大规模最小二乘奇异值分解的并行处理方法

大规模最小二乘问题求解中,直接进行奇异值分解会产生巨大的内存需求以及漫长的计算时间。为解决该问题,提出了一种基于迭代的并行处理方法。该方法利用奇异值分解降维的特性,通过迭代不断减小矩阵规模,直到可以直接使用奇异值分解求解。在迭代过程中,将矩阵分解为许多足够小的子矩阵,并行处理其奇异值分解过程,从而提升运行速度。实验结果表明,该方法即使是串行处理,也使得大规模最小二乘奇异值分解的时间成本及空间成本大大降低;而并行处理在双机条件下加速比接近200％。     

迭代张量高阶奇异值分解的图像恢复方法

为了有效地对图像缺失数据进行恢复, 提出一种迭代张量高阶奇异值分解（HOSVD）图像缺失数据恢复方法。该方法首先利用拉格朗日乘子方法将张量核范数目标函数进行子问题分解操作, 简化了求解过程, 然后迭代地采用张量高阶奇异值分解阈值方法进行子问题求解, 最终得到恢复后的图像缺失数据。将矩阵奇异值阈值算法进行扩展而得的HOSVD阈值方法充分利用了图像内部和图像与图像之间的多重约束关系, 大大提高了恢复精度。模拟实验和真实图像实验结果显示该方法具有良好的缺失数据的恢复性能。     

二值分解压缩和Consensus算法

奇异值分解(SVD)是一种流行的用于高维数据压缩的方法,二值分解是奇异值分解的一种简化形式.实现二值分解的主要算法有两种:迭代启发式算法和贪婪算法.但这两种算法都不是很理想的算法:迭代启发式算法在很多情况下不能保证收敛性,贪婪算法不满足大型数值矩阵分解的需要.采用了一种新的算法来实现二值分解:Consensus的算法.Consensus算法可在渐进多项式时间内找到一般图中的极大二分团.对于某些二分图,该算法的复杂度是多项式时间的.实验结果表明,当迭代启发式算法不起作用时,Consensus算法是一种很好的求解二值分解的方法.该算法远比贪婪算法的效率高,且具有稳定收敛性.     

三分支机器人协调操作及关节力矩优化

针对三分支机器人协调运动，采用分离影响系数法分离各个分支的雅可比矩阵和惯性矩阵，再重新组合成整个系统的雅可比矩阵和惯性矩阵，建立三分支机器人运动学和动力学方程．应用乘子罚函数方法，对三分支机器人基于最小关节驱动力矩优化设计,避免矩阵的奇异值分解，提高计算的稳定性，应用迭代方法，简化了问题的求解．     

分布式奇异值分解最小平方估计算法

对奇异值（SVD）分解求解最小平方估计的问题进行了研究。提出迭代式分割与合并的算法（IDMSVD）,目的是改善奇异值分解在估计参数时非常耗费时间以及内存空间的问题。基于IDMSVD提出了分布式迭代式分割与合并算法（MRDSVD）,使用Hadoop平台的MapReduce来实现,实验结果显示,IDMSVD可以有效改善SVD求最小平方解耗费运行时间与内存空间的问题,MRDSVD算法可进一步改善IDMSVD的运行时间。     

基于SVD 方法的简约型区间二型模糊模型构建

针对奇异值-QR分解方法存在有效奇异值难以确定的问题,采用奇异值分解方法分析从区间二型模糊模型抽取的两个激活强度矩阵,提出了奇异值归一化差值的概念以描述相邻奇异值的变化情况,从而反映了重要规则和冗余规则在奇异值变化上的本质差异;进而根据其临界点确定有效奇异值个数,并利用QR分解得到有效奇异值所对应的重要规则构建简约型区间二型模糊结构。仿真实例验证了所提出方法的有效性和可行性。     

基于随机奇异值分解的快速矩阵补全算法及其应用

为了在保证结果精度的情况下加快运算速度,改进了矩阵补全的代表性算法——奇异值门限(SVT)算法.首先对于输入矩阵进行规整化处理,之后在每一步的迭代中使用奇异值分解算法对矩阵进行恢复.由于每个迭代步中奇异值分解的计算量很大,文中借鉴随机矩阵奇异值分解算法,提出使用块克雷洛夫迭代近似奇异值分解算法和子空间复用技术的快速SVT算法.使用彩色图像和电影评分矩阵对算法进行实验的结果表明,快速SVT算法在不影响图像恢复和评分数据预测效果的同时显著地缩短了计算时间;在图像恢复和电影评分预测的实验中,分别取得了高达7.1倍和3.2倍的加速比.     

一种近似奇异值分解的观测矩阵优化方法

压缩感知观测矩阵的优化通常采用迭代或最优化的思想,其主要缺点是运算复杂度高。针对这种情况,提出一种基于奇异值分解的观测矩阵优化方法。首先对随机矩阵进行奇异值分解,其次减小随机矩阵的奇异值到适定的范围,进而得到条件数相对小的观测矩阵。理论分析和实验结果表明,该方法得到的观测矩阵与稀疏基的互干性较小,能够精确重构信号。与现有的其他优化方法相比,该方法具有实现简单,计算复杂度低和重构精度高的特点。     

基于GPU的并行奇异值分解最小平方估计算法

对奇异值分解(SVD)求解最小平方估计的问题进行了研究。提出迭代式分割与合并的算法(IDMSVD),目的是解决奇异值分解在估计参数时非常耗费内存空间的问题。基于IDMSVD提出了并行IDMSVD算法,并使用GPU实现之。实验结果显示,IDMSVD可以有效地解决SVD求最小平方解耗费运行时间与内存空间的问题,并行IDMSVD算法可进一步改善IDMSVD的运行时间。     

基于图像序列的非刚体三维运动恢复

研究图像序列中非刚体的三维运动重建问题。介绍非刚体运动重建的两种重要算法:奇异值分解法和线性迭代法。对迭代算法所用的重构方法进行修改,在迭代过程中应用结果更为精确的重构方法来求解非刚体的模型和旋转矩阵。对真实图像序列的实验结果验证了该算法的有效性和精确性。     

求解非线性最小二乘问题的实用型方法

１．引言对于非线性最小二乘问题其中,为残差向量且,这里是指通常意义下的范数,即二范数．目标函数的梯度和Ｈｅｓｓｅ矩阵为其中　矩阵, 求解非线性最小二乘问题（１．１）的最基本方法是Ｇａｕｓｓ－Ｎｅｗｔｏｎ法,迭代格式为其中ｄｋ为线性方程组的解,这． 当人为满秩矩阵时,线性方程组（１．５）有唯一解,即并且有如下不等式：其中　是矩阵　的最小特征值．当　人接近奇异时,因此有可能存在着 ｄｋ,使得,即某一步迭代的步长太大,导致 Ｇａｕｓｓ－Ｎｅｗｔｏｎ法迭代失败． 另外,当　为奇异矩阵时,线性方程组（１．５）…     

An efficient method based on the second kind Chebyshev wavelets for solving variable-order fractional convection diffusion equations

In this paper, a class of variable-order fractional convection diffusion equations have been solved with assistance of the second kind Chebyshev wavelets operational matrix. The operational matrix of variable-order fractional derivative is derived for the second kind Chebyshev wavelets. By implementing the second kind Chebyshev wavelets functions and also the associated operational matrix, the considered equations will be reduced to the corresponding Sylvester equation, which can be solved by some appropriate iterative solvers. Also, the convergence analysis of the proposed numerical method to the exact solutions and error estimation are given. A variety of numerical examples are considered to show the efficiency and accuracy of the presented technique.     

Finite-difference solution of the incompressible three-dimensional boundary layer equations for a blunt body

The governing equations for a laminar flow are solved in terms of an orthogonal surface coordinate system. One of the coordinate is determined by the intersection with the body surface of meridional planes which pass through an axis containing the stagnation point. The other coordinate is obtained numerically from the orthogonality condition. The momentum equations have been written in a standard from which allows additional equations of this form to be added with a small modification of the computer code. This equation is replaced with a nonlinear finite-difference equation which is solved as an iterative solution of linear tridiagonal equations. The special form of the governing equations at the stagnation point and the plane of symmetry is determined and the solution of these equations is obtained to provide a unified code. Numerical solutions have been obtained for several special cases and compared to results of other authors. New results are presented for an ellipsoid ar angle of attack and an elliptic-paraboloid at zero incidence.     

拟牛顿法求解化工过程数学模型

使用不需求取偏导数的拟牛顿法,求解化工过程模拟中产生的非线性方程组形式的数学模型。当未知数各分量间绝对值相差较大时,提出了改善收敛性的几种方法,即：（1）加入阻尼因子,以减少迭代值的震荡、（2）将方程适当降价;（3）将差商的绝对步长改为相对步长;（4）新迭代值超来其物理意义范围时,强制其回至其初始值。计算结果表明,与牛顿－拉夫森法相比,拟牛顿法不需求偏导数,对初值要求低,较雅可比迭代法收敛速度快,可用于求解化工过程的非线性方程组。     

An explicit Chebyshev pseudospectral multigrid method for incompressible Navier-Stokes equations

The two-dimensional steady incompressible Navier-Stokes equations in the form of primitive variables have been solved by Chebyshev pseudospectral method. The pressure and velocities are coupled by artificial compressibility method and the NS equations are solved by pseudotime method with an explicit four-step Runge-Kutta integrator. In order to reduce the computational time cost, we propose the spectral multigrid algorithm in full approximation storage (FAS) scheme and implement it through V-cycle multigrid and full multigrid (FMG) strategies. Four iterative methods are designed including the single grid method; the full single grid method; the V-cycle multigrid method and the FMG method. The accuracy and efficiency of the numerical methods are validated by three test problems: the modified one-dimensional Burgers equation; the Taylor vortices and the two-dimensional lid driven cavity flow. The computational results fit well with the exact or benchmark solutions. The spectral accuracy can be maintained by the single grid method as well as the multigrid ones, while the time cost is greatly reduced by the latter. For the lid driven cavity flow problem, the FMG is proved to be the most efficient one among the four iterative methods. A speedup of nearly two orders of magnitude can be achieved by the three-level multigrid method and at least one order of magnitude by the two-level multigrid method.     

LQ控制区段混合能矩阵的微分方程及其应用

本文根据计算结构力学与线性二次控制的对应关系,提出了连续时间有限区段的混合能 分块子矩阵Q2,G2及Φ2.推导出适用于LQ控制非定常课题的二区段连接的凝聚消元公式及 这些子矩阵的微分方程,可用级数展开求解这些方程.当△t很小时,这些分块子矩阵的高次 近似可以大大加速里卡提代数方程算法的收敛性.     

Three-dimensional Poisson solver for a charged beam with large aspect ratio in a conducting pipe

In this paper, we present a three-dimensional Poisson equation solver for the electrostatic potential of a charged beam with large longitudinal to transverse aspect ratio in a straight and a bent conducting pipe with open-end boundary conditions. In this solver, we have used a Hermite-Gaussian series to represent the longitudinal spatial dependence of the charge density and the electric potential. Using the Hermite-Gaussian approximation, the original three-dimensional Poisson equation has been reduced into a group of coupled two-dimensional partial differential equations with the coupling strength proportional to the inverse square of the longitudinal-to-transverse aspect ratio. For a large aspect ratio, the coupling is weak. These two-dimensional partial differential equations can be solved independently using an iterative approach. The iterations converge quickly due to the large aspect ratio of the beam. For a transverse round conducting pipe, the two-dimensional Poisson equation is solved using a Bessel function approximation and a Fourier function approximation. The three-dimensional Poisson solver can have important applications in the study of the space-charge effects in the high intensity proton storage ring accelerator or induction linear accelerator for heavy ion fusion where the ratio of bunch length to the transverse size is large.     

线性矩阵方程异类约束最小二乘解的迭代算法

多矩阵变量线性矩阵方程(LME)约束解的计算问题在参数识别、结构设计、振动理论、自动控制理论等领域都有广泛应用。本文借鉴求线性矩阵方程(LME)同类约束最小二乘解的迭代算法,通过构造等价的线性矩阵方程组,建立了求多矩阵变量LME的一种异类约束最小二乘解的迭代算法,并证明了该算法的收敛性。在不考虑舍入误差的情况下,利用该算法不仅可在有限步计算后得到LME的一组异类约束最小二乘解,而且选取特殊初始矩阵时,可求得LME的极小范数异类约束最小二乘解。另外,还可求得指定矩阵在该LME的异类约束最小二乘解集合中的最佳逼近解。算例表明,该算法是有效的。     

Efficient use of iterative solvers in nested topology optimization

In the nested approach to structural optimization, most of the computational effort is invested in the solution of the analysis equations. In this study, it is suggested to reduce this computational cost by using an approximation to the solution of the analysis problem, generated by a Krylov subspace iterative solver. By choosing convergence criteria for the iterative solver that are strongly related to the optimization objective and to the design sensitivities, it is possible to terminate the iterative solution of the nested equations earlier compared to traditional convergence measures. The approximation is computationally shown to be sufficiently accurate for the purpose of optimization though the nested equation system is not necessarily solved accurately. The approach is tested on several large-scale topology optimization problems, including minimum compliance problems and compliant mechanism design problems. The optimized designs are practically identical while the time spent on the analysis is reduced significantly.     

Solution of Electromagnetic Scattering Problems Involving Three-Dimensional Homogeneous Dielectric Objects by the Single Integral Equation Method

The problem of electromagnetic scattering by a homogeneous dielectric object is usually formulated as a pair of coupled integral equations involving two unknown currents on the surface S of the object. In this paper, however, the problem is formulated as a single integral equation involving one unknown current on S. Unique solution at resonance is obtained by using a combined field integral equation. The single integral equation is solved by the method of moments using a Galerkin test procedure. Numerical results for a dielectric sphere are in good agreement with the exact results. Furthermore, the single integral equation method is shown to have superior convergence speed of iterative solution compared with the coupled integral equations method.