Convex Aggregation Research Articles

A mixed uncertain structural reliability analysis method considering random and convex set variables with correlation

AbstractIn this paper, a mixed model reliability analysis method is put forward for the problem of assessing the reliability of complex engineering structures containing both random and convex aggregate variables. By integrating the ellipsoidal model with correlation and the interval model, the uncertainty region characterized by the ellipsoidal model with correlation is optimized with full consideration of the limited amount of engineering structure sample data, and the risk region represented by the reliability model is redefined, a new mixed reliability assessment criterion is established, and the minimum safe nonprobability reliability index of the structure is built. A key reference is offered by the safety limit diagram of constant reliability indexes set for the first time for the optimal design of engineering structure reliability. The proposed mixed reliability model is compared and analyzed with three classical models. The sensitivity of nonprobability reliability indexes, influenced by random variable parameters and correlation coefficients, is analyzed. This verification confirms that the new reliability model not only provides an accurate assessment of engineering structures' reliability but also lowers the computational demands of engineering design. In this paper, aero‐engine blades and wind turbine blades are taken as examples to expound the validity of the reliability model built using this method and its importance to the structural safety analysis of actual engineering.

Surface subdivision-based method for modeling three-dimensional aggregate with controllable concavity and texture in asphalt concrete mesostructure

Aggregate shape properties can significantly affect the mesoscopic behavior of asphalt concrete. This study proposed a subdivision-based aggregate generation method that can model three-dimensional (3D) aggregates with controllable surface concavity and texture, allowing for investigations on the aggregate shape related mechanisms of asphalt concrete. This method first establishes basic convex aggregates and then repeatedly performs a subdivision-based reshaping procedure to continuously incorporate geometric details into the aggregates. Aggregates with different levels of concavity and texture were generated to form asphalt concrete mesostructures. Based on finite element simulations, the effect of the aggregate concavity on the mechanical properties of asphalt concrete was investigated. The results indicated that both the concavity and texture can be effectively and readily controlled by this method. Increasing aggregate concavity can improve the dynamic modulus of asphalt concrete especially at high temperatures, and may enhance its overall crack resistance but cause more severe local damages.

Generalizing Level Ranking Constraints for Monotone and Convex Aggregates

Generalizing Level Ranking Constraints for Monotone and Convex Aggregates

Reconstruction of Convex Sets from One or Two X-rays

We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called the bad guy and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the digital convex sets which are not fat remains an open question.

An Efficient Construction Method of the 3D Random Asphalt Concrete Model Based on the Background Grid and the Moving-and-Densifying Algorithm

In order to avoid the tedious and time-consuming measuring process for thermal conductivity, many random models have been proposed, but the construction of those random models is still inefficient, which limits the further application. In this paper, a construction method of three-dimensional random asphalt models for predicting thermal conductivity based on the background grid and the moving-and-densifying algorithm was proposed which greatly improves construction efficiency. The influence of the random factors on models’ stability was studied and the range of the key factors within all random factors was restricted. Further, a conflict judgment method for the convex aggregate and the improved take-and-place method based on the background grid method and the moving-and-densifying algorithm was realized by MATLAB code to construct aggregate mixture models. Finally, the aggregate mixtures model was imported into ABAQUS 2022 to predict the thermal conductivity based on the steady-state plate method, and the validity of the predicting result was verified by experimental result. With this construction method, the stability index is improved by more than 80.71%, and packing efficiency is 198.98% higher than before. Additionally, the 3D random model showed a smaller prediction error range (less than 5%) than the 2D models (more than 10%) and was more accurate than the 2D prediction model. This research focused on improving the construction efficiency of the 3D random asphalt concrete model which contributes to full utilization and laying a foundation for further improvement.

Variance function estimation in regression model via aggregation procedures

In the regression problem, we consider the problem of estimating the variance function by means of aggregation methods. We focus on two particular aggregation setting: Model Selection aggregation (MS) and Convex aggregation (C), where the goal is to select the best candidate and to build the best convex combination of candidates, respectively, among a collection of candidates. In both cases, the construction of the estimator relies on a two-step procedure and requires two independent samples. The first step exploits the first sample to build the candidate estimators for the variance function by the residual-based method and then the second dataset is used to perform the aggregation step. We show the consistency of the proposed method with respect to the -error both for MS and C aggregations. We evaluate the performance of these two methods in the heteroscedastic model and illustrate their interest in both the regression problem with reject option and the quantile regression.

Spectral Clustering via Adaptive Layer Aggregation for Multi-Layer Networks

One of the fundamental problems in network analysis is detecting community structure in multi-layer networks, of which each layer represents one type of edge information among the nodes. We propose integrative spectral clustering approaches based on effective convex layer aggregations. Our aggregation methods are strongly motivated by a delicate asymptotic analysis of the spectral embedding of weighted adjacency matrices and the downstream k-means clustering, in a challenging regime where community detection consistency is impossible. In fact, the methods are shown to estimate the optimal convex aggregation, which minimizes the misclustering error under some specialized multi-layer network models. Our analysis further suggests that clustering using Gaussian mixture models is generally superior to the commonly used k-means in spectral clustering. Extensive numerical studies demonstrate that our adaptive aggregation techniques, together with Gaussian mixture model clustering, make the new spectral clustering remarkably competitive compared to several popularly used methods. Supplementary materials for this article are available online.

Convex Aggregation for Opinion Summarization

Recent advances in text autoencoders have significantly improved the quality of the latent space, which enables models to generate grammatical and consistent text from aggregated latent vectors. As a successful application of this property, unsupervised opinion summarization models generate a summary by decoding the aggregated latent vectors of inputs. More specifically, they perform the aggregation via simple average. However, little is known about how the vector aggregation step affects the generation quality. In this study, we revisit the commonly used simple average approach by examining the latent space and generated summaries. We found that text autoencoders tend to generate overly generic summaries from simply averaged latent vectors due to an unexpected L2-norm shrinkage in the aggregated latent vectors, which we refer to as summary vector degeneration. To overcome this issue, we develop a framework Coop, which searches input combinations for the latent vector aggregation using input-output word overlap. Experimental results show that Coop successfully alleviates the summary vector degeneration issue and establishes new state-of-the-art performance on two opinion summarization benchmarks. Code is available at https://github.com/megagonlabs/coop.

New Insights From the Shapley-Folkman Lemma on Dispatchable Demand in Energy Markets

In this paper we consider the aggregation of common convex and non-convex individual Demand Response (DR) models for responsive loads, and apply the Shapley-Folkman (SF) lemma to show that such an aggregate is approximately convex in its action space and cost, and strictly convex under mild conditions. We then discuss how reduced order convex aggregate models can be passed on to System Operators for inclusion in economic and operational models, including a novel polytope approximation as well as an ellipsoidal model derived in a distributed fashion. Numerical simulations confirm our application of the Shapley-Folkman lemma and show that the new reduced order models outperform conventional virtual generator approximations.

Three-dimensional mesoscale modeling of concrete with convex aggregate based on motion simulation

Numerical simulation using a mesoscale model is an effective method to determine the complex mechanical behaviors of concrete. Establishing a numerical sample with a mesostructure close to realistic concrete is crucial for mesoscale simulation. This paper presents a heuristic packing algorithm for constructing 3D random aggregate models with high aggregate content. In the algorithm, the aggregate packing problem is considered as a problem of optimizing the potential energy of the mesoscale configuration. In the initial state, all aggregate particles are randomly placed inside the specimen container; then, the particles move in the container according to the proposed motion simulation strategy until a feasible spatial configuration is obtained. The framework is suitable for various types of aggregates, including spherical, ellipsoidal, polyhedral, and mixed-type aggregates, that have a support mapping representation. In this study, numerous samples with different types of aggregates were established, and the influence of the aggregate gradation on the packing density was examined. The results verified the efficiency and robustness of the proposed algorithm.

Random generation of convex aggregates for DEM study of particle shape effect

In this study, the random generation of 3D convex aggregates is proposed for the numerical study of the particle shape effect in DEM simulation. First, the Bezier curve fitting technique is employed to construct a curve skeleton inside a randomly generated polyhedron. Then, a solid surface is generated based on the curve skeleton using the Gregory surface interpolation algorithm. Next, convex aggregates are created inside both regular and irregular polyhedrons to demonstrate the capability of the proposed algorithm. Finally, DEM simulation of convex aggregates with different roundness features is performed to study the effect of roundness on the shear behavior of granular aggregates.

Three-dimensional mesoscale computational modeling of soil-rock mixtures with concave particles

Soil-rock mixtures (SRMs) are the main unfavorable geologic bodies in Southwest China. This paper presents a novel mesoscale computational modeling study of SRMs with concave aggregates. An efficient 3D mesoscale SRM generation method is proposed by combining the Gilbert–Johnson–Keerthi (GJK)-based collision detection technique, the border placement algorithm and the particle position selection method. A periodic mesh is generated based on the mesh mapping technique. A numerical homogenization analysis of an SRM with a large number of elements is realized, and the estimated parameters are validated by the experimental test results. The results indicate that SRMs with concave aggregates have a higher elastic modulus than those with convex aggregates. This method is helpful for predicting the physical properties of SRMs and has promising applications in engineering.

Impact of the properties of old mortar on creep prediction model of recycled aggregate concrete

By changing the replacement percentage of recycled coarse aggregate (RCA) (0%, 50%, 100%), the basic creep performance and total creep performance of recycled aggregate concrete (RAC) were studied. The experimental results reveal that increasing the RCA replacement percentage can both increase the basic creep deformation and total creep deformation of RAC with the RCA having a bigger impact on the basic creep of RAC. Based on the experimental research and studies on the interaction mechanism of natural coarse aggregate (NCA), old mortar and new mortar in the creep process of RAC, a calculation model of the relationship between the creep of the old mortar, new mortar and the RAC (ONRM) was established. The creep of the old mortar in the RAC was calculated by using the ONRM. Based on this, a two-dimensional random convex aggregate model of the RAC was established with ANSYS finite element software, and the creep of the RAC was calculated. The results of the calculation showed that the ONRM could well reflect the relationship between the creep of the old mortar, new mortar, and the RAC. From above analysis, with ACI 209(92) model as the benchmark, a creep prediction model reflecting the content and performance of the old mortar was built. It is proven that this model can calculate the creep of RAC well.

Localized Gaussian width of $M$-convex hulls with applications to Lasso and convex aggregation

Upper and lower bounds are derived for the Gaussian mean width of a convex hull of $M$ points intersected with a Euclidean ball of a given radius. The upper bound holds for any collection of extreme points bounded in Euclidean norm. The upper bound and the lower bound match up to a multiplicative constant whenever the extreme points satisfy a one sided Restricted Isometry Property. An appealing aspect of the upper bound is that no assumption on the covariance structure of the extreme points is needed. This aspect is especially useful to study regression problems with anisotropic design distributions. We provide applications of this bound to the Lasso estimator in fixed-design regression, the Empirical Risk Minimizer in the anisotropic persistence problem, and the convex aggregation problem in density estimation.

Mesoscopic Numerical Simulation of Fracture Process and Failure Mechanism of Concrete Based on Convex Aggregate Model

To investigate the fracture process and failure mechanism of concrete subjected to uniaxial compressive loading, a new finite element method—the base force element method (BFEM)—was adopted in the modeling of numerical simulation. At mesoscale, concrete is considered as a three-phase heterogeneous material composed of aggregate particles, cement mortar, and the interfacial transition zones between the two phases. A two-dimensional random convex aggregate model was established using the principle of the area equivalence method. A multistage linear damage constitutive model that can describe nonlinear behavior of concrete under mechanical stress was proposed. The mechanical properties of concrete mesoscopic components are determined. The numerical simulation results indicate that the base force element method can be applied to predict the failure pattern of concrete under compressive loading, which have a good accordance with the available experiment data. The stress contour plots were given and used to analyze the failure mechanism of concrete. The effects of specimen size on the strength of concrete material were studied. It is found that compressive strength of concrete decreases as the specimen size increases. In addition, the influences of aggregate distribution, coarse aggregate content, and end friction on concrete performance are explored.

A novel numerical scheme for random parameterized convex aggregate models with a high-volume fraction of aggregates in concrete-like granular materials

Concrete at a mesoscopic scale is normally regarded as a three-phase composite consisting of cement paste, aggregates and their surrounding interfacial transition zones (ITZs). Establishing mesostructure model close to realistic concrete is very crucial to precisely evaluate its mechanical properties. The preponderance of previous investigations has focused on aggregates as circles, ellipses or polygons constructed by the assembly of sides-sides with a low packing density, and little is known about precise mathematical characterizations for polygonal aggregates with a high packing density more than 60% and their surrounding ITZs. In this work, a novel numerical framework that adopts the deformation of a rhombus to mathematically provide a parametric equation of convex polygon characterizing the geometrical morphology of aggregates, is proposed to generate random polygonal aggregate models (RPAMs) with a high packing density. In this framework, a fast-random packing algorithm (FRPA) is developed to generate a high packing density of aggregates of 70%. Based on the parametric equation of polygonal aggregates, the geometrical topology of ITZs is mathematically realized with a convenient manner, rather than those cumbersome approximate operations reported in the literature. Moreover, the present numerical framework can be extended to the three-dimensional case.

Optimal Kullback–Leibler aggregation in mixture density estimation by maximum likelihood

We study the maximum likelihood estimator of density of n independent observations, under the assumption that it is well approximated by a mixture with a large number of components. The main focus is on statistical properties with respect to the Kullback–Leibler loss. We establish risk bounds taking the form of sharp oracle inequalities both in deviation and in expectation. A simple consequence of these bounds is that the maximum likelihood estimator attains the optimal rate ((\mathrm {log} K)/n)^{1/2} , up to a possible logarithmic correction, in the problem of convex aggregation when the number K of components is larger than n^{1/2} . More importantly, under the additional assumption that the Gram matrix of the components satisfies the compatibility condition, the obtained oracle inequalities yield the optimal rate in the sparsity scenario. That is, if the weight vector is (nearly) D -sparse, we get the rate (D\mathrm {log} K)/n . As a natural complement to our oracle inequalities, we introduce the notion of nearly- D -sparse aggregation and establish matching lower bounds for this type of aggregation.

Fast adaptive estimation of log-additive exponential models in Kullback-Leibler divergence

We study the problem of nonparametric estimation of probability density functions (pdf) with a product form on the domain $\triangle =\{(x_{1},\ldots ,x_{d})\in{\mathbb{R}} ^{d},0\leq x_{1}\leq \dots\leq x_{d}\leq 1\}$. Such pdf’s appear in the random truncation model as the joint pdf of the observations. They are also obtained as maximum entropy distributions of order statistics with given marginals. We propose an estimation method based on the approximation of the logarithm of the density by a carefully chosen family of basis functions. We show that the method achieves a fast convergence rate in probability with respect to the Kullback-Leibler divergence for pdf’s whose logarithm belong to a Sobolev function class with known regularity. In the case when the regularity is unknown, we propose an estimation procedure using convex aggregation of the log-densities to obtain adaptability. The performance of this method is illustrated in a simulation study.

Convergence rate analysis of distributed optimization with projected subgradient algorithm

In this paper, we revisit the consensus-based projected subgradient algorithm proposed for a common set constraint. We show that the commonly adopted non-summable and square-summable diminishing step sizes of subgradients can be relaxed to be only non-summable, if the constrained optimum set is bounded. More importantly, for a strongly convex aggregate cost with different types of step sizes, we provide a systematical analysis to derive the asymptotic upper bound of convergence rates in terms of the optimum residual, and select the best step sizes accordingly. Our result shows that a convergence rate of O(1∕k) can be achieved with a step size O(1∕k).

Empirical entropy, minimax regret and minimax risk

We consider the random design regression model with square loss. We propose a method that aggregates empirical minimizers (ERM) over appropriately chosen random subsets and reduces to ERM in the extreme case, and we establish sharp oracle inequalities for its risk. We show that, under the $\varepsilon^{-p}$ growth of the empirical $\varepsilon$-entropy, the excess risk of the proposed method attains the rate $n^{-2/(2+p)}$ for $p\in(0,2)$ and $n^{-1/p}$ for $p>2$ where $n$ is the sample size. Furthermore, for $p\in(0,2)$, the excess risk rate matches the behavior of the minimax risk of function estimation in regression problems under the well-specified model. This yields a conclusion that the rates of statistical estimation in well-specified models (minimax risk) and in misspecified models (minimax regret) are equivalent in the regime $p\in(0,2)$. In other words, for $p\in(0,2)$ the problem of statistical learning enjoys the same minimax rate as the problem of statistical estimation. On the contrary, for $p>2$ we show that the rates of the minimax regret are, in general, slower than for the minimax risk. Our oracle inequalities also imply the $v\log(n/v)/n$ rates for Vapnik-Chervonenkis type classes of dimension $v$ without the usual convexity assumption on the class; we show that these rates are optimal. Finally, for a slightly modified method, we derive a bound on the excess risk of $s$-sparse convex aggregation improving that of Lounici [Math. Methods Statist. 16 (2007) 246-259] and providing the optimal rate.