Hammersley Sequence Research Articles

Comparing sampling techniques to chart parameter space of 21 cm global signal with Artificial Neural Networks

Understanding the first billion years of the universe requires studying two critical epochs: the Epoch of Reionization (EoR) and Cosmic Dawn (CD). However, due to limited data, the properties of the Intergalactic Medium (IGM) during these periods remain poorly understood, leading to a vast parameter space for the global 21cm signal. Training an Artificial Neural Network (ANN) with a narrowly defined parameter space can result in biased inferences. To mitigate this, the training dataset must be uniformly drawn from the entire parameter space to cover all possible signal realizations. However, drawing all possible realizations is computationally challenging, necessitating the sampling of a representative subset of this space. This study aims to identify optimal sampling techniques for the extensive dimensionality and volume of the 21cm signal parameter space. The optimally sampled training set will be used to train the ANN to infer from the global signal experiment. We investigate three sampling techniques: random, Latin hypercube (stratified), and Hammersley sequence (quasi-Monte Carlo) sampling, and compare their outcomes. Our findings reveal that sufficient samples must be drawn for robust and accurate ANN model training, regardless of the sampling technique employed. The required sample size depends primarily on two factors: the complexity of the data and the number of free parameters. More free parameters necessitate drawing more realizations. Among the sampling techniques utilized, we find that ANN models trained with Hammersley Sequence sampling demonstrate greater robustness compared to those trained with Latin hypercube and Random sampling.

Active Learning Processes for Smart Inspection, Verification Operations and Modelling of Surfaces with Geometrical Deviations

The ISO 17450-2:2012 defines the verification operation. The ISO 17450-1:2011 defines the extracted features and the extraction operations on the “non-ideal surface model or on the real surface of the workpiece”. The ISO 12781-2:2011 and 12180-2:2011 provide dedicated extraction strategies for the flatness and cylindricity. But no real measurement process is explicit here. However, one of the key factors in an inspection plan is the choice of the measurement strategy. Some sampling strategies, such as Hammersley sequence, offer robust and efficient technics to ensure good distribution of measurement points. But measurements are always defined with a “blind sampling strategy”. So, in this paper, the author proposed to use adaptative sampling strategies, based on the Active learning method based on Kriging for the Inspection of Large Surfaces (AK-ILS) and developed an original Active Learning method based on Multi-Layer-Perceptron (AL-MLP), to classify the conform / non-conform parts and to ensure the reconstruction of the surfaces, with smart Skin Models Shapes and a minimal sampling size.

A Novel Wind Power Outlier Detection Method with Support Vector Machine Optimized by Improved Harris Hawk

The accurate detection of wind power outliers plays a crucial role in wind power forecasting, while the inherited strong randomness and high fluctuations bring great challenges to this issue. This work investigates the way to improve the outlier detection accuracy based on support vector machine (SVM). Although SVM can achieve good results for outlier detection in theory, its performance is heavily dependent on the hyper-parameters. Parameter optimization is not an easy task due to its complex nonlinear multi-optimum nature; an improved Harris hawk optimization (IHHO) is proposed to optimize the parameters of SVM for more accurate outlier detection. HHO takes the cooperative behavior and chasing style of Harris’ hawks in nature called surprise pounce and can effectively search the optimal one in large parameter space, but it tends to fall into local optimum. To solve this issue, an improved Harris hawk optimization algorithm (IHHO) was proposed to obtain the optimal parameters of SVM. First, Hammersley sequence initialization is carried out to acquire good initial solutions. Then, a nonlinear factor control mode and an adaptive Gaussian–Cauchy mutation perturbation strategy are proposed to avoid getting trapped in local optima. In this way, a novel wind power outlier detection method named IHHO-SVM was constructed. The results on several wind power data with outliers show that IHHO-SVM outperforms SVM and HHO-SVM, which achieves the highest average F1 score of 96.63% and exhibits the smallest standard deviation. Compared to commonly used models for detecting outliers in wind power, such as isolation forest (IF), local outlier factor (LOF), SVM with grey wolf optimization (GWO-SVM), and SVM with particle swarm optimization (PSO-SVM), the proposed IHHO-SVM model shows the best overall performance with precision, recall, and F1 scores of 95.76%, 96.94%, and 96.35%, respectively.

Data-Driven State Transition Algorithm for Fuzzy Chance-Constrained Dynamic Optimization.

Many actual industrial production processes are dynamic and uncertain. When uncertain information are described by subjective experience and experts' knowledge based on scanty or vague information, fuzzy uncertainty exists. Fuzzy chance-constrained dynamic programming are applicable to industrial production modeling accompanied by fuzzy uncertainty and dynamics, where constraints need not or cannot be completely satisfied. In this article, a fuzzy chance-constrained dynamic optimization (FCCDO) formulation on the basis of credibility theory is established, in which, the credibility is used to measure the fuzzy uncertainty level of constraints. To solve the FCCDO problem (FCCDOP), an improved fuzzy simulation technique based on Hammersley sequence sampling is raised to transform fuzzy chance constraints to their deterministic equivalents, and then a data-driven state transition algorithm (DDSTA) using deep neural networks (DNNs) is put forward to achieve a stable, global and robust optimization performance. Finally, the successful applications of the FCCDO method to industrial studies demonstrate its advantages.

Method study on failure probability and uncertainty analysis of stress corrosion cracking in fixed parts of PWR

To estimate the failure probability of stress corrosion cracking (SCC) in the fixed parts of pressurized water reactors (PWR), an approach based on the Paris model was developed and its uncertainty analysis was performed as well. In the study of SCC failure, the uncertainty of fracture toughness was translated into integral form to improve calculation efficiency. Moreover, uncertainty parameters such as crack size and Paris model coefficients were studied in the uncertainty analysis. On this basis, the thermowell of the pressurizer of PWR as an example was analyzed in this study. The Monte Carlo (MC) method and polynomial chaos expansion (PCE) approach were used to calculate the mean value, relative uncertainty, upper limit of tolerance interval of failure probability, and sensitivity coefficient of input parameter to compare the advantages and disadvantages of these methods. According to the results, compared with the other method, 300 sets of sampling with Latin Hypercube sampling (LHS) and Sobol sequences can converge the mean, and 300 sets of sampling with Halton sequence and Hammersley sequence can converge the relative uncertainty. When LHS with the 4th order Wilks' formula and Halton sequence with the 1st order Wilks' formula to compute the upper limit of tolerance interval, the accuracy and cost of the computation can be balanced. According to the results of the sensitivity analysis, crack aspect ratio is the most sensitive parameter. PCE surrogate model can accurately calculate the mean, relative uncertainty, the upper limit of tolerance interval and sensitivity coefficient, thereby reducing the computational cost.

A probabilistic study into the influences of fiber angle orientations on energy absorption performance of carbon fiber-reinforced polymer composite tubes

The influence of fiber orientations on the crashworthiness of carbon fiber-reinforced polymer (CFRP) composite tubes was investigated to fill in the knowledge gap regarding the effects of fiber angle on the energy-absorbing potential of these tubes. This study adopted the Hammersley sequence sampling (HSS) method to generate a quasi-random distribution of the design objectives. Then, LS-DYNA was utilized for numerical quasi-static crushing tests to measure the specific energy absorption (SEA) and the crushing force efficiency (CFE) of a quasi-static CFRP tube at different fiber angles. The Kriging method was implemented to generate a surrogate model. Finally, Pareto results were extracted deterministically and probabilistically using the NSGAII algorithm. Fiber orientation angles are naturally exposed to uncertainties in the manufacturing process; thus, the main contribution of the study is illustrating the effects of fiber angle orientations on the energy-absorbing characteristics of CFRP tubes while incorporating uncertainties into the optimization procedure. Interestingly, the optimum fiber angle orientations revealed the potential advantage of using non-conventional angle-ply layouts by improving the CFE by up to 4% compared with the initial design configuration. The results also demonstrated the capability and superior performance of the optimization procedure in finding the global optima, and the necessity of conducting a probabilistic examination in similar studies.

A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

Physics-informed neural networks (PINNs) have shown to be effective tools for solving both forward and inverse problems of partial differential equations (PDEs). PINNs embed the PDEs into the loss of the neural network using automatic differentiation, and this PDE loss is evaluated at a set of scattered spatio-temporal points (called residual points). The location and distribution of these residual points are highly important to the performance of PINNs. However, in the existing studies on PINNs, only a few simple residual point sampling methods have mainly been used. Here, we present a comprehensive study of two categories of sampling for PINNs: non-adaptive uniform sampling and adaptive nonuniform sampling. We consider six uniform sampling methods, including (1) equispaced uniform grid, (2) uniformly random sampling, (3) Latin hypercube sampling, (4) Halton sequence, (5) Hammersley sequence, and (6) Sobol sequence. We also consider a resampling strategy for uniform sampling. To improve the sampling efficiency and the accuracy of PINNs, we propose two new residual-based adaptive sampling methods: residual-based adaptive distribution (RAD) and residual-based adaptive refinement with distribution (RAR-D), which dynamically improve the distribution of residual points based on the PDE residuals during training. Hence, we have considered a total of 10 different sampling methods, including six non-adaptive uniform sampling, uniform sampling with resampling, two proposed adaptive sampling, and an existing adaptive sampling. We extensively tested the performance of these sampling methods for four forward problems and two inverse problems in many setups. Our numerical results presented in this study are summarized from more than 6000 simulations of PINNs. We show that the proposed adaptive sampling methods of RAD and RAR-D significantly improve the accuracy of PINNs with fewer residual points for both forward and inverse problems. The results obtained in this study can also be used as a practical guideline in choosing sampling methods.

On-site calibration method of the AACMM based on a high-precision CNC machine tool

An articulated arm coordinate measuring machine (AACMM) is a kind of instrument used to test workpieces in the industrial field, and its measurement uncertainty depends on the on-site environmental factors to a certain extent. If there is a problem with the AACMM during on-site testing, it needs to be re-calibrated to ensure the measurement uncertainty. In this work, a fast on-site calibration method for the AACMM is proposed using a high-precision computer numerical control (CNC) machine tool. The AACMM is fixed on the machine tool table. The calibration artifact is clamped on the machine tool spindle. This calibration artifact can move to any point in the global space of the AACMM along with the operation of the spindle. Then, a virtual calibration artifact with calibration points is achieved using the CNC machine tool. The positions of these points are calculated using the improved Hammersley sequence. The calibration method using a high-precision CNC machine tool is experimentally verified through the single-point articulation performance test of ASME B89.4.22-2004 and the length measurement test of ISO 10360-12. The calibration method proposed in this work only uses an on-site high-precision CNC machine tool, one of the most common machines in many industrial fields, as the calibration instrument for the AACMM calibration. No extra equipment is required, which is convenient and beneficial to calibrate the AACMM. In addition, the CNC machine tool can construct different kinds of virtual structures for the AACMM calibration, such as virtual rods, virtual circles, virtual balls and even non-uniform 3D artifacts with complex curved surfaces. This may give rise to many novel calibration methods and further improve the calibration accuracy.

Output-only structural damage identification using hybrid Jaya and differential evolution algorithm with reference-free correlation functions

To solve the optimization-based structural damage identification problem, a novel hybrid algorithm based on Jaya and differential evolution algorithm (HJDEA) is proposed to detect, locate and quantify structural damages by effectively incorporating the powerful local exploitation capacity of Jaya algorithm and global exploration capability of differential evolution. Meanwhile, Hammersley sequence initialization and Lévy flight search mechanism are introduced into HJDEA to further improve convergence rate and refining the quality of the best solution. Four different algorithms, genetic algorithm, particle swarm optimization, Jaya and the proposed HJDEA are employed for comparative study. In addition, the objective function is established by adjacent acceleration correlation function so as to avoid false identification caused by defining improper reference point. The performance of the proposed damage identification strategy based on HJDEA and adjacent acceleration correlation function is investigated with numerical examples involving an 8-DOF lumped mass model and a cantilever beam, as well as an experimental study of the ASCE benchmark structure under white noise excitation. Results show that the proposed hybrid identification method is accurate, efficient and robust in the identification of the damage existence, location and severity of stiffness and mass parameters even with partial output-only responses and 20% noise-polluted measurements.

Trapezoidal type-2 fuzzy inference system with tensor unfolding structure learning method

In the paper, trapezoidal type-2 fuzzy inference system (TT2FIS) is introduced. To construct the type-2 fuzzy inference system, trapezoidal type-2 fuzzy sets are adopted to construct the antecedent part of the fuzzy rules, the centers of the fuzzy sets are roughly clustered by an evolving autonomous data partitioning algorithm. To further eliminate the effect of the centers that are generated by dataset itself, and it can be easily impacted by the unbalance data, the data clouds that are generated by autonomous data partitioning algorithm are filtered by Hammersley sequence, and symmetrical trapezoidal type-2 fuzzy sets are generated by the generated cluster centers and problem-free standard deviation generating method. To the consequent part, generalized type-2 TSK consequent is formulated by tensor, which is obtained via three constituent parts of the trapezoidal type-2 fuzzy sets (lower membership function, upper membership function and type-reduction set of data samples), the parameters of consequent part are the iterative results of a matrix equation that is unfolded from the tensor. Finally, simulation results are carried out to verify the effectiveness of the proposed trapezoidal type-2 fuzzy inference system. Simulation results show that the generalization of the TT2FIS is better than the coincide type-2 fuzzy inference system or adaptive type-1 fuzzy inference system.

Structural identification using improved butterfly optimization algorithm with adaptive sampling test and search space reduction method

Swarm intelligence optimization algorithms have been widely used in structural identification due to its powerful search ability, while an unpleasant identification result is regularly achieved with predefined wide search range. To address this issue, a hybrid strategy, initially reducing search space by adaptive sampling test and then identifying the actual structural parameters with improved butterfly optimization algorithm (IBOA) in the reduced search range by adaptive search space reduction method, is proposed and employed in this paper. In one aspect, the clustering competition learning mechanism and the chaotic elite learning mechanism are introduced to improve the performance of butterfly optimization algorithm (BOA). In the other aspect, adaptive sampling test and search space reduction method are also developed for search space reduction of unknown parameters. Genetic algorithm, BOA, IBOA, and proposed hybrid sampling and IBOA (sampling-IBOA) are evaluated by numerical examples of simply-supported beam and truss structure, as well as an experimental test of steel grid benchmark structure for comparative study. In addition, the effect of the window width, four different sampling methods, hybrid Ham-IBOA and gradient search on application of the proposed sampling-IBOA method are further illustrated. The numerical and experimental results demonstrate that the proposed sampling-IBOA method can significantly improve computational efficiency and identification accuracy, especially for Hammersley sequence sampling among the four sampling methods.

Probabilistic power flow calculation based on importance-Hammersley sampling with Eigen-decomposition

This paper presents a novel probabilistic power flow calculation method for power systems with integrated wind farms, based on importance sampling and Hammersley sequence with eigen-decomposition. The method proposed in this paper adopts importance sampling to build probability density functions in a desired range to compress the sampling space and hence reduce the calculation burden. The Hammersley sequence, a kind of low discrepancy sequence, is used to obtain uniform samples for improving the sampling efficiency. In addition, since the traditional Cholesky method is unable to decompose the non-positive definite correlation matrix, this paper applies the eigen-decomposition to solve this problem for multiple correlated wind sources. Case studies are conducted on modified IEEE test systems, where the advantages of the proposed method are verified. According to the simulation results, the proposed method shows greater accuracy and efficiency, compared to the traditional methods.

Equidistributed Search+Probability Based Tracking Strategy to Locate an Air Pollutant Source With Two UAVs

As the atmospheric pollution becomes an increasingly serious problem, finding accurately the location of pollutant sources is still challenging. In the present work, a probability-based tracking strategy is proposed for guiding two cooperative unmanned aerial vehicles (UAVs) within a quest area to find an atmospheric pollutant source. This tracking strategy implies deploying algorithmically two phases: exploration and exploitation. During the exploration phase each vehicle follows a trajectory based on plane coordinates generated from a Hammersley sequence. The overlapping between UAVs’ trajectories is avoided by splitting guidance points into two groups by using the k-means algorithm. The navigation trajectories are smoothed by an TSP solver and a cubic spline planning algorithm. The exploitation phase redirects the search to specific locations where the probability of finding the source is higher. This is achieved by considering the quest area as a mesh, where each cell is assigned a probability computed with information collected by the UAVs measurement system. Every time a high concentration is found, the probabilities are recalculated, and flight trajectories are adjusted. The trajectories are semicircular, and the radius is decreased when a new high concentration is found. Simulation data of the proposed tracking strategy shows promising results on the accuracy achieved in the finding of the pollutant source, in comparison with three other tracking strategies: leader-follower, random walk with particle swarm optimization, and a hill climb traceability algorithm.

A sample construction method in kinematics characteristics domain to identify the feed drive model

The modeling accuracy of feed drive is mainly affected by two factors: the accuracy of model form and the completeness of sample data which is used to identify the parameters of the established model form. One of the main works of this study is to construct the experimental trajectories which are used to obtain the sample data. Different from the trajectories selected based on the researcher's experience in previous literatures which lacks theoretical evidence to confirm the completeness of the sample data, in this paper, a reverse construction method of experimental trajectory is proposed based on the required sample data. First, the completeness of sample data is analyzed, and the samples are generated by using the Hammersley sequence method. Then, the experimental trajectories are derived. In addition, considering that the data of the feed system has temporal correlations between the adjacent motion states in time sequence, this study made some improvements on the modeling framework composed of the first-principle model and machine learning model. The experiments conducted in the actual feed drive system showed that the maximum prediction error of the tracking error is 7.8%, which confirmed the effectiveness of the designed trajectories and the high accuracy of the proposed model. Furthermore, the advantage of the proposed trajectory is confirmed by the comparison experiment. At the same time, the same trajectories are used to train different model forms to verify the advantage of the proposed model. The proposed feed drive modeling method can be used for designing a high-performance feed drive control system or compensating errors when generating motion commands, thereby improving the machining accuracy of machine tool.

An adaptive sampling method for accurate measurement of aeroengine blades

The surface of aeroengine blades has geometrical complexity and its measurement accuracy directly affect the assessment correction of the different machining processes. A NURBS based parametric model was employed to represent the geometry of blade surface for evaluating deviation between ideal model and measurement model. In view of the various curvature at the different blade region, an adaptive surface sampling strategy with bending moment theory is proposed. The distribution and number of the check points can be determined adaptively and accurately in accordance with geometrical feature altering. The accuracy and effectiveness of the proposed method were validated through simulations. Case studies involving comparing with bidirectional isoparametric method, equal area method and hammersley sequence method showed that the deviation between the measured blade surface and its reconstructed model using the developed method is the least among all the methods while the numbers of the selected check points are the same.

Multi-objective optimization operation of the green energy island based on Hammersley sequence sampling

The Green Energy Island (GEI) is a system that supplies heating, cooling and electricity energy. This system includes various supply-side technologies, which introduces a large amount of degrees of freedom in the design and operation phases. Therefore, the optimal operation parameters should be determined to ensure continuous and stable heating, cooling and power supply of the GEI system. In this paper, the ε-constraints method based on Hammersley sequence sampling (HSS) method is proposed. Comparison with the traditional ε-constraints method shows that the ε-constraints method based on HSS method can save a lot of time in solving the multi-objective optimization problem. Moreover, an innovative multi-objective optimization framework that integrates the ε-constraints method, HSS method and TOPSIS approach is established in economic, energy and environmental benefit. Through the optimization calculation for three typical days, the approximate Pareto frontiers with possible operation parameters were presented, which provided a reference for decision makers. Finally, by using TOPSIS approach, the optimal energy dispatch was obtained for three typical days. The results show that the energy and environmental benefit of the GEI system is significant with the approximate 5–35% of the fuel energy saving ratio and the approximate 10–65% of carbon dioxide emission reduction ratio.

Response surface modeling of CO2dynamic storage efficiency factor in high permeability thick sandstones

AbstractIn order to utilize as much of the pore space for CO2storage in high permeability thick saline aquifers, it is vital to investigate the interactions of injected CO2with formation brine and rock. In order to quantify the displacement process, we investigate the dynamic storage efficiency factor (DSEF) for saline aquifers where pressure increase is minimal during the injection phase. Dimensionless numbers are derived from basic governing equations, constitutive equations, initial and boundary conditions using the inspection analysis. Then using the Hammersley sequence sampling, 178 numerical experiments are designed, and a compositional reservoir simulator is used to perform these simulations. In the next step, response surface regression analysis is used to establish a relationship between DSEF obtained from the numerical simulations and the corresponding dimensionless numbers. The simulation results show that for the studied conditions the underground dynamics is mostly influenced by the gravity number, followed by effective aspect ratio and dip numbers. The results from the response surface regression analysis are used to develop a correlation, which can be used to estimate the dynamic CO2storage capacity of relevant zones. This study provides quantitative measures for the different competing mechanisms involved in underground displacement of fluids in CO2geological storage, which can serve as a useful tool during planning phase of storage projects. © 2019 Society of Chemical Industry and John Wiley & Sons, Ltd.

Impact of Sampling Technique on the Performance of Surrogate Models Generated with Artificial Neural Network (ANN): A Case Study for a Natural Gas Stabilization Unit

Data-driven models are essential tools for the development of surrogate models that can be used for the design, operation, and optimization of industrial processes. One approach of developing surrogate models is through the use of input–output data obtained from a process simulator. To enhance the model robustness, proper sampling techniques are required to cover the entire domain of the process variables uniformly. In the present work, Monte Carlo with pseudo-random samples as well as Latin hypercube samples and quasi-Monte Carlo samples with Hammersley Sequence Sampling (HSS) are generated. The sampled data obtained from the process simulator are fitted to neural networks for generating a surrogate model. An illustrative case study is solved to predict the gas stabilization unit performance. From the developed surrogate models to predict process data, it can be concluded that of the different sampling methods, Latin hypercube sampling and HSS have better performance than the pseudo-random sampling method for designing the surrogate model. This argument is based on the maximum absolute value, standard deviation, and the confidence interval for the relative average error as obtained from different sampling techniques.

Approximate and exact optimal designs for $$2^k$$ factorial experiments for generalized linear models via second order cone programming

Model-based optimal designs of experiments (M-bODE) for nonlinear models are typically hard to compute. The literature on the computation of M-bODE for nonlinear models when the covariates are categorical variables, i.e. factorial experiments, is scarce. We propose second order cone programming (SOCP) and Mixed Integer Second Order Programming (MISOCP) formulations to find, respectively, approximate and exact A- and D-optimal designs for $$2^k$$ factorial experiments for Generalized Linear Models (GLMs). First, locally optimal (approximate and exact) designs for GLMs are addressed using the formulation of Sagnol (J Stat Plan Inference 141(5):1684–1708, 2011). Next, we consider the scenario where the parameters are uncertain, and new formulations are proposed to find Bayesian optimal designs using the A- and log det D-optimality criteria. A quasi Monte-Carlo sampling procedure based on the Hammersley sequence is used for computing the expectation in the parametric region of interest. We demonstrate the application of the algorithm with the logistic, probit and complementary log–log models and consider full and fractional factorial designs.

Efficient sampling algorithm for large-scale optimization under uncertainty problems

Uncertainty is part of the real-world optimization problems. The major bottleneck in solving large-scale stochastic optimization problems is the computational intensity of scenarios or samples. To this end, this research presents a novel sampling approach. This sampling called LHS-SOBOL combines one-dimensional uniformity of LHS and d-dimensional uniformity of Sobol. This paper analyzes existing and novel sampling techniques by conducting large-scale experiments with different functions. The sampling techniques which are analyzed are Monte Carlo Sampling (MCS), Latin Hypercube Sampling (LHS), Hammersley Sequence Sampling (HSS), Latin Hypercube-Hammersley Sequence Sampling (LHS-HSS), Sobol Sampling, and the proposed novel Latin Hypercube-Sobol Sampling (LHS-SOBOL). It was found that HSS performs better up to 40 uncertain variables, Sobol up to 100 variables, LHS-HSS up to 250 variables, and LHS-SOBOL for large-scale uncertainties for larger than 100 variables.