Parallel eigenvalue calculation based on multiple shift–invert Lanczos and contour integral based spectral projection method

We discuss the possibility of using multiple shift–invert Lanczos and contour integral based spectral projection method to compute a relatively large number of eigenvalues of a large sparse and symmetric matrix on distributed memory parallel computers. The key to achieving high parallel efficiency in this type of computation is to divide the spectrum into several intervals in a way that leads to optimal use of computational resources. We discuss strategies for dividing the spectrum. Our strategies make use of an eigenvalue distribution profile that can be estimated through inertial counts and cubic spline fitting. Parallel sparse direct methods are used in both approaches. We use a simple cost model that describes the cost of computing k eigenvalues within a single interval in terms of the asymptotic cost of sparse matrix factorization and triangular substitutions. Several computational experiments are performed to demonstrate the effect of different spectrum division strategies on the overall performance of both multiple shift–invert Lanczos and the contour integral based method. We also show the parallel scalability of both approaches in the strong and weak scaling sense. In addition, we compare the performance of multiple shift–invert Lanczos and the contour integral based spectral projection method on a set of problems from density functional theory (DFT).     

Tracking and identification of regime-switching systems using binary sensors

This work is concerned with tracking and system identification for time-varying parameters. The parameters are Markov chains and the observations are binary valued with noise corruption. To overcome the difficulties due to the limited measurement information, Wonham-type filters are developed first. Then, based on the filters, two popular estimators, namely, mean squares estimator (MSQ) and maximum posterior (MAP) estimator are constructed. For the mean squares estimator, we derive asymptotic normality in the sense of weak convergence and in the sense of strong approximation. The asymptotic normality is then used to derive error bounds. When the Markov chain is infrequently switching, we derive error bounds for MAP estimators. When the Markovian parameters are fast varying, we show that the averaged behavior of the parameter process can be derived from the stationary measure of the Markov chain and that can be estimated using empirical measures. Upper and lower error bounds on estimation errors are also established.     

Richardson Extrapolation of Iterated Discrete Galerkin Method for Eigenvalue Problem of a Two Dimensional Compact Integral Operator

We consider approximation of eigenelements of a two-dimensional compact integral operator with a smooth kernel by discrete Galerkin and iterated discrete Galerkin methods. By choosing numerical quadrature appropriately, we obtain superconvergence rates for eigenvalues and iterated eigenvectors, and for gap between the spectral subspaces. We propose an asymptotic error expansions of the iterated discrete Galerkin method and asymptotic error expansion of approximate eigenvalues. We then apply Richardson extrapolation to obtain improved error bounds for the eigenvalues. Numerical examples are presented to illustrate theoretical estimate.     

压差矢量水听器阵波束形成的优化

以各向同性噪声作为背景噪声场,对矢量水听器阵波束形成的优化方法进行了系统的研究.在分析各向同性噪声场中声压、振速相关性的基础上,利用最大信噪比准则推导了基阵的最佳增益与噪声场特性和阵列流形之间的关系,提出利用最佳增益权对阵列接收信号进行波束形成处理.通过对矢量水听器阵进行分解,将最佳增益波束形成引入矢量阵信号处理中.以二维压差矢量水听器阵作为接收基阵进行了仿真计算和湖上试验,结果表明,矢量阵的最佳增益波束形成可以获得性能更优的波束和更大的空间处理增益.     

Robust adaptive beamforming method using principal eigenpairs with modification of PASTd

Adaptive arrays suffer from performance degradation in the presence of steering vector errors. The doubly constrained robust Capon beamformer (DCRCB) can deal with the problem, utilizing all the eigenvalues and eigenvectors of the covariance matrix, which leads to high computational complexity. This paper presents a robust beamforming method which is computationally efficient, exploiting principal eigenpairs only. The eigenpairs can be estimated based on the projection approximation subspace tracking with deflation (PASTd). The original PASTd algorithm, which does not provide orthonormal eigenvectors in general, is modified so that the orthonormalization of eigenvectors can be efficiently made using the structure of the modified algorithm. The proposed beamforming method significantly reduces the computational load, particularly when the number of the directional signals is much less than that of sensor elements, and substantially has the same performance as the conventional one utilizing all the eigenpairs.     

Approximate maximum-likelihood approach to ARMA spectral estimation

A three-step approximate maximum-likelihood method for ARMA spectral estimation is derived, based on an idea due to Walker. The asymptotic properties of the proposed estimator are investigated and an explicit expression for its asymptotic covariance matrix is presented. The estimator provides the asymptotic accuracy of a maximum-likelihood technique, at a modest computational cost.     

有色噪声背景下的正交子空间辨识

针对传统子空间辨识中存在的有色噪声干扰问题,本文提出一种正交子空间辨识方法.首先,根据子空间辨识算法机制构建含有色噪声的扩展状态空间模型.然后,结合有色噪声的相关性分析,研究了传统子空间辨识方法的有偏性问题,并重新设计了投影向量和正交投影方式,用以消除有色噪声干扰.最后,对投影后的数据矩阵进行奇异值分解,获取广义能观测矩阵,进而求得系统的状态空间模型参数.仿真结果表明该方法在有色噪声干扰下是一致无偏的,并且具有渐进二阶统计特性.结合陀螺仪的具体实验结果表明,该算法在实际应用中具有比传统子空间辨识法更高的辨识精度.     

On output-error methods for system identification

In this paper are derived consistency and asymptotic normality results for the output-error method of system identification. The output-error estimator has the advantage over the prediction-error estimator of being more easily computable. However, it is shown that the output-error estimator can never be more efficient than the prediction-error estimator. The main result of the paper provides necessary and sufficient conditions for the output-error estimator and the prediction-error estimator to have the same efficiency, irrespective of the spectral density of the noise process.     

基于子空间逼近的特征空间波束形成算法

特征空间波束形成（ESB）算法为了得到信号子空间需要对采样协方差矩阵进行特征值分解,运算量十分巨大,这大大限制了其应用。为了减低ESB算法的运算量,利用有理子空间逼近的原理,提出一种不需要估计信号源个数的快速ESB算法。该方法利用一个介于信号和噪声特征值之间的分界值将特征空间分成两个子空间,并用矩阵幂乘和此分界值的有理式逼近这两个子空间的投影矩阵,将此投影矩阵代入到ESB算法的权值求解式中,在不降低性能的前提下,可大大提高波束形成的运算速度。计算机仿真验证了该算法的有效性,并分析了分界值取值方法的不同对子空间划分及波束形成性能的影响。     

Efficient Two-Sided Estimates for the Spectrum of Some Elliptic Operators

Using the principle of maximum, we establish the upper and lower bounds for the spectrum of some elliptic operators and their grid analogs. More accurate estimates of the spectrum of differential operators are obtained from the exact formulas for the error of the eigenvalues by the finite-difference method. Two-sided estimates of the eigenvalues of difference analogs of spectral problems give a majorant and a minorant for the error of the phase velocities of grid waves in vibration problems for various objects.

     

Stability criteria of matrix polynomials

In this paper, the stability of matrix polynomials is investigated. First, upper and lower bounds are derived for the eigenvalues of a matrix polynomial. The bounds are based on the spectral radius and the norms of the related matrices, respectively. Then, by means of the argument principle, stability criteria are presented which are necessary and sufficient conditions for the stability of matrix polynomials. Furthermore, a numerical algorithm is provided for checking the stability of matrix polynomials. Numerical examples are given to illustrate the main results.     

On new estimates for Lyapunov exponents of discrete time varying linear systems

In this paper, we propose certain new bounds for the Lyapunov exponents of discrete time varying linear systems. The bounds are expressed in terms of spectral radii of matrix coefficients and therefore may be used to establish the exponential stability of time varying system on the basis of eigenvalues of individual coefficient. This approach is known in the literature as frozen time method.     

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

The paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The conclusion is that if local approximation properties of nonconforming finite element spaces are better than total errors (sums of global approximation errors and consistency errors) of nonconforming finite element methods, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we analyze abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. We show that this condition hold for most low-order nonconforming finite elements in literature. In addition, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we also propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.     

General framework for asymptotic properties of generalized weightednonlinear least-squares estimators with deterministic and stochasticweighting

This paper studies the asymptotic properties (strong consistency, convergence rate, asymptotic normality) of a generalized weighted nonlinear least-squares estimator under weak noise assumptions. Both deterministic and stochastic weighting are handled and the presence of model errors is considered. For particular models, estimators, and noise assumptions the general framework boils down to known time and frequency-domain estimators     

Algebras with compatible quantale-enriched structures

Based on quantale-enriched category, we consider algebras with compatible quantale-enriched structures, which can be viewed as fuzzification of ordered algebraic structures. We mainly study groupoids and semigroups with compatible quantale-enriched structures from this viewpoint. Some basic concepts such as ideals, homomorphisms, residuated quantale-enriched groupoids are developed and some examples of them are given. Our approach gives a complement to the approach initiated by Rosenfeld to study fuzzy abstract algebra, and these two approaches are combined in the present paper to study fuzzy aspects of abstract algebra structures.     

Kernel selection with spectral perturbation stability of kernel matrix

Kernel selection is one of the key issues both in recent research and application of kernel methods. This is usually done by minimizing either an estimate of generalization error or some other related performance measure. Use of notions of stability to estimate the generalization error has attracted much attention in recent years. Unfortunately, the existing notions of stability, proposed to derive the theoretical generalization error bounds, are difficult to be used for kernel selection in practice. It is well known that the kernel matrix contains most of the information needed by kernel methods, and the eigenvalues play an important role in the kernel matrix. Therefore, we aim at introducing a new notion of stability, called the spectral perturbation stability, to study the kernel selection problem. This proposed stability quantifies the spectral perturbation of the kernel matrix with respect to the changes in the training set. We establish the connection between the spectral perturbation stability and the generalization error. By minimizing the derived generalization error bound, we propose a new kernel selection criterion that can guarantee good generalization properties. In our criterion, the perturbation of the eigenvalues of the kernel matrix is efficiently computed by solving the derivative of a newly defined generalized kernel matrix. Both theoretical analysis and experimental results demonstrate that our criterion is sound and effective.     

Z基于传感器网络的远程状态估计

针对传感器网络中的远程状态估计, 提出一种多传感器切换的卡尔曼滤波器. 通过分析估计误差的统计特性, 证明估计误差的协方差具有边界, 采用线性矩阵不等式的形式给出了边界的收敛条件. 研究测量数据丢失对估计器性能的影响, 使用临界到达概率作为估计器的稳定性判据, 得到采用线性矩阵不等式求解临界到达概率的方法. 数值仿真证实了结论的正确性.     

Bounds for uncertain matrix root-clustering in a union of subregions

The research for robustness bounds for systems whose behaviour is described by a linear state-space model is addressed. The paper lays stress on the location of the eigenvalues of the state matrix when this matrix is subject either to an unstructured additive uncertainty or to a structured additive uncertainty. In the first case, upper bounds on the spectral norm of the uncertainty matrix are determined whereas in the second case, upper bounds on the maximal real perturbation in the state matrix are derived. In both cases, the fact that these bounds are not exceeded ensures that the eigenvalues of the uncertain state matrix lie in a specified region 𝒟 of the complex plane in which those of the nominal state matrix already lie. These bounds are obtained through a linear matrix inequalities approach. This approach allows to specify 𝒟, not only as a simple convex region, symmetric with respect to the real axis, but also as a non-convex (but symmetric with respect to the real axis) region defined itself as a union of convex subregions, each of them being not necessarily symmetric with respect to the real axis. Copyright © 1999 John Wiley & Sons, Ltd.     

Robust adaptive beamforming using a novel signal power estimation algorithm

Recently, many robust adaptive beamforming (RAB) methods based on covariance matrix reconstruction have been proposed. Motivated by the idea, in this paper, a novel and efficient signal power estimator is devised to reconstruct the interference-plus-noise covariance (INC) matrix, with the corresponding RAB algorithm proposed. Firstly, the steering vectors of the incoming sources are derived using the Capon spatial spectrum and known array geometry. Secondly, a set of linear equations is established based on the signal subspace projection, from which the powers of the incoming sources are estimated. Based on the presumed angular sector of the signal-of-interest (SOI), the steering vectors and powers of the SOI and interferences are distinguished, and the INC matrix is then reconstructed. Finally, the beamformer is determined by the estimated INC matrix and SOI steering vector. The proposed algorithm is computationally more efficient than other reconstruction-based methods because there are closed-form solutions for the signal powers. Simulation results indicate that our proposed algorithm performs better than the existing methods at high signal-to-noise ratios (SNRs), and achieves nearly optimal performance across a wide range of SNR.     

Reduced order state estimators for discrete-time stochastic systems

A reduced order, least squares, state estimator is developed for linear discrete-time systems having both input disturbance noise and output measurement noise with no output being free of measurement noise. The order reduction is achieved by using a Luenberger observer in connection with some of the system outputs and a Kalman filter to estimate the state of the Luenberger observer. The order of the resulting state estimator is reduced from the order of the usual Kalman filter system state estimator by the number of system outputs selected for use as inputs to the Luenberger Observer. The manner in which the noise associated with the selected system outputs affects the state estimation error covariance provides considerable insight into the compromise being attempted.