Distributed Least-Squares Iterative Methods in Large-Scale Networks:A Survey

Many science and engineering applications involve solvinga linear least-squares system formed from some field measurements. In the distributed cyber-physical systems(CPS),each sensor node used for measurement often only knowspartial independent rows of the least-squares system. To solve the least-squares all the measurements must be gathered at a centralized location and then perform the computa-tion. Such data collection and computation are inefficient because of bandwidth and time constraints and sometimes areinfeasible because of data privacy concerns. Iterative methods are natural candidates for solving the aforementionedproblem and there are many studies regarding this. However,most of the proposed solutions are related to centralized/parallel computations while only a few have the potential to beapplied in distributed networks. Thus distributed computations are strongly preferred or demanded in many of the realworld applications, e.g. smart-grid, target tracking, etc. Thispaper surveys the representative iterative methods for distributed least-squares in networks.

Low-complexity signal detection for massive MIMO systems via trace iterative method

Linear minimum mean square error(MMSE)detection has been shown to achieve near-optimal performance for massive multiple-input multiple-output(MIMO)systems but inevitably involves complicated matrix inversion,which entails high complexity.To avoid the exact matrix inversion,a considerable number of implicit and explicit approximate matrix inversion based detection methods is proposed.By combining the advantages of both the explicit and the implicit matrix inversion,this paper introduces a new low-complexity signal detection algorithm.Firstly,the relationship between implicit and explicit techniques is analyzed.Then,an enhanced Newton iteration method is introduced to realize an approximate MMSE detection for massive MIMO uplink systems.The proposed improved Newton iteration significantly reduces the complexity of conventional Newton iteration.However,its complexity is still high for higher iterations.Thus,it is applied only for first two iterations.For subsequent iterations,we propose a novel trace iterative method(TIM)based low-complexity algorithm,which has significantly lower complexity than higher Newton iterations.Convergence guarantees of the proposed detector are also provided.Numerical simulations verify that the proposed detector exhibits significant performance enhancement over recently reported iterative detectors and achieves close-to-MMSE performance while retaining the low-complexity advantage for systems with hundreds of antennas.

Separating iterative solution model of generalized nonlinear dynamic least squares for data processing in building of digital earth

Data coming from different sources have different types and temporal states. Relations between one type of data and another ones, or between data and unknown parameters are almost nonlinear. It is not accurate and reliable to process the data in building the digital earth with the classical least squares method or the method of the common nonlinear least squares. So a generalized nonlinear dynamic least squares method was put forward to process data in building the digital earth. A separating solution model and the iterative calculation method were used to solve the generalized nonlinear dynamic least squares problem. In fact, a complex problem can be separated and then solved by converting to two sub problems, each of which has a single variable. Therefore the dimension of unknown parameters can be reduced to its half, which simplifies the original high dimensional equations.

ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS

The Galerkin and least-squares methods are two classes of the most popular Krylov subspace methOds for solving large linear systems of equations. Unfortunately, both the methods may suffer from serious breakdowns of the same type: In a breakdown situation the Galerkin method is unable to calculate an approximate solution, while the least-squares method, although does not really break down, is unsucessful in reducing the norm of its residual. In this paper we first establish a unified theorem which gives a relationship between breakdowns in the two methods. We further illustrate theoretically and experimentally that if the coefficient matrix of a lienar system is of high defectiveness with the associated eigenvalues less than 1, then the restarted Galerkin and least-squares methods will be in great risks of complete breakdowns. It appears that our findings may help to understand phenomena observed practically and to derive treatments for breakdowns of this type.

Solving method of generalized nonlinear dynamic least squares for data processing in building of digital mine

Data are very important to build the digital mine. Data come from many sources, have different types and temporal states. Relations between one class of data and the other one, or between data and unknown parameters are more nonlinear. The unknown parameters are non random or random, among which the random parameters often dynamically vary with time. Therefore it is not accurate and reliable to process the data in building the digital mine with the classical least squares method or the method of the common nonlinear least squares. So a generalized nonlinear dynamic least squares method to process data in building the digital mine is put forward. In the meantime, the corresponding mathematical model is also given. The generalized nonlinear least squares problem is more complex than the common nonlinear least squares problem and its solution is more difficultly obtained because the dimensions of data and parameters in the former are bigger. So a new solution model and the method are put forward to solve the generalized nonlinear dynamic least squares problem. In fact, the problem can be converted to two sub problems, each of which has a single variable. That is to say, a complex problem can be separated and then solved. So the dimension of unknown parameters can be reduced to its half, which simplifies the original high dimensional equations. The method lessens the calculating load and opens up a new way to process the data in building the digital mine, which have more sources, different types and more temporal states.

A Low-Complexity Signal Detection Utilizing AOR Iterative Method for Massive MIMO Systems

Massive multiple-input multiple-output(MIMO) system is capable of substantially improving the spectral efficiency as well as the capacity of wireless networks relying on equipping a large number of antenna elements at the base stations. However, the excessively high computational complexity of the signal detection in massive MIMO systems imposes a significant challenge for practical hardware implementations. In this paper, we propose a novel minimum mean square error(MMSE) signal detection using the accelerated overrelaxation(AOR) iterative method without complicated matrix inversion, which is capable of reducing the overall complexity of the classical MMSE algorithm by an order of magnitude. Simulation results show that the proposed AOR-based method can approach the conventional MMSE signal detection with significant complexity reduction.

Least-Squares Solutions of Generalized Sylvester Equation with Xi Satisfies Different Linear Constraint

In this paper, an iterative method is constructed to find the least-squares solutions of generalized Sylvester equation , where is real matrices group, and satisfies different linear constraint. By this iterative method, for any initial matrix group within a special constrained matrix set, a least squares solution group with satisfying different linear constraint can be obtained within finite iteration steps in the absence of round off errors, and the unique least norm least-squares solution can be obtained by choosing a special kind of initial matrix group. In addition, a minimization property of this iterative method is characterized. Finally, numerical experiments are reported to show the efficiency of the proposed method.

Multi-loop Constrained Iterative Model Predictive Control Using ARX -PLS Decoupling Structure

A multi-loop constrained model predictive control scheme based on autoregressive exogenous-partial least squares(ARX-PLS) framework is proposed to tackle the high dimension, coupled and constraints problems in industry processes due to safety limitation, environmental regulations, consumer specifications and physical restriction. ARX-PLS decoupling character enables to turn the multivariable model predictive control(MPC) controller design in original space into the multi-loop single input single output(SISO) MPC controllers design in latent space.An idea of iterative method is applied to decouple the constraints latent variables in PLS framework and recursive least square is introduced to identify ARX-PLS model. This algorithm is applied to a non-square simulation system and a stirred reactor for ethylene polymerizations comparing with adaptive internal model control(IMC) method based on ARX-PLS framework. Its application has shown that this method outperforms adaptive IMC method based on ARX-PLS framework to some extent.

An Iterative Learning Approach to Identify Fractional Order KiBaM Model

This paper discusses the parameter and differentiation order identification of continuous fractional order KiBaM models in ARX (autoregressive model with exogenous inputs) and OE (output error model) forms. The least squares method is applied to the identification of nonlinear and linear parameters, in which the Grünwald-Letnikov definition and short memory principle are applied to compute the fractional order derivatives. An adaptive P-type order learning law is proposed to estimate the differentiation order iteratively and accurately. Particularly, a unique estimation result and a fast convergence speed can be arrived by using the small gain strategy, which is unidirectional and has certain advantages than some state-of-art methods. The proposed strategy can be successfully applied to the nonlinear systems with quasi-linear characteristics. The numerical simulations are shown to validate the concepts. © 2017 Chinese Association of Automation.

基于ILS的航向空域仿真系统研究

仪表着陆系统(ILS)为飞行器着陆的最后阶段提供安全保障。在分析某机场仪表着陆系统工作原理的基础上,对调制深度差(DDM)进行理论研究,应用控制变量法和最小二乘法原理建立在DDM线性变化范围内SBO信号的幅值与航道宽度的关系模型。与实际情况的对比分析验证了模型的正确性。基于Lab VIEW软件,实现了航道宽度仿真软件的设计。软件具有对虚拟航向面三维显示及航道宽度模拟调节功能。通过与机场实际设备相连接,可实时显示当前航向空域信息。软件较传统的二维显示更具有直观性,可以作为机场工作人员的教学工具,提升机场工作人员对仪表着陆系统的认识水平。

A MODEL IDENTIFICATION METHOD OF VIBRATING STRUCTURES FROM INCOMPLETE MODAL INFORMATION

The accurate mathematical models for complicated structures are verydifficult to construct.The work presented here provides an identification method for estimating the mass, damping , and stiffness matrices of linear dynamical systems from incompleteexperimental data. The mass, stiffness, and damping matrices are assumed to be real,symmetric, and positive definite. The partial set of experimental complex eigenvalues and corresponding eigenvectors are given. In the proposed method the least squaresalgorithm is combined with the iteration technique to determine systems identified matrices and corresponding design parameters. several illustrative examples, are presented to demonstrate the reliability of the proposed method .It is emphasized thatthe mass, damping and stiffness martices can be identified simultaneously.

A MODEL IDENTIFICATION METHOD OF VIBRATING STRUCTURES FROM INCOMPLETE MODAL INFORMATION

The accurate mathematical models for complicated structures are very difficult to construct.The work presented here provides an identification method for estimating the mass.damping,and stiffness matrices of linear dynamical systems from incomplete experimental data.The mass,stiffness and damping matrices are assumed to be real,symmetric,and positive definite The partial set of experimental complex eigenvalues and corresponding eigenvectors are given.In the proposed method the least squares algorithm is combined with the iteration technique to determine systems identified matrices and corresponding design parameters.Seeveral illustative examples,are presented to demonstrate the reliability of the proposed method .It is emphasized that the mass,damping and stiffness matrices can be identified simultaneously.

Newton, Halley, Pell and the Optimal Iterative High-Order Rational Approximation of √<span style='margin-left:-2px;margin-right:2px;border-top:1px solid black'>N</span>

In this paper we examine single-step iterative methods for the solution of the nonlinear algebraic equation f (x) = x2 - N = 0 , for some integer N, generating rational approximations p/q that are optimal in the sense of Pell’s equation p2 - Nq2 = k for some integer k, converging either alternatingly or oppositely.

The Hidden Geometry of the Babylonian Square Root Method

We propose and demonstrate an original geometric argument for the ancient Babylonian square root method, which is analyzed and compared to the Newton-Raphson method. Based on simple geometry and algebraic analysis the former original iterated map is derived and reinterpreted. Time series, fixed points, stability analysis and convergence schemes are studied and compared for both methods, in the approach of discrete dynamical systems.

基于ico_HEALPix网格的超高阶地球重力场建模方法

本文针对传统地理网格数据剖分的重力场模型构建在高纬度区域出现数据冗余的问题,将分层等积等纬度像素化(HEALPix)网格结构引入地球重力场解算中,提出了利用二十面体HEALPix(ico_HEALPix)网格的超高阶地球重力场建模方法,实现了全球3600阶次球谐位系数的高效构建,同时针对ico_HEALPix网格在球谐分析过程中法矩阵不是严格块对角化结构的问题,设计了迭代算法,有效提高模型构建的精度。试验表明,ico_HEALPix网格数据在数据量小于地理网格500万的前提下,通过迭代方法构建的全球地球重力场模型精度可达到优于地理网格的效果,球谐位系数误差阶RMS提升1~2个数量级,还解决了地理网格南北极点畸变和数据冗余的问题,提高了网格的数据利用率。

基于机器视觉的软磨片定位测量系统

针对软磨片外轮廓定位测量过程中,边缘较难提取,轮廓圆心与半径拟合精度较低的问题,提出通过中值滤波和灰度变换改善图像质量,使用灰度投影与灰度方差统计进行边缘筛选,利用圆心约束最小二乘法拟合外轮廓,通过高斯—牛顿法得到轮廓圆心坐标与半径,完成高精度定位和测量。通过实验对比,系统定位与测量误差保持在0.2mm以内,检测速度达到每片0.8s,在生产应用中具有实用性和高效性,提高了软磨片的检测效率和生产自动化程度。

抗间歇采样转发干扰的发射波形与失配滤波器联合优化算法

间歇采样转发干扰(interrupted sampling repeater jamming,ISRJ)与发射信号之间具有强相关性,若在接收端采用匹配滤波处理则会产生多个数量可控的高逼真假目标,对雷达的检测性能具有极大的干扰。针对上述问题,提出了一种抗ISRJ的发射波形和接收滤波器联合优化算法,将失配滤波体制下的信噪比(signal to noise ratio,SNR)损失、发射信号恒模约束以及滤波器能量约束考虑在内,以最小化干扰信号归一化脉压后峰值、失配滤波输出信号积分副主比以及输出干信比,并采用主分量最小化(majorization-minimization,MM)方法与平方迭代加速算法提高算法运行速度。仿真结果表明,与其他同类方法相比,所提算法能在保证可靠抗干扰性能的同时,极大缩短运行时间,具有较好的实时性。

DQ变换和MUSIC算法在ITER磁体电源信号间谐波检测中的应用

随着国际热核聚变实验堆(International Thermonuclear Experimental Reactor,ITER)计划的逐步开展,保证ITER磁体电源系统的稳定运行显得尤为重要。文章采用将DQ变换和多信号分类(multiple signal classification,MUSIC)算法相结合的方法进行间谐波频率检测,信号的幅度和相位由最小二乘法来估计。DQ变换可以消除大幅度ITER基波分量,MUSIC算法可以通过矩阵特征分解检测出短数据条件下的谐波和间谐波,适用短时平稳的间谐波检测,两者相结合可以有效检测出大幅度基波附近存在小幅度间谐波。仿真实验表明,计算经DQ变换后检测出的ITER信号谐波频率时,取中间信号计算真实频谱较为正确,两侧信号则有较大的误差。

加权总体最小二乘法在三维框架转换中的应用

国际地球参考框架转换是小角度直角坐标转换,一般采用布尔莎七参数模型通过最小二乘方法求解。针对坐标转换中存在系数矩阵有常数元素和重复元素的情况,提出适用于ITRF转换的加权结构总体最小二乘方法。该方法考虑坐标先验精度,构造了顾及系数矩阵结构性的矩阵,确保重复元素具有相同的改正数,并使用拉格朗日乘数法迭代求解。实际数据解算转换参数结果显示,该算法的精度和准确度优于传统方法。

表征土壤水分持留曲线的几种模型的适应性研究

土壤水分持留曲线指的是土壤体积水分含量与压力水头之间的关系,在研究土壤水分流动和溶质运移中有着非常重要的作用。由于它们之间的关系复杂,难以从理论上推导出确切的关系式;但通过大量的试验研究,人们已提出了许多经验公式来描述它,其中比较常用的有:Brooks-Corey（1964）模型,Gardner（1970）模型,van Genuchten（1980）模型和 Gardner-Russo（1988）模型等。在这些模型中都含有许多待求的参数。本文借助于最小二乘法,形成了确定这四个模型中的参数所对应的非线性方程组,并用Picard迭代求解它们。最后,用数值例子说明了这四种模型对不同类型土壤的适应性。