Set Of Representative Points Research Articles

Quantitative Property of MF-Discrepancy and Efficient Point-Selection Strategy for the Nonlinear Stochastic Response Analysis of Structures with Random Parameters

The response analysis of high-dimensional and strongly nonlinear systems with random parameters remains a significant challenge in stochastic computational mechanics. To address this challenge, some methods based on the high-efficacy point sets have been developed, in which efficient global-point-set methods represented by low-discrepancy are of paramount importance in generating representative point sets. Several discrepancies including the extended F-discrepancy (EF-discrepancy) and the generalized F-discrepancy (GF-discrepancy) have been introduced to assess the uniformity and the efficacy of a representative point set. In such context, a maximal marginal EF-discrepancy (MF-discrepancy), which is an extended form of the GF-discrepancy, is proposed in this paper and then the properties of the MF-discrepancy are studied in detail. The probability distribution of the MF-discrepancy is derived, including a rigorous proof for random point sets and a model based on an assumption for some generic point sets. A generalized Koksma-Hlawka inequality is established accordingly to govern the worst error estimate. The lowest bound of the MF-discrepancy is given, and two intuitive quantitative indices are proposed to measure the goodness of the MF-discrepancy. Based on the lowest bound, an enhanced point-selection strategy with a unified theoretical framework for minimizing the MF-discrepancy is proposed. In this framework, locally minimizing the MF-discrepancy yields the two-step point-selection method, and a new point-selection strategy is proposed based on the global minimization of the MF-discrepancy, which is verified to be efficient and robust, especially in high-dimensional cases. Several numerical examples, including a 2-story shear frame, a 10-story shear frame, and a 10-story reinforced concrete frame structure modeled by the finite element method, are studied, verifying the efficiency and the robustness of the proposed point-selection strategy.

A clustering-based partially stratified sampling for high-dimensional structural reliability assessment

Assessing structural reliability problem with high-dimensional random inputs is still challenging due to the “curse of dimensionality”. In this paper, this challenge is addressed by extending the Generalized Distribution Reconstruction Method via Characteristic Function Inversion (GDRM-CFI). Specifically, a clustering-based partially stratified sampling method is proposed for selecting high-dimensional points to numerically evaluate the characteristic function (CF) curve of complex high-dimensional problems. An improved number-theoretical method (i-NTM) is used to establish a uniform, efficient point set, ensuring determinism and reducing variability. Subsequently, a partial stratification approach partitions the high-dimensional space into orthogonal two-dimensional subspaces. The fundamental point set is projected into each subspace, and the k-means clustering algorithm identifies centroids within each, acting as representative points. The complete set of representative points from all subspaces formulates the high-dimensional point set. Numerical examples are investigated, which demonstrate the proposed method is effective for high-dimensional structural reliability assessment.

Reliability analysis of structures using probability density evolution method and stochastic spectral embedding surrogate model

AbstractThis study presents an efficient reliability analysis method using probability density evolution method (PDEM) and stochastic spectral embedding (SSE) based surrogate model. The PDEM is used to estimate the structural response's probability density function (PDF). The PDEM is derived based on the principle of probability conservation where generalized density evolution equations (GDEEs) are decoupled from the physical system. The GDEEs are solved using finite difference method coupled with total variation diminishing, in which a set of representative points of random parameters are generated using the generalized F‐discrepancy scheme. To obtain satisfactory accuracy of the numerical solution, representative points are needed, which becomes computationally expensive for complex structures. To reduce the computation burden, the SSE is used, which approximates the original response surface. The SSE is a class of supervised machine learning algorithm where it is trained by few observations and enables output prediction as spectral representation. This is achieved by minimizing the residual using domain decomposition technique. To illustrate the proposed SSE‐based PDEM, three numerical examples are investigated, including the reliability analysis of four‐branch problem and shear building frame subjected to ground acceleration, and the reliability‐based design optimization of a moment‐resisting frame coupled with the nonlinear energy sink with negative stiffness and sliding friction. Numerical results show that the proposed SSE‐based PDEM can estimate failure probability using a very small number of representative points without compromising accuracy compared with Monte Carlo simulation, which leads to a reduction in computational costs.

A non-linear Kalman filter for track parameters estimation in high energy physics

The Kalman Filter is a widely used approach for the linear quadratic estimation of dynamical systems and is frequently employed within nuclear and particle physics experiments for the reconstruction of charged particle trajectories, known as tracks. Implementations of this formalism often make assumptions on the linearity of the underlying dynamic system and the Gaussian nature of the process noise and measurement model, which are violated in a number of track reconstruction applications. This paper introduces an implementation of a Non-Linear Kalman Filter (NLKF) within the ACTS track reconstruction toolkit. The NLKF addresses the issue of non-linearity by using a set of representative sample points during the projection of the track state to the measurement. In a typical use case, the NLKF outperforms the so-called Extended Kalman Filter in the accuracy and precision of the track parameter estimates obtained, with an increase in CPU time below a factor of two. It is therefore a promising approach for use in applications where precise estimation of track parameters is a key concern.

Reliability Analysis of Stochastic System using Stochastic Spectral Embedding based Probability Density Evolution Method

This study presents a reliability analysis of stochastic system using the probability density evolution method (PDEM). The PDEM is formulated according to the principle of probability conservation where generalized density evolution equations (GDEEs) are completely decoupled. To estimate the probability density function accurately, a set of representative points of random variables are generated using the GF-discrepancy scheme. A large number of representative points is needed to obtain satisfactory accuracy, which becomes computationally expensive. To reduce the computation burden, a stochastic spectral embedding (SSE) is used as a surrogate model which approximates the original response surface. To illustrate the proposed SSE-based PDEM, two numerical examples are investigated, including the reliability analysis of four-branch problem, and the reliability-based design optimization of a shape memory alloy based damped outrigger tall timber building. Numerical results show that the proposed SSE-based PDEM can estimate failure probability using a very small number of representative points without compromising accuracy compared with Monte Carlo simulation, which leads to a reduction in computational costs.

Multi-view Image Color Consistency Enhancement using 3D Point Cloud Registration

Color inconsistency in multi-view images degrades the performance of various consumer electronics applications that use multi-view images, such as panoramic image stitching and 3D reconstruction for VR contents. In this paper, a novel approach to multi-view image color consistency enhancement is proposed. In contrast to conventional methods that try to find a nonlinear transfer function for each color channel, in the proposed method, the color distribution of the source image is non-rigidly transformed into that of the reference image using a 3D point set registration algorithm. In order to significantly reduce the computation of the 3D registration process, representative color points are utilized for the 3D registration instead of all the pixel color points. The number of representatives is much less than the number of image pixels, but they effectively approximate the color distribution of the image. Through extensive experiments, we show that the proposed method outperforms conventional methods in color consistency enhancement both subjectively and objectively, and the proposed representative color point set approximation improves the computational efficiency.

Random vibration analysis of the maglev vehicle-guideway system using a probability density evolution method with uncertain parameters

Random dynamic analyses of maglev vehicle-guideway coupled systems are of significant interest but have yet been comprehensively performed. This paper presents a time-dependent coupled random model of a maglev vehicle-controller-guideway-pier to investigate random vibration characteristics of the dynamic system. In the study, a 12-degree-of-freedom vehicle model is considered, a three-dimensional detailed guideway model is established using the finite element method, and the electromagnetic levitation force is controlled with a current feedback controller. Then, the generalized F-discrepancy representative point set selection strategy and probability density evolution method, which can consider multiple random parameters, are combined to analyze the dynamic system. Random vehicle parameters and irregularity are introduced and the mean value, standard deviation, limits satisfying different probability guarantee rates, and probabilistic information are calculated. The model was validated with field measurement data from an electromagnetic suspension (EMS) maglev line. In addition, the Monte Carlo method was used to assess the accuracy and efficiency of the proposed method. The results demonstrate that the proposed method has good accuracy and efficiency. The effects of vehicle running speed, random irregularity, and random vehicle loads on the dynamic vibration of the system are discussed.

HCDC: A novel hierarchical clustering algorithm based on density-distance cores for data sets with varying density

Cluster analysis is a crucial data mining technology widely used in image segmentation, language processing, and pattern recognition. Most existing clustering algorithms cannot identify complex shapes in manifold data sets and data sets with varying-density distribution, especially when clusters with significant differences in density are close to each other. Hierarchical clustering algorithms can identify data sets of arbitrary shapes. However, hierarchical clustering algorithms not only cannot cluster datasets with significant density variations but also have a high time cost. So in this paper, we propose a novel hierarchical clustering algorithm based on density-distance cores, called HCDC. It first selects the density-distance representative points for each point from the set of candidate representative points. Then it selects density-distance cores from all density-distance representatives. And it replaces the whole data set with density-distance cores and uses a new distance between them to apply hierarchical clustering. To avoid the influence of noise points in the dataset when finding density-distance cores, we also propose the noise point detection method and verify the feasibility of this method. In this paper, we compare our proposed algorithm with existing classical and novel algorithms on synthetic and real datasets. Experiments show that our algorithm clusters better than existing algorithms on complex-shaped datasets and datasets with different densities. On datasets with sparse and dense clusters close to each other, the ARI score of HCDC is more than 0.1 higher than that of LDP-MST. In particular, on the grid dataset, HCDC’s ARI score is 0.997 higher than LDP-MST. On DS3 and DS8, HCDC’s ARI score is more than 0.14 higher than the second-best algorithm, RNN-DBSCAN. Moreover, on the zoo dataset, HCDC’s ARI score is 0.15 and 0.6 higher than RNN-DBSCAN and LDP-MST, respectively. On the olivetti face dataset, HCDC is the only algorithm with an NMI score above 0.9 on photo1 and photo2 datasets.

AK-PDEMi: A failure-informed enrichment algorithm for improving the AK-PDEM in reliability analysis

A failure-informed enrichment algorithm is devised to improve the performance of the existing adaptive Kriging-probability density evolution method (AK-PDEM) for reliability analysis. This improved method is named the AK-PDEMi. Contrary to empirically prescribing the sample size of representative points in the existing AK-PDEM, the representative point set in the AK-PDEMi is sequentially enriched by new sets of representative points generated by a failure-informed enrichment scheme, which aims to sequentially making fine partitions of the key sub-regions where the representative points make critical contributions to the failure probability. In this regard, a double-loop configuration is devised: the inner loop adaptively refines the accuracy of Kriging model to reduce the Kriging-induced error, and the outer loop involves the failure-informed enrichment process to alleviate the PDEM-associated discretization error. The outer and inner loops are complementary and proceed sequentially until both of their convergence criteria are satisfied. Three numerical examples are studied and comprehensive comparisons are made between the proposed AK-PDEMi and other conventional reliability algorithms. Results show that the AK-PDEMi shows remarkable advantage over the existing AK-PDEM.

A downsampling method enables robust clustering and integration of single-cell transcriptome data

The random noises, sampling biases, and batch effects often confound true biological variations in single-cell RNA-sequencing (scRNA-seq) data. Adjusting such biases is key to the robust discoveries in downstream analyses, such as cell clustering, gene selection and data integration. Here we propose a model-based downsampling algorithm based on minimal unbiased representative points (MURPXMBD). MURPXMBD is designed to retrieve a set of representative points by reducing gene-wise random independent errors, while retaining the covariance structure of biological origin hence provide an unbiased representation of the cell population. Subsequent validation using benchmark datasets shows that MURPXMBD can improve the quality and accuracy of clustering algorithms, and thus facilitate the discovery of new cell types. Besides, MURPXMBD also improves the performance of dataset integration algorithms. In summary, MURPXMBD serves as a useful noise-reduction method for single-cell sequencing analysis in biomedical studies.

Utilizing differential characteristics of high dimensional data as a mechanism for dimensionality reduction

• A generally applicable dimensionality reduction method namely CKD has been proposed. • CKD leverages the differential characteristics of the data with respect to a reference data set. • Reference data can be obtained through an independent measurement or from the data themselves. • CKD shows superior performance over existing techniques. Dimensionality reduction and visualization of high dimensional data are fundamental tasks across scientific disciplines. Current dimensionality reduction methods rely on the use of some restrictive mathematical rules in projecting the data onto lower dimensions, which often leads to results of limited accuracy. Here, we present a generally applicable strategy of analyzing high-dimensional data by leveraging the differential characteristics of the data with respect to a reference data set. Depending on the problem, the reference data can be obtained either through an independent measurement or from the data themselves by finding a set of representative data points. The differentiating characteristics between the two sets of data are optimized by correlation transformation and scaling of the covariance matrices . We show the superior performance of the proposed method over existing techniques for a variety of data sets from different biomedical applications .

Structural design optimization under dynamic reliability constraints based on the probability density evolution method and highly-efficient sensitivity analysis

The present contribution investigates the feasibility of solving a class of dynamic-reliability-based design optimization problems in the framework of the probability density evolution method (PDEM). The PDEM combined with the extreme value distribution strategy is employed for the dynamic reliability assessment. Based on this information, a small set of important representative points (IRPs) that have a relatively large impact on the dynamic reliability is identified. Then, the sensitivity of the dynamic reliability constraints with respect to the design variables is estimated with the set of IRPs. Since the number of IRPs is usually small, the numerical effort associated with the sensitivity analysis is considerably reduced without compromising the accuracy of the results. By embedding the PDEM-based dynamic reliability assessment and sensitivity analysis into a class of first-order optimization algorithms, the design optimization problem under dynamic reliability constraints is efficiently solved. Two numerical examples involving nonlinear structures are presented to demonstrate the effectiveness and efficiency of the proposed method.

Stochastic Analysis on the Resonance of Railway Trains Moving over a Series of Simply Supported Bridges

Resonance problems encountered in vehicle-bridge interaction (VBI) have attracted widespread concern over the past decades. Due to system random characteristics, the prediction of resonant speeds and responses will become more complicated. To this end, this study presents stochastic analysis on the resonance of railway trains moving over a series of simply supported bridges with consideration of the randomness of system parameters. A train-slab track-bridge (TSB) vertically coupled dynamics model is established following the basic principle of vehicle-track-coupled dynamics. The railway train is composed of multiple vehicles, and each of them is built by seven rigid parts assigned with a total of 10 degrees of freedom. The rail, track slab, and bridge are considered as Euler–Bernoulli beams, and the vibration equations of which are established by the modal superposition method (MSM). Except for the nonlinear wheel-rail interaction based on the Hertz contact theory, the other coupling relations between each subsystem are assumed to be linear elastic. The number theory method is employed to obtain the representative sample point sets of the random parameters, and the flow trajectories of probabilities for the TSB dynamics system are captured by a probability density evolution method (PDEM). Numerical results indicate that the maximum bridge and vehicle responses are mainly dominated by the primary train-induced resonant speed; the last vehicle of a train will be more seriously excited when the bridges are set in resonance by the train; the resonant speeds and responses are rather sensitive to the system randomness, and the possible maximum amplitudes predicted by the PDEM are significantly underestimated by the traditional deterministic method; optimized parameters of the TSB system are preliminary obtained based on the representative point sets and imposed screening conditions.

Efficient reliability analysis based on deep learning-enhanced surrogate modelling and probability density evolution method

An improved method, termed as the AL-DLGPR-PDEM, is presented to address high-dimensional reliability problems. The novelty of this work lies in developing a complete framework for combining the deep learning (DL) architectures, serving as the utility of dimension reduction, and the Gaussian process regression (GPR), resulting in the so-called DLGPR model. First, the parameters of both the DL and the GPR are inferred using a joint-optimization scheme, rather than the traditional two-step, separate-training scheme. Second, the network configuration of the DLGPR is optimally determined by using a grid-search procedure involving cross-validation, instead of an empirical setting manner. On this basis, the DLGPR is adaptively refined via an active learning (AL)-based sampling strategy, so as to gain the desired DLGPR using as fewer training samples as possible. Eventually, the finalized DLGPR is evaluated at the whole representative point set, thereby the probability density evolution method (PDEM) is conducted accordingly. Two numerical examples are investigated. The first one tackles with the static reliability analysis of a planner steel frame, where the case of small failure probabilities is also considered; the second one addresses the dynamic reliability analysis of the steel frame under fully non-stationary stochastic seismic excitation. Comparisons against other existing reliability methods are conducted as well. Results demonstrate that the proposed AL-DLGPR-PDEM achieves a fair tradeoff between accuracy and efficiency for dealing with high-dimensional reliability problems in both static and dynamic analysis examples.

Augmented low-rank methods for gaussian process regression

This paper presents techniques to improve the prediction accuracy of approximation methods used in Gaussian process regression models. Conventional methods such as Nystrom and subset of data methods rely on low-rank approximations to the kernel matrix derived from a set of representative data points. Prediction accuracy suffers when the number of representative points is small or when the length scale is small. The techniques proposed here augment the set of representative points with neighbors of each test input to improve accuracy. Our approach leverages the general structure of the problem through the low-rank approximation and improves its accuracy further by exploiting locality at each test input. Computations involving neighbor points are cast as updates to the base approximation which result in significant savings. To ensure numerical stability, prediction is done via orthogonal projection onto the subspace of the kernel approximation derived from the augmented set. Experiments on synthetic and real datasets show that our approach is robust with respect to changes in length scale and matches the prediction accuracy of the full kernel matrix while using fewer points for kernel approximation. This results in faster and more accurate predictions compared to conventional methods.

Surrogate modeling immersed probability density evolution method for structural reliability analysis in high dimensions

In conjunction with advanced surrogate modeling methods, an improved scheme of probability density evolution method (PDEM) is presented to tackle with the challenge inherent in high-dimensional structural reliability analysis. In this method, the KPCA-GPR model is developed, where the kernel principal component analysis (KPCA)-based nonlinear dimension reduction and the Gaussian process regression (GPR) surrogate model are combined via a joint-training scheme. In this regard, the identified KPCA-based subspace is optimal to the approximation accuracy of the resultant GPR model. Then, the KPCA-GPR model is constructed using the active learning (AL)-based sampling strategy, so as to accurately approximate the equivalent extreme-value (EEV) of structural responses at the whole representative point set involved in the PDEM with as fewer samples as possible. Finally, the reliability is readily evaluated by the one-dimensional integral of the EEVs’ probability density function derived from the PDEM. To illustrate the effectiveness of the proposed AL-KPCA-GPR-PDEM, two numerical examples are studied, involving the reliability analysis of both nonlinear analytical functions with different dimensions and shear-frame structures under earthquake ground motions. Numerical results indicate that massive computational cost savings and desirable accuracy enhancement are achieved by the AL-KPCA-GPR-PDEM when dealing with the reliability problems in high dimensions.

Active learning and active subspace enhancement for PDEM-based high-dimensional reliability analysis

An efficient reliability method, termed as the AL-AS-GPR-PDEM, is proposed in this study, which effectively combines the probability density evolution method (PDEM) and the Gaussian process regression (GPR) surrogate model enhanced by both the active subspace (AS)-based dimension reduction and the active learning (AL)-based sampling strategy. In this method, the AS-GPR model is developed, where the AS and the GPR are jointly calibrated so as to bypass the tricky prerequisite of gradient information to discover the latent AS of the original parameter space; then, the AL-based sampling strategy is employed to construct a satisfactorily accurate AS-GPR model using as fewer training samples as possible. This treatment allows that the newly-added training sample at each iteration can be simultaneously used to both update the discovered low-dimensional AS and refine the constructed GPR model within the AS. Lastly, the finalized AS-GPR model is employed to predict the equivalent extreme values (EEVs) of structural responses at the representative point set, thereby the PDEM-based reliability analysis can be readily performed based on the estimated EEVs. To testify the effectiveness of the proposed method, four numerical examples are addressed, involving both surrogate modelling of analytical functions with different characteristics and dynamic reliability analysis of linear/nonlinear engineering structures under seismic excitations. It is demonstrated that the proposed method is of satisfactory accuracy and efficiency for structural reliability analysis in high dimensions.

Pure discrete spectrum and regular model sets in d-dimensional unimodular substitution tilings.

Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

An improved sieve point method for the reliability analysis of structures

The efficiency of existing stochastic analysis method depends on the discretization of the random variables domain. The number theoretical method has been proposed to discretize the random variable space and solve the generalized density evolution equation via sampling strategy. This method traditionally involves hyper-ball sieving (HS) algorithm to sample the representative point set. However, the sieving radius of the hyper-ball is determined subjectively, and the efficiency and accuracy of the analysis depend on the selected radius. To avoid this subjective selection, an equal volume hyper-ball sieving method is presented in this paper. By transforming the hypercube spatial volume of random variables into that of an equivalent hyper-ball, the radius of the equal volume hyper-ball is obtained analytically. This radius is further optimized with a minimum star discrepancy in the representative point set. The performance and accuracy of the proposed method are checked in four numerical examples, and the representative point set such obtained is more uniform with smaller NRP leading to more accurate and efficient subsequent stochastic analysis than the HS method.

Big data based research on the management system framework of ideological and political education in colleges and universities

Artificial intelligence model combined with data mining technology can mine useful data from college ideological and political education management, and conduct process evaluation and teaching management. Therefore, based on the superiority of data mining technology and artificial intelligence system, this paper improves the traditional algorithm and constructs a university ideological and political education management model based on big data artificial intelligence. Moreover, this study uses a local sensitive hash function to generate representative point sets and uses the generated representative point sets for clustering operations. In order to verify the performance of the algorithm model, a control experiment is designed to compare the algorithm of this paper with traditional data mining methods. It can be seen from the research results that the algorithm model constructed in this paper has good performance and can be applied to practice.