quasi-Monte Carlo Algorithms Research Articles

A multipoint prediction model for the deformation of concrete dams considering climatic features of high-altitude regions

High-altitude regions are characterized by natural climatic features such as low temperatures and high radiation. Concrete dams built in these regions face more complex operating conditions. Traditional single-output deformation prediction models ignore the correlations between different monitoring points, making it difficult to assess the overall deformation behavior of concrete dams. The black-box nature of some machine learning models can lead to a lack of interpretability, affecting the practical application. Therefore, a multipoint prediction model (MPM) is proposed to analyze the deformation behavior of concrete dams in high-altitude regions. Firstly, the deformation monitoring points are divided based on the climatic features of high-altitude regions. Subsequently, the MPM is constructed using the multi-output CatBoost model and Quasi-Monte Carlo algorithm to achieve deformation prediction across different partitioned monitoring points. Finally, the elucidation of the MPM optimization procedure in a transparent manner alongside the quantification of the influence exerted by input variables on output outcomes notably augments the interpretive capacity of the MPM. The proposed model is validated using monitoring data from a concrete dam in a high-altitude region. The disparities in chaotic and entropy variation features among deformation monitoring points in different zones corroborate the appropriateness of the zoning in this study. Comparisons between the single-output CatBoost model and MPM verify the feasibility and efficiency of multipoint predictions. Comparisons with other multi-output prediction models validate the superior predictive performance of the MPM. Additionally, the proposed model's generalization performance is validated using data from another concrete arch dam in high-altitude regions.

A search for short-period Tausworthe generators over F b with application to Markov chain quasi-Monte Carlo

A one-dimensional sequence u 0 , u 1 , u 2 , … ∈ [ 0 , 1 ) is said to be completely uniformly distributed (CUD) if overlapping s-blocks ( u i , u i + 1 , … , u i + s − 1 ) , i = 0 , 1 , 2 , … , are uniformly distributed for every dimension s ≥ 1 . This concept naturally arises in Markov chain quasi-Monte Carlo (QMC). However, the definition of CUD sequences is not constructive, and thus there remains the problem of how to implement the Markov chain QMC algorithm in practice. Harase [A table of short-period Tausworthe generators for Markov chain quasi-Monte Carlo. J Comput Appl Math. 2021;384:Paper No. 113136, 12.] focussed on the t-value, which is a measure of uniformity widely used in the study of QMC, and implemented short-period Tausworthe generators (i.e. linear feedback shift register generators) over the two-element field F 2 that approximate CUD sequences by running for the entire period. In this paper, we generalize a search algorithm over F 2 to that over arbitrary finite fields F b with b elements and conduct a search for Tausworthe generators over F b with t-values zero (i.e. optimal) for dimension s = 3 and small for s ≥ 4 , especially in the case where b = 3, 4, and 5. We provide a parameter table of Tausworthe generators over F 4 , and report a comparison between our new generators over F 4 and existing generators over F 2 in numerical examples using Markov chain QMC.

Perfect prosthetic heart valve: generative design with machine learning, modeling, and optimization.

Majority of modern techniques for creating and optimizing the geometry of medical devices are based on a combination of computer-aided designs and the utility of the finite element method This approach, however, is limited by the number of geometries that can be investigated and by the time required for design optimization. To address this issue, we propose a generative design approach that combines machine learning (ML) methods and optimization algorithms. We evaluate eight different machine learning methods, including decision tree-based and boosting algorithms, neural networks, and ensembles. For optimal design, we investigate six state-of-the-art optimization algorithms, including Random Search, Tree-structured Parzen Estimator, CMA-ES-based algorithm, Nondominated Sorting Genetic Algorithm, Multiobjective Tree-structured Parzen Estimator, and Quasi-Monte Carlo Algorithm. In our study, we apply the proposed approach to study the generative design of a prosthetic heart valve (PHV). The design constraints of the prosthetic heart valve, including spatial requirements, materials, and manufacturing methods, are used as inputs, and the proposed approach produces a final design and a corresponding score to determine if the design is effective. Extensive testing leads to the conclusion that utilizing a combination of ensemble methods in conjunction with a Tree-structured Parzen Estimator or a Nondominated Sorting Genetic Algorithm is the most effective method in generating new designs with a relatively low error rate. Specifically, the Mean Absolute Percentage Error was found to be 11.8% and 10.2% for lumen and peak stress prediction respectively. Furthermore, it was observed that both optimization techniques result in design scores of approximately 95%. From both a scientific and applied perspective, this approach aims to select the most efficient geometry with given input parameters, which can then be prototyped and used for subsequent in vitro experiments. By proposing this approach, we believe it will replace or complement CAD-FEM-based modeling, thereby accelerating the design process and finding better designs within given constraints. The repository, which contains the essential components of the study, including curated source code, dataset, and trained models, is publicly available at https://github.com/ViacheslavDanilov/generative_design.

The curse of dimensionality for the Lp-discrepancy with finite p

The Lp-discrepancy is a quantitative measure for the irregularity of distribution of an N-element point set in the d-dimensional unit-cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. It's inverse for dimension d and error threshold ε∈(0,1) is the minimal number of points in [0,1)d such that the minimal normalized Lp-discrepancy is less or equal ε. It is well known, that the inverse of L2-discrepancy grows exponentially fast with the dimension d, i.e., we have the curse of dimensionality, whereas the inverse of L∞-discrepancy depends exactly linearly on d. The behavior of inverse of Lp-discrepancy for general p∉{2,∞} has been an open problem for many years. In this paper we show that the Lp-discrepancy suffers from the curse of dimensionality for all p in (1,2] which are of the form p=2ℓ/(2ℓ−1) with ℓ∈N.This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite Lq-norm, where q is an even positive integer satisfying 1/p+1/q=1.

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.

Adaptive Quasi-Monte Carlo method for nonlinear function error propagation and its application in geodetic measurement

The standard deviation of the nonlinear functional value can be obtained by the law of covariance propagation. Existing covariance propagation methods of the nonlinear model contain the following problems: the approximate function method requires complicated derivative operation; the Monte Carlo method has a high simulated burden and low convergence effectiveness. To overcome these disadvantages, we introduce the Quasi-Monte Carlo (QMC) method and design the implementation process of the QMC method for covariance propagation with independent or correlated observations. Considering that the QMC method cannot balance the number of simulations and the accuracy of the results, a novel QMC algorithm for small numbers of batches simulation is proposed, namely, Adaptive Quasi-Monte Carlo (AQMC). The QMC method and the AQMC algorithm are applied in the forward intersection and covariance propagation of the GNSS baseline vector in geodetic measurement. The results verify the effectiveness of the QMC method and the AQMC algorithm. Compared with the adaptive Monte Carlo method, the AQMC method can improve the computational efficiency by almost 84.4%. The proposed approach provides a new idea for the covariance propagation of the nonlinear model.

A quasi-Monte Carlo data compression algorithm for machine learning

We introduce an algorithm to reduce large data sets using so-called digital nets, which are well distributed point sets in the unit cube. These point sets together with weights, which depend on the data set, are used to represent the data. We show that this can be used to reduce the computational effort needed in finding good parameters in machine learning algorithms. To illustrate our method we provide some numerical examples for neural networks.

Secure pseudorandom bit generators and point sets with low star-discrepancy

The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are nowadays also applied to very high-dimensional integration problems. Hence multi-dimensional point sets of reasonable size with low discrepancy are badly needed. A seminal result from Heinrich, Novak, Wasilkowski and Woźniakowski shows the existence of a positive number C such that for every dimension d there exists an N-element point set in [0,1)d with star-discrepancy of at most Cd∕N. This is a pure existence result and explicit constructions of such point sets would be very desirable. The proofs are based on random samples of N-element point sets which are difficult to realize for practical applications.In this paper we propose to use secure pseudorandom bit generators for the generation of point sets with star-discrepancy of order O(d∕N). This proposal is supported theoretically and by means of numerical experiments.

Quasi-Monte Carlo Finite Element Analysis for Wave Propagation in Heterogeneous Random Media

We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractive index is random and spatially heterogenous. Our focus is on the case in which the region can contain multiple wavelengths. We bypass the usual sign-indefiniteness of the Helmholtz problem by switching to an alternative sign-definite formulation recently developed by Ganesh and Morgenstern Numer. Algorithms, 83 (2020), pp. 1441--1487. The price to pay is that the regularity analysis required for QMC methods becomes much more technical. Nevertheless we obtain a complete analysis with error comprising stochastic dimension truncation error, finite element error, and cubature error, with results comparable to those obtained for the diffusion problem.

Modeling of wave fields generated by ultrasonic transducers using a quasi-Monte Carlo method.

The sound fields generated by ultrasonic transducers are modeled using the quasi-Monte Carlo (QMC) method, which is found to overcome the conflict between accuracy and efficiency that occurs in existing wave field calculation methods. The RI equation, which is frequently used as a model equation in ultrasonic field calculation, is used here as an exact method and for comparison purposes. In the QMC method, the judgment sampling method and Halton sequence are used for pseudo-random sampling from the sound source, and then the sound field distributions are found by solving the integral solution using the sample mean. Numerical examples and results are presented when modeling unfocused, focused, and steered and focused beam fields. The accuracy and efficiency of the QMC method are discussed by comparing the results obtained using different modeling methods. The results show that the proposed method has a high level of efficiency due to the nature of the QMC algorithm and a high level of accuracy because no approximation is required. In addition, wave fields can be modeled with the QMC method as long as sound sources can be effectively pseudo-randomly sampled, allowing the proposed method to be applied to various types of transducers.

ADVANCED STOCHASTIC METHODS FOR MULTIDIMENSIONAL INTEGRALS AND APPLICATIONS

The problem for accurate evaluation of multidimensional integrals related to different areas has been presented. A comprehensive experimental study based on Faure and Hammersley low discrepancy sequences and Fibonacci based lattice rule has been done. Our tests show that the quasi-Monte Carlo algorithms under consideration are efficient tool for computing multidimensional integrals. This will be important in order to obtain more reliable results in some applications.

Grouped Normal Variance Mixtures

Grouped normal variance mixtures are a class of multivariate distributions that generalize classical normal variance mixtures such as the multivariate t distribution, by allowing different groups to have different (comonotone) mixing distributions. This allows one to better model risk factors where components within a group are of similar type, but where different groups have components of quite different type. This paper provides an encompassing body of algorithms to address the computational challenges when working with this class of distributions. In particular, the distribution function and copula are estimated efficiently using randomized quasi-Monte Carlo (RQMC) algorithms. We propose to estimate the log-density function, which is in general not available in closed form, using an adaptive RQMC scheme. This, in turn, gives rise to a likelihood-based fitting procedure to jointly estimate the parameters of a grouped normal mixture copula jointly. We also provide mathematical expressions and methods to compute Kendall’s tau, Spearman’s rho and the tail dependence coefficient λ. All algorithms presented are available in the R package nvmix (version ≥ 0.0.5).

Machining accuracy improvement for a dual-spindle ultra-precision drum roll lathe based on geometric error analysis and calibration

This paper performs a comprehensive analysis and calibration on the geometric error of the ultra-precision drum roll lathe with dual-spindle symmetrical structure and cross slider layout. Firstly, the volumetric error model which contains all geometric errors of the dual-spindle ultra-precision drum roll lathe (DSUPDRL) is developed based on the combination of the homogenous transfer matrix (HTM) and multi-body system (MBS) theory. Secondly, sensitivity analysis for the volumetric error model is conducted to identify the sensitive geometric error components of the DSUPDRL using an improved Sobol method based on the quasi-Monte Carlo algorithm. The result of sensitivity analysis laid the foundation for the subsequent geometric error calibration. Then, some sensitive error components along the X and Z directions are calibrated using a laser interferometer and a pair of inductance displacement probes. Besides the volumetric error model, the concentricity error caused by dual-spindle symmetrical structure is proposed and calibrated by the on-machine measurement using a classic reversal method. Finally, a large-scale roller mold with a diameter of 250 mm and a length of 600 mm is machined using the DSUPDRL after calibration. The experimental result shows that 1.4 μm/600 mm generatrix accuracy is obtained, which validate the effectiveness of the geometric error analysis and calibration.

Deep learning observables in computational fluid dynamics

Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large number of expensive (forward) numerical solutions of the corresponding PDEs. We propose a machine learning algorithm, based on deep artificial neural networks, that predicts the underlying input parameters to observable map from a few training samples (computed realizations of this map). By a judicious combination of theoretical arguments and empirical observations, we find suitable network architectures and training hyperparameters that result in robust and efficient neural network approximations of the parameters to observable map. Numerical experiments are presented to demonstrate low prediction errors for the trained network networks, even when the network has been trained with a few samples, at a computational cost which is several orders of magnitude lower than the underlying PDE solver.Moreover, we combine the proposed deep learning algorithm with Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods to efficiently compute uncertainty propagation for nonlinear PDEs. Under the assumption that the underlying neural networks generalize well, we prove that the deep learning MC and QMC algorithms are guaranteed to be faster than the baseline (quasi-) Monte Carlo methods. Numerical experiments demonstrating one to two orders of magnitude speed up over baseline QMC and MC algorithms, for the intricate problem of computing probability distributions of the observable, are also presented.

Derandomised lattice rules for high dimensional integration

We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider 'half-shifted' rules in which each component of the shift is an odd multiple of \(1/(2N)\) where \(N\) is the number of points in the lattice. By applying the principle that there is always at least one choice as good as the average, we show that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of only order \({1/N^2}\). We carry out numerical experiments where the generating vector is chosen component-by-component (CBC), as for randomly shifted lattices, and where the shift is chosen by a new `CBC for shift' algorithm. The numerical results are encouraging. 
 
 References J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: The quasi-Monte Carlo way. Acta Numer., 22:133–288, 2013. doi:10.1017/S0962492913000044. J. Dick, D. Nuyens, and F. Pillichshammer. Lattice rules for nonperiodic smooth integrands. Numer. Math., 126(2):259–291, 2014. doi:10.1007/s00211-013-0566-0. T. Goda, K. Suzuki, and T. Yoshiki. Lattice rules in non-periodic subspaces of sobolev spaces. Numer. Math., 141(2):399–427, 2019. doi:10.1007/s00211-018-1003-1. F. Y. Kuo. Lattice rule generating vectors. URL http://web.maths.unsw.edu.au/ fkuo/lattice/index.html. D. Nuyens and R. Cools. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput., 75:903–920, 2006. doi:10.1090/S0025-5718-06-01785-6. I. H. Sloan and S. Joe. Lattice methods for multiple integration. Oxford Science Publications. Clarendon Press and Oxford University Press, 1994. URL https://global.oup.com/academic/product/lattice-methods-for-multiple-integration-9780198534723. I. H. Sloan and H. Wozniakowski. When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex., 14(1):1–33, 1998. doi:10.1006/jcom.1997.0463. I. H. Sloan, F. Y. Kuo, and S. Joe. On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput., 71:1609–1641, 2002. doi:10.1090/S0025-5718-02-01420-5.

Metrical Star Discrepancy Bounds for Lacunary Subsequences of Digital Kronecker-Sequences and Polynomial Tractability

Abstract The star discrepancy D N * ( 𝒫 ) $D_N^* \left( {\cal P} \right)$ is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1) d with |𝒫| = N and D N * ( 𝒫 ) = O ( ( log N ) d - 1 / N ) $D_N^* \left( {\cal P} \right) = O\left( {\left( {\log \,N} \right)^{d - 1} /N} \right)$ . However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed −1). In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1) d with |𝒫| = N and D N * ( 𝒫 ) ≤ C d / N $D_N^* \left( {\cal P} \right) \le C\sqrt {d/N}$ . Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d. Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (n α)-sequence and showed a metrical discrepancy bound of the form C d ( log d ) / N $C\sqrt {d\left({\log \,d} \right)/N}$ with implied absolute constant C> 0 independent of N and d. In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

Computation of intersection volume using discrete quadrature algorithm

Quasi Monte Carlo algorithm is one of the most famous particle method, which is used to solve intersection volume computation problem. Inspired by the quasi Monte Carlo algorithm, a discrete quadrature algorithm is proposed to calculate the intersection volume. First, the three dimensional entity with unknown function is divided into a series of random sequences points using the quasi Monte Carlo algorithm. Second, calculating the common point set in terms of a given function three dimensional entity. Finally, both the intersection volume and irregularity intersection volume are approximated by the number of the common point set. Simulation shows that the proposed method has the advantages in estimation accuracy and computational load.

Convergence of sequential quasi-Monte Carlo smoothing algorithms

Gerber and Chopin [J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 (2015) 509–579] recently introduced Sequential quasi-Monte Carlo (SQMC) algorithms as an efficient way to perform filtering in state–space models. The basic idea is to replace random variables with low-discrepancy point sets, so as to obtain faster convergence than with standard particle filtering. Gerber and Chopin (2015) describe briefly several ways to extend SQMC to smoothing, but do not provide supporting theory for this extension. We discuss more thoroughly how smoothing may be performed within SQMC, and derive convergence results for the so-obtained smoothing algorithms. We consider in particular SQMC equivalents of forward smoothing and forward filtering backward sampling, which are the most well-known smoothing techniques. As a preliminary step, we provide a generalization of the classical result of Hlawka and Muck [Computing (Arch. Elektron. Rechnen) 9 (1972) 127–138] on the transformation of QMC point sets into low discrepancy point sets with respect to non uniform distributions. As a corollary of the latter, we note that we can slightly weaken the assumptions to prove the consistency of SQMC.

COMPUTING HIGH DIMENSIONAL INTEGRALS WITH MONTE CARLO METHODS

High dimensional integrals are usually solved with Monte Carlo algorithms and quasi Monte Carlo algorithms. We are doing numerical testing which compare low discrepancy and Monte Carlo algorithms. It is well known that Sobol algorithm has some advantageous over the other low discrepancy sequences, that’s why we use this algorithm for our numerical example. The obtained relative error confirms this superiority of the presented Monte Carlo and quasi Monte Carlo algorithms even when small number of sample points are used. It is very interesting that the presented high dimensional integral gives very low relative error even for computational time less than one second which shows the great importance of the developed algorithms.

Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?

Quasi-Monte Carlo algorithms are studied for designing discrete approximations of two-stage linear stochastic programs with random right-hand side and continuous probability distribution. The latter should allow for a transformation to a distribution with independent marginals. The two-stage integrands are piecewise linear, but neither smooth nor lie in the function spaces considered for QMC error analysis. We show that under some weak geometric condition on the two-stage model all terms of their ANOVA decomposition, except the one of highest order, are continuously differentiable and that first and second order ANOVA terms have mixed first order partial derivatives and belong to $$L_{2}$$L2. Hence, randomly shifted lattice rules (SLR) may achieve the optimal rate of convergence $$O(n^{-1+\delta })$$O(n-1+ź) with $$\delta \in (0,\frac{1}{2}]$$źź(0,12] and a constant not depending on the dimension if the effective superposition dimension is at most two. We discuss effective dimensions and dimension reduction for two-stage integrands. The geometric condition is shown to be satisfied almost everywhere if the underlying probability distribution is normal and principal component analysis (PCA) is used for transforming the covariance matrix. Numerical experiments for a large scale two-stage stochastic production planning model with normal demand show that indeed convergence rates close to the optimal are achieved when using SLR and randomly scrambled Sobol' point sets accompanied with PCA for dimension reduction.