On the finite element approximation of elliptic QVIs with noncoercive operators

In this paper, we extend the approach developed by the author for the standard finite element method in the L^∞‐norm of the noncoercive variational inequalities (VI) (Numer Funct Anal Optim.2015;36:1107‐1121.) to impulse control quasi‐variational inequality (QVI). We derive the optimal error estimate, combining the so‐called Bensoussan‐Lions Algorithm and the concept of subsolutions for VIs.     

User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient

In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide non asymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.     

Statistical inference for coherent systems with Weibull distributed component lifetimes under complete and incomplete information

Point estimators for the parameters of the component lifetime distribution in coherent systems are evolved assuming to be independently and identically Weibull distributed component lifetimes. We study both complete and incomplete information under continuous monitoring of the essential component lifetimes. First, we prove that the maximum likelihood estimator (MLE) under complete information based on progressively Type‐II censored system lifetimes uniquely exists and we present two approaches to compute the estimates. Furthermore, we consider an ad hoc estimator, a max‐probability plan estimator and the MLE for the parameters under incomplete information. In order to compute the MLEs, we consider a direct maximization of the likelihood and an EM‐algorithm–type approach, respectively. In all cases, we illustrate the results by simulations of the five‐component bridge system and the 10‐component parallel system, respectively.     

面向应急对地观测任务的多平台资源部署优化研究

提出了一种基于遗传算法的面向应急对地观测任务的多平台资源部署优化方法。该方法通过把观测区域离散化为网格点的集合,将多平台资源部署问题形式化为一个组合优化问题,其目标是在一定响应时间约束下最大化观测区域覆盖率。设计的求解算法采用整数编码表示各平台资源的部署位置,使用精英保留策略加快算法收敛速度。仿真结果表明,该方法能够快速获得满意的卫星、飞艇、无人机多平台资源部署方案。     

Nonsingular systems of generalized Sylvester equations: An algorithmic approach

We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ?‐Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ?‐Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition and leads to a backward stable O(n³r) algorithm for computing the (unique) solution.     

A Penalized Likelihood Method for Classification With Matrix-Valued Predictors

We propose a penalized likelihood method to fit the linear discriminant analysis model when the predictor is matrix valued. We simultaneously estimate the means and the precision matrix, which we assume has a Kronecker product decomposition. Our penalties encourage pairs of response category mean matrix estimators to have equal entries and also encourage zeros in the precision matrix estimator. To compute our estimators, we use a blockwise coordinate descent algorithm. To update the optimization variables corresponding to response category mean matrices, we use an alternating minimization algorithm that takes advantage of the Kronecker structure of the precision matrix. We show that our method can outperform relevant competitors in classification, even when our modeling assumptions are violated. We analyze three real datasets to demonstrate our method’s applicability. Supplementary materials, including an R package implementing our method, are available online.     

An Asynchronous Distributed Expectation Maximization Algorithm for Massive Data: The DEM Algorithm

The family of expectation--maximization (EM) algorithms provides a general approach to fitting flexible models for large and complex data. The expectation (E) step of EM-type algorithms is time-consuming in massive data applications because it requires multiple passes through the full data. We address this problem by proposing an asynchronous and distributed generalization of the EM called the distributed EM (DEM). Using DEM, existing EM-type algorithms are easily extended to massive data settings by exploiting the divide-and-conquer technique and widely available computing power, such as grid computing. The DEM algorithm reserves two groups of computing processes called workers and managers for performing the E step and the maximization step (M step), respectively. The samples are randomly partitioned into a large number of disjoint subsets and are stored on the worker processes. The E step of DEM algorithm is performed in parallel on all the workers, and every worker communicates its results to the managers at the end of local E step. The managers perform the M step after they have received results from a γ-fraction of the workers, where γ is a fixed constant in (0, 1]. The sequence of parameter estimates generated by the DEM algorithm retains the attractive properties of EM: convergence of the sequence of parameter estimates to a local mode and linear global rate of convergence. Across diverse simulations focused on linear mixed-effects models, the DEM algorithm is significantly faster than competing EM-type algorithms while having a similar accuracy. The DEM algorithm maintains its superior empirical performance on a movie ratings database consisting of 10 million ratings. Supplementary material for this article is available online.     

一种结构动力响应分析的快速高阶算法

为了提高基于高阶格式的结构动力响应微分求积分析方法的计算效率,发展了一种求解动力方程的快速算法．利用微分求积原理将结构动力方程转化为标准Sylvester方程的形式,通过对系数矩阵进行矩阵分解,进而将动力响应Sylvester方程化为一系列标准线性方程组,采用相关成熟算法求解这些线性方程组后即可获得结构动力时程响应的全部解答．结构动力响应微分求积分析方法为高阶数值方法,一步计算可以获得多个时点处的动力响应．基于本文快速算法,不必直接对矩阵方程进行求解．数值算例表明,本文快速算法能够准确地计算出结构动力响应,具有数值精度高、收敛性好的优点．     

单轴压缩条件下溶蚀礁灰岩细观变形破坏特征研究

礁灰岩的溶蚀对海洋工程和岛礁工程的稳定性有明显的影响.孔隙型溶蚀是礁灰岩常见的溶蚀方式.本文通过构建随机溶蚀孔洞的离散元模型,模拟孔洞型溶蚀礁灰岩的细观变形破坏特征,分析其变形破坏规律与裂隙演化特征.结果 表明,基于随机聚类算法构建的溶蚀礁灰岩离散元模型能够很好地模拟孔洞型溶蚀作用,且其变形破坏特征具有明显的规律;随着溶蚀率的增加,溶蚀礁灰岩单轴应力应变曲线由单峰型逐渐转变为双峰型或多峰型,破坏特征由脆性破坏逐渐转变为塑性破坏;随着溶蚀率的增加,礁灰岩单轴抗压强度和弹性模量都逐渐减小,抗压强度呈指数函数下降,弹性模量呈线性函数下降,而泊松比则表现为先增后降,说明试样在高溶蚀率时主要发生结构性破坏;随着溶蚀率增加,礁灰岩的宏观破裂面逐渐消失,由于溶蚀孔洞的相互作用,导致了起源于孔洞周围的裂隙开始萌生,并逐渐贯通,最终导致试样发生整体宏观破坏.本研究成果可为深入认识海洋工程和岛礁工程中溶蚀礁灰岩的变形破坏特征提供理论参考.     

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.