DIRECT PERTURBATION METHOD FOR REANALYSIS OF MATRIX SINGULAR VALUE DECOMPOSITION

The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturbation method has previously been proposed by the author in this journal, and now the direct perturbation method has also been presented in this paper. The second-order perturbation results of non-repeated singular values and the corresponding left and right singular vectors are obtained. The results can meet the general needs of most problems of various practical applications. A numerical example is presented to demonstrate the effectiveness of the direct perturbation method.

PERTURBATION METHOD FOR REANALYSIS OF THE MATRIX SINGULAR VALUE DECOMPOSITION

The perturbation method for the reanalysis of the singular value decomposition (SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of great worth to enhance computational efficiency of the iterative analysis problems that require matrix singular value decomposition repeatedly. The asymptotic estimate formulas for the singular values and the corresponding left and right singular vectors up to second-order perturbation components are derived. At the end of the paper the way to extend the perturbation method to the case of general complex matrices is advanced.

A TRUST-REGION METHOD FOR SOLVING TRUNCATED COMPLEX SINGULAR VALUE DECOMPOSITION

The truncated singular value decomposition has been widely used in many areas of science including engineering,and statistics,etc.In this paper,the original truncated complex singular value decomposition problem is formulated as a Riemannian optimiza-tion problem on a product of two complex Stiefel manifolds,a practical algorithm based on the generic Riemannian trust-region method of Absil et al.is presented to solve the underlying problem,which enjoys the global convergence and local superlinear conver-gence rate.Numerical experiments are provided to illustrate the efficiency of the proposed method.Comparisons with some classical Riemannian gradient-type methods,the existing Riemannian version of limited-memory BFGS algorithms in the MATLAB toolbox Manopt and the Riemannian manifold optimization library ROPTLIB,and some latest infeasible methods for solving manifold optimization problems,are also provided to show the merits of the proposed approach.

On the Application of Adomian Decomposition Method to Special Equations in Physical Sciences

The current manuscript makes use of the prominent iterative procedure, called the Adomian Decomposition Method (ADM), to tackle some important special differential equations. The equations of curiosity in this study are the singular equations that arise in many physical science applications. Thus, through the application of the ADM, a generalized recursive scheme was successfully derived and further utilized to obtain closed-form solutions for the models under consideration. The method is, indeed, fascinating as respective exact analytical solutions are accurately acquired with only a small number of iterations.

Extracting the interference components of normal modes in shallow water waveguide using singular value decomposition method

The normal mode interference characteristic in shallow water waveguide is a valu- able topic in the fields of underwater acoustic. A method for extracting the interference components of normal modes from broadband acoustic propagation data recorded by a single hy- drophone without any prior information is present in this paper. First, a Hermitian matrix is formed by the power spectral density. Second, a singular value decomposition （SVD） is performed on the Hermitian matrix to obtain the orthonormal eigenvectors, which are proportional to the interference components of normal modes. The fundamental equations of the new extracting method are derived based on normal mode and waveguide invariant theory. And the validity of the present method is verified by the numerical simulation and experimental results. In addition, the extracted results of normal-mode interference components are intended to be used for passive ranging of broadband sources.

基于压缩感知的缺失机械振动信号重构新方法

针对工业机械设备实时监测中不可控因素导致的振动信号数据缺失问题,提出一种基于自适应二次临近项交替方向乘子算法(adaptive quadratic proximity-alternating direction method of multipliers, AQ-ADMM)的压缩感知缺失信号重构方法。AQ-ADMM算法在经典交替方向乘子算法算法迭代过程中添加二次临近项,且能够自适应选取惩罚参数。首先在数据中心建立信号参考数据库用于构造初始字典,然后将K-奇异值分解(K-singular value decomposition, K-SVD)字典学习算法和AQ-ADMM算法结合重构缺失信号。对仿真信号和两种真实轴承信号数据集添加高斯白噪声后作为样本,试验结果表明当信号压缩率在50%~70%时,所提方法性能指标明显优于其它传统方法,在重构信号的同时实现了对含缺失数据机械振动信号的快速精确修复。

An Approximate Linear Solver in Least Square Support Vector Machine Using Randomized Singular Value Decomposition

In this paper, we investigate the linear solver in least square support vector machine（LSSVM） for large-scale data regression. The traditional methods using the direct solvers are costly. We know that the linear equations should be solved repeatedly for choosing appropriate parameters in LSSVM, so the key for speeding up LSSVM is to improve the method of solving the linear equations. We approximate large-scale kernel matrices and get the approximate solution of linear equations by using randomized singular value decomposition（randomized SVD）. Some data sets coming from University of California Irvine machine learning repository are used to perform the experiments. We find LSSVM based on randomized SVD is more accurate and less time-consuming in the case of large number of variables than the method based on Nystrom method or Lanczos process.

基于同态加密的隐私保护主成分分析方法

在现实生活中,不同的行业之间,甚至同行业不同部门之间的数据并不互通,随着计算机算力的提升,制约模型训练效果的不是算力而是数据量。因此,想要得到更好的算法模型,仅靠某一方的数据是不够的,需要两方或者多方的参与,这就要求对各方的数据进行隐私保护。除此之外,随着收集的数据越来越详细,数据的维数也越来越大。面对高维的数据,数据降维是不可缺少的环节,而在数据降维方面,主成分分析(Principal Component Analysis,PCA)是常用的手段。当拥有数据的两方想要合作进行隐私保护的数据降维时,同态加密技术是一种解决办法。同态加密技术可以在保护数据隐私的前提下对加密数据进行计算,可以用在加密数据的PCA上。针对上述应用场景,利用CKKS同态加密方案,通过幂法迭代的SVD技术设计了一种两方加密数据进行PCA的方案,在保护两方数据隐私的前提下实现数据降维的目的;通过改进传统幂法迭代步骤,避免了代价高昂的同态密文除法运算,使得在选取较小的加密参数时,也能支持更多的幂法迭代次数,从而在缩短同态计算时间的同时提高计算精度。在公共数据集上进行测试,并与现有方案进行对比,该方案在计算耗时上缩短了约80%,与明文计算结果的均方误差缩减到1%以内。

基于增广SVD-MWKF的激励识别与结构响应重构

针对传统卡尔曼滤波算法在结构响应重构应用中需要外部激励及测量噪声方差先验已知的问题,提出一种基于增广SVD-MWKF(Singular Value Decomposition-Moving Window Kalman Filter)的激励识别与结构响应重构方法。首先,引入奇异值分解降噪技术以优化移动窗口法对测量噪声方差的实时估计。随后使用基于增广状态空间方程的卡尔曼滤波算法并结合部分测点的加速度测量数据,实现对结构外部激励的识别及各位置的速度、加速度响应的重构。最后,对起重机桁架和简支梁分别进行数值模拟和试验分析,结果表明,相较于移动窗口法,所提方法对测量噪声方差估计更加准确,且对外部激励能进行有效识别。

基于比例边界有限元法计算应力强度因子的不确定量化分析

应力强度因子是预测荷载作用下结构中裂纹产生和扩展的重要参数。半解析的比例边界有限元法结合了有限元和边界元法的优势,在裂纹尖端或存在奇异应力的区域不需要局部网格细化,可以直接提取应力强度因子。在比例边界有限元法计算应力强度因子的框架下,引入随机参数进行蒙特卡罗模拟(Monte Carlo simulation, MCS),并提出一种新颖的基于MCS的不确定量化分析。与直接的MCS不同,采用奇异值分解构造低阶的子空间,降低系统的自由度,并使用径向基函数对子空间进行近似,通过子空间的线性组合获得新的结构响应,实现基于MCS的快速不确定量化分析。考虑不同荷载状况下,结构形状参数和材料属性参数对应力强度因子的影响,使用改进的MCS计算应力强度因子的统计特征,量化不确定参数对结构的影响。最后通过若干算例验证了该算法的准确性和有效性。

基于奇异值分解张拉整体结构找形方法

张拉整体结构是由一组不连续的受压单元包含于连续受拉单元组成的稳定自平衡结构,为了寻找自平衡状态下的构型,引入奇异值分解的方法,将寻找张拉整体结构的自平衡构型问题转化为最小奇异值的判定。通过广义节点坐标和构件之间的连接关系建立结构的数学模型;引入力密度的概念,对张拉整体结构进行受力分析,列写包含平衡矩阵的平衡方程;对平衡矩阵进行奇异值分解,利用分解获得的最小奇异值判断平衡方程是否有解,对张拉整体结构是否存在自平衡构型进行初步判定,再依据获得的力密度的均匀性(同组构件力密度大小相等)和正负属性(杆的力密度小于0,索的力密度大于0)对结构自平衡状态进一步判断;通过实例分析对该找形方法的可行性进行了验证,结果表明:该方法可以找到张拉整体结构自平衡构型。本文为寻找自平衡张拉整体结构提供了一种思路。

基于多滤波降噪法的桥梁应变监测信号处理

针对桥梁应变监测信号存在多源噪声,以及现有降噪法难以准确选取有效固有模态函数(Intrinsic Mode Function,IMF)分量的问题,提出一种有效的多滤波降噪法。首先,采用自适应噪声抵消器对含噪信号进行预处理,以滤除低频噪声,再对其进行自适应噪声完备集合经验模态分解(Complete Ensemble Empirical Mode Decomposition with Adaptive Noise,CEEMDAN)得到IMF分量,并利用频谱分析-自相关函数双重判断准则选取IMF分量;随后,对其进行奇异值分解,利用奇异值差分谱确定各分量有效阶数;最后,将各有效IMF分量的有效阶重构得到去噪信号。通过模拟试验验证上述方法的合理性,并将其应用于桥梁应变监测信号处理。结果表明:采用上述双重判断准则选取有效IMF分量具有较好效果,且提出的多滤波降噪法在桥梁应变监测信号处理中具有显著的优越性。

基于近红外光谱技术的气体浓度检测研究

气体浓度在各领域分析中具有重要意义,由于在气体浓度检测过程中,受光谱维度的影响导致检测结果出现较大的误差,为了降低在测量过程中产生的不利影响,提出基于近红外光谱技术的气体浓度检测研究。通过去势-标准正态变换,校正近红外光谱基线。联合广义S变换和奇异值分解共同去噪近红外光谱,提升光谱质量。基于主成分分析提出偏最小二乘降维法用于降维近红外光谱。以朗伯比尔定律为基础,引入Lorenz线性拟合近红外光谱吸收谱线,采用梯度下降法直接拟合预处理近红外光谱吸收信号,计算得到最终气体浓度检测结果。实验结果表明,所提方法在检测甲苯、丙烷和丙烯气体浓度时,检测结果与实际气体浓度基本一致,有效降低了残差平方和与均方根误差,且检测时间低于2.3 s。

求解一类平行四边形域上热传导反问题的无网格方法

基于基本解和径向基函数相结合的方法,对采用不同径向基作为近似解的基底,用平行四边形域反演一类热传导反问题的初值和热源项进行研究.因插值构造的方程系数矩阵是病态的,即系数矩阵的条件数较大,本文基于GCV准则的奇异值分解正则化方法求解线性方程组,选取不同噪声级,通过数值算例显示所给方法是稳定有效的.

Fault diagnosis for gearboxes based on Fourier decomposition method and resonance demodulation

Condition monitoring and fault diagnosis of gearboxes play an important role in the maintenance of mechanical systems.The vibration signal of gearboxes is characterized by complex spectral structure and strong time variability,which brings challenges to fault feature extraction.To address this issue,a new demodulation technique,based on the Fourier decomposition method and resonance demodulation,is proposed to extract fault-related information.First,the Fourier decomposition method decomposes the vibration signal into Fourier intrinsic band functions(FIBFs)adaptively in the frequency domain.Then,the original signal is segmented into short-time vectors to construct double-row matrices and the maximum singular value ratio method is employed to estimate the resonance frequency.Then,the resonance frequency is used as a criterion to guide the selection of the most relevant FIBF for demodulation analysis.Finally,for the optimal FIBF,envelope demodulation is conducted to identify the fault characteristic frequency.The main contributions are that the proposed method describes how to obtain the resonance frequency effectively and how to select the optimal FIBF after decomposition in order to extract the fault characteristic frequency.Both numerical and experimental studies are conducted to investigate the performance of the proposed method.It is demonstrated that the proposed method can effectively demodulate the fault information from the original signal.

附参数约束的方程赋权值解法研究

针对附参数约束等式方程的函数模型解算问题,提出一种对约束方程赋权值再联合观测方程统一解算的约束方程赋权统一解法。为验证此方法的可靠性和效率,基于矩阵分析法推导了适用于附参数约束方程解算的正交三角分解法和广义奇异值分解法。通过算例对3种解算法进行比较分析,结果表明:当给约束方程附加合适范围内的权值时,约束方程赋权统一解法获得的参数估计值与矩阵分解算法的参数估计结果在要求的精度范围内一致;赋权统一解法与正交三角分解法的计算效率接近,广义奇异值分解法计算效率较低;约束方程赋权统一解法更符合参数模型统一表达和平差准则设定,有利于平差算法设计和误差特性分析和研究。

非负约束修正阻尼算法求解核磁共振测井T_(2)谱反演问题

为了优化核磁共振测井技术,加快核磁共振测井数据处理速度,提出了一类非负约束修正阻尼算法求解核磁共振T_(2)谱反演问题。通过数学建模及理论推导,证明该算法可以更好地保留T_(2)谱的有效信息,同时也能更有效地减少反演问题的噪声对解的影响。数值算例表明新算法相较于修正的截断奇异值算法和阻尼算法具有更好、更稳定的反演效果,尤其是低信噪比情况下新算法的优势更加明显,可以进一步优化核磁共振测井技术。

基于前后向矩阵束方法的线性排布无人机集群目标规模估算

利用前后向矩阵束方法,提出了一种有效的“蜂群”无人机群目标个数估计方法用于目标检测。与传统的参数估算方法相比,所提方法对于稀疏信号也能准确地进行估算。首先,在一定的角度或频率下,获取线阵排列群目标的雷达散射截面;然后,根据已知的雷达回波构造Hankel-Toeplitz矩阵,并对该矩阵进行奇异值分解;最后,通过设置合适的阈值选取比例较大的奇异值,以此准确地估算线阵排列群目标的个数。此外,文中还讨论了不同信噪比对估算结果的影响。为了提高精度,文中采用多组不同方位角度的回波信号来联合估算群目标的个数。仿真结果表明,该方法能够有效、准确地估算线阵排列群目标的个数,而且估计结果不受噪声的影响,具有较好的鲁棒性。

基于深度神经网络的存储压缩方法研究

由于深度神经网络存在内存密集和计算密集两大特征,其无法满足移动设备存储的需求。为此,提出一种基于深度神经网络的存储压缩方法,该方法通过SVD矩阵奇异值分解方法,并辅以网络剪枝技术,在深度神经网络运行过程中引入少量神经元,并设定神经元数连接数,降低权重规模。通过一系列方法对策,尽可能在不损失精度的前提下,对深度神经网络模型大幅度压缩,其压缩程度达到了5×、15×。

基于奇异值分解的柔性直流输电系统短路故障识别方法

常规的输电系统短路故障识别方法以故障波形变化识别为主,受到输电系统运行状态的影响,故障识别存在一定的误差,为此设计了基于奇异值分解的柔性直流输电系统短路故障识别方法。提取柔性直流输电系统短路故障特征,包括正常运行状态的正序分量特征与故障状态下的负序分量特征,保留负序分量转化为频域形式,便于后续故障识别。使用奇异值分解识别输电系统短路故障区间,将故障区间分为区外故障识别与区内故障识别两部分,利用奇异值的稳定性作为故障区间的识别依据,从而实现短路故障的精准识别。经仿真实验验证,该方法具有较高的识别精准度。