Numerical experiments of preconditioned Krylov subspace methods solving the dense non-symmetric systems arising from BEM

Discretization of boundary integral equations leads, in general, to fully populated non-symmetric linear systems of equations. An inherent drawback of boundary element method (BEM) is that, the non-symmetric dense linear systems must be solved. For large-scale problems, the direct methods require expensive computational cost and therefore the iterative methods are perhaps more preferable. This paper studies the comparative performances of preconditioned Krylov subspace solvers as bi-conjugate gradient (Bi-CG), generalized minimal residual (GMRES), conjugate gradient squared (CGS), quasi-minimal residual (QMR) and bi-conjugate gradient stabilized (Bi-CGStab) for the solution of dense non-symmetric systems. Several general preconditioners are also considered and assessed. The results of numerical experiments suggest that the preconditioned Krylov subspace methods are effective approaches solving the large-scale dense non-symmetric linear systems arising from BEM.     

GM模型的几何性质及统计检验

以空间向量几何理论为基础，详细地分析了Ｇａｕｓｓ－Ｍａｒｋｏｖ线性模型的几何性质；在实现了模型图形化描述的基础上，利用各解向量的几何及统计性质构造出了不同情形下的统计检验方法和相应的检验统计量，为全面、深入地研究和灵活运用该模型提出了一种简洁直观的数学方法     

基于一次投影子空间追踪的压缩感知信号重构

为了降低信号重构算法的复杂度,实现对稀疏度未知信号的重构,提出了一种基于一次投影子空间追踪(OPSP)的信号重构方法。首先根据约束等距性质确定信号稀疏度的上下界,并将最接近上下界中值的整数作为稀疏度的估计值;然后在子空间追踪(SP)算法的框架下,去掉了迭代中观测向量在支撑集上的投影过程,降低了算法的复杂度。为了更准确地衡量算法的重构性能,提出用完整信号的重构概率作为衡量算法重构性能的指标。与传统的SP算法相比,所提算法可以重构稀疏度未知的信号,且重构时间短,重构概率高。仿真结果验证了该算法的有效性。     

Preconditioning the solution of the time-dependent neutron diffusion equation by recycling Krylov subspaces

Spectral preconditioners are based on the fact that the convergence rate of the Krylov subspace methods is improved if the eigenvalues of the smallest magnitude of the system matrix are ‘removed’. In this paper, two preconditioning strategies are studied to solve a set of linear systems associated with the numerical integration of the time-dependent neutron diffusion equation. Both strategies can be implemented using the matrix–vector product as the main operation and succeed at reducing the total number of iterations needed to solve the set of systems.     

Adaptive detectors in the Krylov subspace

The validity of the application of the Krylov subspace techniques in adaptive filtering and detection is investigated. A new verification of the equivalence of two well-known methods in the Krylov subspace, namely the multistage Wiener filters （MWF） and the auxiliary-vector filtering （AVF）, is given in this paper. The MWF and AVF are incorporated into two well-known detectors, namely, the adaptive matched filter （AMF） and Kelly＇s generalized likelihood ratio test （CLRT） including their diagonally loaded versions, which form new detectors. Compared to the conventional AMF, CLRT, and their diagonally loaded versions as well as the reduced-rank AMF and GLRT, the probabilities of detection （PDs） of the new detectors are improved especially when the sample support is low. More importantly, the new detectors are robust of the rank selection of the clutter subspace compared to the reduced-rank AMF and GLRT. These new detectors all possess asymptotic constant false alarm rate （CFAR） property.     

基于MapReduce与相关子空间的局部离群数据挖掘算法

针对高维海量数据,在MapReduce编程模型下,提出了一种基于相关子空间的局部离群数据挖掘算法.该算法首先利用属性维上的局部稀疏程度,重新定义了相关子空间,从而能够有效地刻画各种局部数据集上的分布特征;其次,利用局部数据集的概率密度,给出了相关子空间中的局部离群因子计算公式,有效地体现了相关子空间中数据对象不服从局部数据集分布特征的程度,并选取离群程度最大的N个数据对象定义为局部离群数据;在此基础上,采用LSH分布式策略,提出了一种MapReduce编程模型下的局部离群数据挖掘算法;最后,采用人工数据集和恒星光谱数据集,实验验证了该算法的有效性、可扩展性和可伸缩性.     

基于最近邻子空间搜索的两类文本分类方法

在文本分类中,最近邻搜索算法具有思想简单、准确率高等优点,但通常在分类过程中的计算量较大。为克服这一不足,提出了一种基于最近邻子空间搜索的两类文本分类方法。首先提取每一类样本向量组的特征子空间,并通过映射将子空间变换为高维空间中的点,然后把最近邻子空间搜索转化为最近邻搜索完成分类过程。在Reuters-21578数据集上的实验表明,该方法能够有效提高文本分类的性能,具有较高的准确率、召回率和F1值。     

1D correlation filter based class-dependence feature analysis for face recognition

In this paper, a novel one-dimensional correlation filter based class-dependence feature analysis (1D-CFA) method is presented for robust face recognition. Compared with original CFA that works in the two dimensional (2D) image space, 1D-CFA encodes the image data as vectors. In 1D-CFA, a new correlation filter called optimal extra-class origin output tradeoff filter (OEOTF), which is designed in the low-dimensional principal component analysis (PCA) subspace, is proposed for effective feature extraction. Experimental results on benchmark face databases, such as FERET, AR, and FRGC, show that OEOTF based 1D-CFA consistently outperforms other state-of-the-art face recognition methods. This demonstrates the effectiveness and robustness of the novel method.     

Spectral preconditioning of Krylov spaces: Combining PLS and PC regression

In regularized regression the vectors that lie in Krylov and eigen subspaces, formed in PLS and PC regressions respectively, provide useful low dimensional approximations for the LS regression coefficient vector. By preconditioning the LS normal equations we provide a framework in which to combine these methods, and so exploit both of their respective advantages. The link between the proposed method to orthogonal signal correction and to cyclic subspace regression is made. The performance of the proposed solution in reducing the dimension of the regression problem, and the shrinkage properties of the resulting coefficient vector, are both examined.     

一种新的基于遗忘因子的递推子空间辨识算法

针对工业系统中广泛存在的时变特性, 提出一种新的递推子空间辨识算法, 实现对系统状态空间模型的在线递推估计. 为更好地跟踪系统时变特性, 研究基于遗忘因子的输入输出数据矩阵构造机制, 以提高递推算法的收敛速度; 针对算法中奇异值分解的求解问题, 将梯度型算法引入基于遗忘因子的状态子空间跟踪中, 实现对广义能观测矩阵的估计, 避免了子空间近似带来的估计有偏性; 该算法计算简单有效, 且对初值具有更高的鲁棒性; 最后给出该递推算法的性能分析, 理论证明其收敛性, 并通过仿真实例验证算法的有效性.