用随机奇异值分解算法求解矩阵恢复问题

本文研究了大型低秩矩阵恢复问题.利用随机奇异值分解（RSVD）算法,对稀疏矩阵做奇异值分解.该算法与Lanczos方法相比,在误差精度一致的同时运算时间大大降低,且该算法对相对低秩矩阵也有效.     

用随机奇异值分解算法求解矩阵恢复问题（英文）

本文研究了大型低秩矩阵恢复问题.利用随机奇异值分解(RSVD)算法,对稀疏矩阵做奇异值分解.该算法与Lanczos方法相比,在误差精度一致的同时运算时间大大降低,且该算法对相对低秩矩阵也有效.     

基于增秩Kalman滤波的动态荷载识别和结构响应重构

针对传统的荷载识别方法受不适定性问题影响导致识别误差较大,且受传感器数上的限制也无法监测所有结构易损伤位置处振动响应的问题,提出了一种基于增秩Kalman滤波（augmented Kalman filter, AKF）算法的动态荷载识别和结构响应重构方法.基于结构状态空间方程,形成由荷载向量和状态向量组成的增秩状态向量（augmented-rank state vector,ASV）,利用Kalman滤波算法获得增秩状态向量的最小方差无偏（minimum variance unbiased, MVU）估计,实现了状态和荷载向量的同时识别.结合最优状态估计和观测矩阵,实现了未布置传感器处的结构动力响应重构.通过三个有限元案例,初步验证了该方法的可行性和有效性.结果表明,当荷载位置固定或移动时,所提方法均能有效地识别荷载和重构响应,精度较高且对测量噪声不敏感.传感器的种类、数量和布置位置对荷载识别和响应重构精度会有一定影响.     

基于块循环矩阵的对称张量的最佳秩-1逼近

对称张量的最佳秩-1问题是张量研究中非常重要的部分.首先,基于三阶张量的块循环矩阵,提出了求解对称张量最佳秩-1逼近问题的一个新方法.其次,针对求解对称张量的最佳秩-1逼近方法,给出了对称张量的最佳秩-1逼近不变性的一个充要条件,以及逼近误差上界的估计.最后,数值算例表明了上述方法的可行性和误差上界的正确性.     

基于自由度缩聚的最小秩修正损伤诊断

为了解决实测模态参数与有限元分析模态参数不匹配对损伤诊断精度影响的问题,推导了基于自由度缩聚法的残余力向量公式及最小秩修正公式.通过对结构自由度缩聚后的损伤前、后残余力向量的运算,可以得出相较于损伤前的残余力变化率向量,将残余力变化率向量元素的绝对值作为改进的残余力向量,通过运用推导出的改进残余力向量,能够较好地解决采用最小秩修正法时所选取模态个数必须等于待修正刚度矩阵秩这一矛盾,并由缩聚后最小秩修正公式计算出损伤程度.研究表明:在考虑噪音干扰下,改进的残余力向量法对自由度缩聚后的受损结构依然具有较高地识别精度.利用推导的最小秩修正公式进行损伤程度识别其结果是可靠的.本文所提方法既可以实现对实测自由度不完备结构的损伤定位,又可进行损伤程度的识别,具有较高的鲁棒性和损伤诊断性能.     

计算函数导数的FODF-SPH方法的修正

在光滑粒子流体动力学(Smooth Particle Hydrodynamics:SPH)核近似方法原理的基础上,通过泰勒级数展开已提出的计算函数导数的FODF-SPH(Frist Order Derivative Free:FODF)方法进行了修正,并分别推导出一元和多元函数的修正公式.用不同的粒子间距和不同的光滑长度计算一元函数的导数和多元函数的偏导数,修正公式与FODF-SPH方法误差进行了对比分析.结果表明,提出的修正措施在提高精度、减少误差及加快收敛速度等方面起了很大的作用.     

分子动力学模拟中非成键相互作用计算的误差估计

高效的非成键相互作用计算对于分子动力学模拟具有核心意义.本文在一个统一的框架下,综述短程相互作用的截断方法、长程静电相互作用的光滑粒子网格Ewald方法和交错网格Ewald方法的误差估计.与传统的误差估计假设体系均匀且无相关性不同,本文介绍的误差估计可以推广到非均匀和有相关性的体系.本文通过具体例子讨论非均匀性和相关性对误差的本质性影响,以及可能的修正方式,并说明误差估计对于提高非成键相互作用的计算精度和速度有重要作用.本文还展示一个针对光滑粒子网格Ewald方法的实用参数优化方法,使得在保证精度的同时选取计算效率近似最优的参数组合成为可能,改善了传统上参数全凭经验选取的局面.     

二维土壤溶质输运问题的CN有限元法

首先给出二维土壤溶质输运问题时间二阶精度的Crank-Nicolson(CN)时间半离散化格式,然后直接从CN时间半离散化格式出发,建立具有时间二阶精度的全离散化CN有限元格式,并给出CN有限元解的误差分析,最后用数值例子验证全离散化CN有限元格式的优越性.这种方法提高了时间离散的精度,并极大地减少时间方向的迭代步,从而减少实际计算中截断误差的积累,提高计算精度和计算效率.而且方法绕开对空间变量半离散化有限元格式的讨论,使得理论研究更简便.     

解一类Hessian矩阵亏秩的修正BFGs算法及其局部Q-超线性收敛性

本文对凸函数在极值点的Hessian矩阵是秩亏一的情况下,给出了一类求解无约束优化问题的修正BFGS算法.算法的思想是对凸函数加上一个修正项,得到一个等价的模型,然后简化此模型得到一个修正的BFGS算法.文中证明了该算法是一个具有超线性收敛的算法,并且把修正的BFGS算法同Tensor方法进行了数值比较,证明了该算法对求解秩亏一的无约束优化问题更有效.     

“超单元”法的误差估计

一、引言 关于平面弹性断裂问题,文献[1]指出采用普通的有限元方法。由于裂缝尖端处奇性的影响,其能量模误差只能得到h~(1/2)阶估计,不管采用多少次插值函数都不能改善其精度。因此用通常的有限元法计算弹性断裂问题,其精度是很差的,文献[2—3]通过在奇     

关于TLS和LS解的扰动分析

1．引言本文采用卜］的记号．最小二乘（LS）和总体最小二乘（TLS）是科学计算中的两种重要方法．尤是TLS，近来已有多篇论文讨论［1－6，8－16］．奇异值分解（SVD）和CS分解是研究TLS和LS的重要工具．令ACm，BCm,C=(A,B)，A和C的SVD分别为（1．1）（1．2）其中P51为某个正整数，U，U，V，V均为西矩阵，UI，UI，VI，VI为上述矩阵的前P列，z1一山。g（。1，…，内），】2＝di。g（内十l，…，。小】1＝dl。g（61；…，站，】2二diag（4＋1；…，dk），。l三··2。120和dl三…三d。20分别为C和A的奇异值，Z＝mhfm．n十以…     

非线性不等式组的光滑近似方法及其收敛性

将非线性不等式组的求解转化成非线性最小二乘问题,利用引入的光滑辅助函数,构造新的极小化问题来逐次逼近最小二乘问题.在一定的条件下,文中所提出的光滑高斯-牛顿算法的全局收敛性得到保证.适当条件下,算法的局部二阶收敛性得到了证明.文后的数值试验表明本文算法有效.     

Deep partial least squares for instrumental variable regression

In this paper, we propose deep partial least squares for the estimation of high-dimensional nonlinear instrumental variable regression. As a precursor to a flexible deep neural network architecture, our methodology uses partial least squares for dimension reduction and feature selection from the set of instruments and covariates. A central theoretical result, due to Brillinger (2012) Selected Works of Daving Brillinger. 589-606, shows that the feature selection provided by partial least squares is consistent and the weights are estimated up to a proportionality constant. We illustrate our methodology with synthetic datasets with a sparse and correlated network structure and draw applications to the effect of childbearing on the mother's labor supply based on classic data of Chernozhukov et al. Ann Rev Econ. (2015b):649–688. The results on synthetic data as well as applications show that the deep partial least squares method significantly outperforms other related methods. Finally, we conclude with directions for future research.     

矩阵方程AXB+CY D=E的三对角中心对称极小范数最小二乘解

本文研究了矩阵方程AXB+CY D=E的三对角中心对称极小范数最小二乘解问题.利用矩阵的Kronecker积和Moore-Penrose广义逆方法,得到了矩阵方程AXB+CY D=E的三对角中心对称极小范数最小二乘解的表达式.     

A NEW PSEUDOSPECTRAL APPROXIMATION FOR THE FOWARD-BACKWARD HEAT EQUATION

ＡＮＥＷＰＳＥＵＤＯＳＰＥＣＴＲＡＬＡＰＰＲＯＸＩＭＡＴＩＯＮＦＯＲＴＨＥＦＯＷＡＲＤ－ＢＡＣＫＷＡＲＤＨＥＡＴＥＱＵＡＴＩＯＮ￥ＹｅＸｉｎｇｄｅ（叶兴德）ＪｉａｎｇＪｉｎｓｈｅｎｇ（江金生）（Ｄｅｐｔ．ｏｆＭａｔｈ．＆Ｉｎｆｏｒ．Ｓｃｉ．，Ｈａｎｇ...     

A weighted least squares method for scattered data fitting

In this paper, we present a weighted least squares method to fit scattered data with noise. Existence and uniqueness of a solution are proved and an error bound is derived. The numerical experiments illustrate that our weighted least squares method has better performance than the traditional least squares method in case of noisy data.     

TLS和LS问题的比较

There are a number of articles discussing the total least squares(TLS) and the least squares(LS) problems.M.Wei(M.Wei, Mathematica Numerica Sinica 20(3)(1998),267-278) proposed a new orthogonal projection method to improve existing perturbation bounds of the TLS and LS problems.In this paper,wecontinue to improve existing bounds of differences between the squared residuals,the weighted squared residuals and the minimum norm correction matrices of the TLS and LS problems.     

Monotonic properties of the least squares mean

We settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual (Loewner) order. Indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature. Our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity. Moreover, we introduce a weighted least squares mean and derive our results in this more general setting.     

Randomized algorithms for total least squares problems

Motivated by the recently popular probabilistic methods for low‐rank approximations and randomized algorithms for the least squares problems, we develop randomized algorithms for the total least squares problem with a single right‐hand side. We present the Nyström method for the medium‐sized problems. For the large‐scale and ill‐conditioned cases, we introduce the randomized truncated total least squares with the known or estimated rank as the regularization parameter. We analyze the accuracy of the algorithm randomized truncated total least squares and perform numerical experiments to demonstrate the efficiency of our randomized algorithms. The randomized algorithms can greatly reduce the computational time and still maintain good accuracy with very high probability.     

Regularization tools and robust optimization for ill-conditioned least squares problem: A computational comparison

Least squares problems arise frequently in many disciplines such as image restorations. In these areas, for the given least squares problem, usually the coefficient matrix is ill-conditioned. Thus if the problem data are available with certain error, then after solving least squares problem with classical approaches we might end up with a meaningless solution. Tikhonov regularization, is one of the most widely used approaches to deal with such situations. In this paper, first we briefly describe these approaches, then the robust optimization framework which includes the errors in problem data is presented. Finally, our computational experiments on several ill-conditioned standard test problems using the regularization tools, a Matlab package for least squares problem, and the robust optimization framework, show that the latter approach may be the right choice.