对称正交矩阵反问题及其最佳逼近

本文主要讨论下面两个问题：问题Ⅰ：给定矩阵X,B∈R~(m×n),求对称正交矩阵A∈SOR~(m×m),使得AX=B．问题Ⅱ：给定矩阵(?)∈R~(m×m),求矩阵A~*∈S_E使得(?)这里S_E问题Ⅰ的解集合,‖·‖指Frobenius范数．本文首先讨论具有k阶对称主子阵的n(n＞k)阶正交矩阵的C-S分解,利用这个结果,得到了问题Ⅰ有解的充要条件和通解的一般形式．然后,对给定矩阵(?)∈R~(m×m),讨论了矩阵(?)在问题Ⅰ的解集合S_E中的最佳逼近,得到了最佳逼近解的表达式．     

基于交替投影算法求解单变量线性约束矩阵方程问题

研究如下线性约束矩阵方程求解问题:给定A∈R~(m×n),B∈R~(n×p)和C∈R~(m×p),求矩阵X∈R(?)R~(n×n)"使得A×B=C以及相应的最佳逼近问题,其中集合R为如对称阵,Toeplitz阵等构成的线性子空间,或者对称半(ε)正定阵,(对称)非负阵等构成的闭凸集.给出了在相容条件下求解该问题的交替投影算法及算法收敛性分析.通过大量数值算例说明该算法的可行性和高效性,以及该算法较传统的矩阵形式的Krylov子空间方法(可行前提下)在迭代效率上的明显优势,本文也通过寻求加速技巧进一步提高算法的收敛速度.     

线性流形上矩阵方程BTXB=F的双对称最小二乘解

1引言令R~(n×m)、OR~(n×n)、SR~(n×n)(SR_0~(n×n))分别表示所有n×m阶实矩阵、n阶实正交阵、n阶实对称矩阵(实对称半正定阵)的全体,A~ 表示A的Moore-Penrose广义逆,I_k表示k阶单位矩阵,S_k表示k阶反序单位矩阵。R(A)表示A的列空间,N(A)表示A的零空间,rank(A)表示矩阵A的秩。对A=(a_(ij)),B=(b_(ij))∈R~(n×m),A*B表示A与     

非负整数对称阵可实现性问题的算法

1引 言 设P是有p个元素,oj,j＝1,…,p,的有限集,{Si｝,I=1,…,n,为P的子集族.记A=(aij)为{Si｝的关联矩阵,其中,当Oj∈Si时aij=1,否则aij=0.若AAT＝B＝(bij),即bij=|Si ∩ Sj|,则B是对称的且bii≥Bij≥0.反过来,已知n阶非负整数对称阵B,是否存在一个n×m的0-1矩阵A使B=AAT,以及如何计算使B=AAT成立的最小的m(即容度),这即是John B Kelly于1968年在文献[1]中讨论的非负整数对称阵的可实现性问题.     

不变子空间上特征值的扰动

1引言设矩阵A∈C~(n×n),B∈C~(m×m),Q∈C~(n×m)为列满秩矩阵,令R=AQ-QB．当R的范数很小的时候,我们分析矩阵B的特征值对A的特征值的逼近性．当A,B都是Hermite阵时,上述问题已经被Kahan解决．近年来,对可对角化矩阵的情形,取得了一些新的成果．[4][5][6]中给出了几个范数不等式,并应用于矩阵特征值     

ЛЯПУНОВ第二方法与矩阵方程 AX XB′=C(Ⅱ)

本文考虑实数域的任一子域Ω上的一般矩阵方程:AX XB’=C 的求解问题,其中A、B、C分别是Ω上的 m×m 阵、n×n 阵与 m×n 阵.讨论了该方程的相容条件以及相容方程的解法;给出了它在矩阵分解理论与多项式理论上的应用,并且得到了Ляпунов方程 AX XA’=-I_n 有解的必要或充分条件,从而明确了何时可以构造Ляпунов函数的一些条件.     

矩阵方程AXAT+BYBT=C的对称与反对称最小范数最小二乘解

对于任意给定的矩阵A∈Rk×m,B∈Rk×n和C∈Rk×k,利用奇异值分解和广义奇异值分解,我们给出了矩阵方程AXAT+BYBT=C的对称与反对称最小范数最小二乘解的表达式.     

p—除环上矩阵秩的恒等式

本文证明了[1]中的猜测:在p—除环上有恒等式r(?)=r(A) r(B) ((I_s-BB~ )C(I_n-A~ A)),并且改进了这个结果,此外还给出了几个关于矩阵秩的恒等式.设Ω是p-除环,A是Ω上的m×n矩阵.μ(A)表示由A的行向量张成的Ω上的左向量空间,N(A)表示满足XA=0的行向量张成的Ω上的左向量空间,则μ(A)(?)Ω,N(A)(?)Ω_m,μ(A)、N(A)、Ω_m、Ω_n都是左Ω—模,并且dim N(A)=m-r(A).引理1 A、B、C分别是Ω上的m×n、m×s和s×n矩阵,那么     

半张量积下矩阵方程组AX=B,XC=D的最小二乘解

本文研究了半张量积下矩阵方程组AX=B,XC=D在不同情况下的最小二乘解X*∈R~(p×q),其中矩阵A∈R~(m×n),B∈R~(h×k),C∈R~(a×b),D∈R~(l×d)给定.根据半张量积的定义将其转变为普通乘积下的矩阵方程组,再结合矩阵奇异值分解及矩阵微分给出该方程组在不同情况下最小二乘解的解析表达式,并用数值算例加以验证.     

Wishart矩阵二次型初步理论及其应用

设A～W_m（n,Σ）,D为m×m对称阵,称B=A＇DA为Wishart矩阵二次型,本文讨论了Wishart矩阵二次型的分布密度及其各种初步性质,并将其推广到椭球等高分布族,最后用二次型理论解决了求正态总体协差阵Σ的某一类Bayes估计的问题。     

Wavelets from the Loop Scheme

Anewwavelet-based geometric mesh compression algorithm was developed recently in the area of computer graphics by Khodakovsky, Schröder, and Sweldens in their interesting article [23]. The new wavelets used in [23] were designed from the Loop scheme by using ideas and methods of [26, 27], where orthogonal wavelets with exponential decay and pre-wavelets with compact support were constructed. The wavelets have the same smoothness order as that of the basis function of the Loop scheme around the regular vertices which has a continuous second derivative; the wavelets also have smaller supports than those wavelets obtained by constructions in [26, 27] or any other compactly supported biorthogonal wavelets derived from the Loop scheme (e.g., [11, 12]). Hence, the wavelets used in [23] have a good time frequency localization. This leads to a very efficient geometric mesh compression algorithm as proposed in [23]. As a result, the algorithm in [23] outperforms several available geometric mesh compression schemes used in the area of computer graphics. However, it remains open whether the shifts and dilations of the wavelets form a Riesz basis of L₂(?²). Riesz property plays an important role in any wavelet-based compression algorithm and is critical for the stability of any wavelet-based numerical algorithms. We confirm here that the shifts and dilations of the wavelets used in [23] for the regular mesh, as expected, do indeed form a Riesz basis of L₂(?²) by applying the more general theory established in this article.     

On the blow up criterion for the 2-D compressible Navier-Stokes equations

Motivated by [10], we prove that the upper bound of the density function j9 controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain.     

Convergence Estimates of Multilevel Additive and Multiplicative Algorithms for Non-symmetric and Indefinite Problems

New uniform estimates for multigrid algorithms are established for certain non-symmetric indefinite problems. In particular, we are concerned with the simple additive algorithm and multigrid (V(1,0)-cycle) algorithms given in [5]. We prove, without full elliptic regularity assumption, that these algorithms have uniform reduction per iteration, independent of the finest mesh size and number of refinement levels, provided that the coarsest mesh size is sufficiently small.     

A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems

A finite difference method for a time-dependent convection-diffusion problem in one space dimension is constructed using a Shishkin mesh. In two recent papers (Clavero et al. (2005) [2] and Mukherjee and Natesan (2009) [3]), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in [2] and [3] and under a slightly less restrictive condition on the mesh.     

Langevin type limiting processes for adaptive MCMC

Adaptive Markov Chain Monte Carlo (AMCMC) is a class of MCMC algorithms where the proposal distribution changes at every iteration of the chain. In this case it is important to verify that such a Markov Chain indeed has a stationary distribution. In this paper we discuss a diffusion approximation to a discrete time AMCMC. This diffusion approximation is different when compared to the diffusion approximation as in Gelman et al. [5] where the state space increases in dimension to ∞. In our approach the time parameter is sped up in such a way that the limiting process (as the mesh size goes to 0) approaches to a non-trivial diffusion process.     

SPIRALS IN 2-D GAS DYNAMICS SYSTEMS

1.Preliminaries(I)ModelsConsiderthetwomodels:iselltropicandadiabaticflows,(a)2-Disentropicflow(b)2-Dadiabaticflowwherep,(u,v)andpisdensity,velocityandpresure,respectively.andwiththe2-DRiemanndatawhere(i)-statesaredescribedtoProblem(1.1)(1.3)and(1.2)...     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

Isotropic refinement and recoarsening in two dimensions

In this paper we describe an implementation of the isotropic red-green refinement technique for two-dimensional triangulations as described in [1] and [5]. In addition, we present a new method for local recoarsening of a previously refined mesh. Furthermore, we explain some techniques to interpolate data while manipulating a grid.     

NONCONFORMING QUADRILATERAL ROTATED Q1 ELEMENT FOR REISSNER－MINDLIN PLATE

In this paper,we extend two rectangular elements for Reissner-Mindlin plate[9] to the quadrilateral case,Optimal H^1 and L^2 error bounds independent of the plate hickness are derived under a mild assumption on the mesh partition.     

Perturbation of the eigenvectors of the graph Laplacian: Application to image denoising

Patch-based denoising algorithms currently provide the optimal techniques to restore an image. These algorithms denoise patches locally in “patch-space”. In contrast, we propose in this paper a simple method that uses the eigenvectors of the Laplacian of the patch-graph to denoise the image. Experiments demonstrate that our denoising algorithm outperforms the denoising gold-standards. We provide an analysis of the algorithm based on recent results on the perturbation of kernel matrices (El Karoui, 2010) [1], [2], and theoretical analyses of patch denoising algorithms (Levin et al., 2012) [3], (Taylor and Meyer, 2012) [4].