Approximating Pareto curves using semidefinite relaxations

We approximate as closely as desired the Pareto curve associated with bicriteria polynomial optimization problems. We use three formulations (including the weighted sum approach and the Chebyshev approximation) and each of them is viewed as a parametric polynomial optimization problem. For each case is associated a hierarchy of semidefinite relaxations and from an optimal solution of each relaxation one approximates the Pareto curve by solving an inverse problem (first two cases) or by building a polynomial underestimator (third case).     

Imposing symmetry in augmented linear systems

This paper discusses the methods of imposing symmetry in the augmented system formulation (ASF) for least‐squares (LS) problems. A particular emphasis is on upper Hessenberg problems, where the challenge lies in leaving all zero‐by‐definition elements of the LS matrix unperturbed. Analytical solutions for optimal perturbation matrices are given, including upper Hessenberg matrices. Finally, the upper Hessenberg LS problems represented by unsymmetric ASF that indicate a normwise backward stability of the problem (which is not the case in general) are identified. It is observed that such problems normally arise from Arnoldi factorization (for example, in the generalized minimal residual (GMRES) algorithm). The problem is illustrated with a number of practical (arising in the GMRES algorithm) and some ‘purpose‐built’ examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Efficient statistical inference for partially nonlinear errors-in-variables models

In this paper, we consider the partially nonlinear errors-in-variables models when the non- parametric component is measured with additive error. The profile nonlinear least squares estimator of unknown parameter and the estimator of nonparametric component are constructed, and their asymptotic properties are derived under general assumptions. Finite sample performances of the proposed statistical inference procedures are illustrated by Monte Carlo simulation studies.     

Convergence of densities of some functionals of Gaussian processes

The aim of this paper is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein?s method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a fixed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some fixed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein–Uhlenbeck processes is discussed.     

求解非线性最小二乘问题的迭代法

基于解线性方程组的Jacobi迭代法的思想，本文提出了一种求解非线性最小二乘问题的迭代法，大量的数值试验表明该迭代法具有较好的收敛性，另外，还将所设计的迭代法进行了并行化。     

天然气消费量的偏最小二乘支持向量机预测

结合偏最小二乘法和支持向量机的优缺点,提出基于偏最小二乘支持向量机的天然气消费量预测模型。首先,利用偏最小二乘法确定影响天然气消费量的新综合变量,建立以新综合变量为输入,天然气消费量为输出的支持向量机模型,对天然气消费量进行了预测;然后,与多元回归、偏最小二乘回归、普通支持向量机做误差检验比较,验证该方法的可行性与正确性。结果表明,此天然气消费量预测模型具有较高的精确度和应用价值。     

Linear Least Squares Estimation of Regression Models for Two-Dimensional Random Fields

We consider the problem of estimating regression models of two-dimensional random fields. Asymptotic properties of the least squares estimator of the linear regression coefficients are studied for the case where the disturbance is a homogeneous random field with an absolutely continuous spectral distribution and a positive and piecewise continuous spectral density. We obtain necessary and sufficient conditions on the regression sequences such that a linear estimator of the regression coefficients is asymptotically unbiased and mean square consistent. For such regression sequences the asymptotic covariance matrix of the linear least squares estimator of the regression coefficients is derived.     

“接触式轮廓仪的自动标注”的问题解析

就2020年"高教社杯"全国大学生数学建模竞赛D题"接触式轮廓仪的自动标注"给出了一种求解方法,并针对学生在参赛论文中出现的问题作了简要的说明与点评.为保证求解的连贯性,论文的前一部分是问题的求解,后一部分是参赛论文的点评.     

A class of discontinuous Petrov–Galerkin methods. II. Optimal test functions

We lay out a program for constructing discontinuous Petrov–Galerkin (DPG) schemes having test function spaces that are automatically computable to guarantee stability. Given a trial space, a DPG discretization using its optimal test space counterpart inherits stability from the well posedness of the undiscretized problem. Although the question of stable test space choice had attracted the attention of many previous authors, the novelty in our approach lies in the fact we identify a discontinuous Galerkin (DG) framework wherein test functions, arbitrarily close to the optimal ones, can be locally computed. The idea is presented abstractly and its feasibility illustrated through several theoretical and numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

拟线性回归预测模型的稳定最小二乘解

可线性化回归预测模型通过换元进行线性回归,换元前后的因变量具有异方差性,致使拟线性回归参数的精度较低.运用GL算法给出了此类模型的稳定最小二乘解,提高了参数的估计精度,最后给出了一个应用实例.