our objective is twofold: (i) to analytically derive and numerically test the fundamental behavioral aspects predicted by the two main available fourth-order models, and to guide the calibration of their unknown parameters; (ii) motivated by the pronounced non-uniqueness of the phase-field solution in the anisotropic case and uncertainties in the material model, to transition from a deterministic to a stochastic model by introducing a material-related random field in the anisotropic phase-field energy functional. We employ Monte Carlo, randomized quasi-Monte Carlo and stochastic spectral methods to estimate statistical moments of the phase-field variable. The two fourth-order phase-field models are shown to predict a very different response both in their deterministic and stochastic versions. For either modeling choice, we believe that the stochastic approach, which captures several possible zigzag crack paths, holds significant promise to enable meaningful predictions of anisotropic fracture with phase-field models.