A multiplicative Gauss-Newton minimization algorithm:Theory and application to exponential functions

Multiplicative calculus(MUC)measures the rate of change of function in terms of ratios,which makes the exponential functions significantly linear in the framework of MUC.Therefore,a generally non-linear optimization problem containing exponential functions becomes a linear problem in MUC.Taking this as motivation,this paper lays mathematical foundation of well-known classical Gauss-Newton minimization(CGNM)algorithm in the framework of MUC.This paper formulates the mathematical derivation of proposed method named as multiplicative Gauss-Newton minimization(MGNM)method along with its convergence properties.The proposed method is generalized for n number of variables,and all its theoretical concepts are authenticated by simulation results.Two case studies have been conducted incorporating multiplicatively-linear and non-linear exponential functions.From simulation results,it has been observed that proposed MGNM method converges for 12972 points,out of 19600 points considered while optimizing multiplicatively-linear exponential function,whereas CGNM and multiplicative Newton minimization methods converge for only 2111 and 9922 points,respectively.Furthermore,for a given set of initial value,the proposed MGNM converges only after 2 iterations as compared to 5 iterations taken by other methods.A similar pattern is observed for multiplicatively-non-linear exponential function.Therefore,it can be said that proposed method converges faster and for large range of initial values as compared to conventional methods.

STAP method based on atomic norm minimization with array amplitude-phase error calibration

In this paper,a space-time adaptive processing(STAP)method is proposed for the airborne radar with the array amplitude-phase error considered,which is based on atomic norm minimization(ANM).In the conventional ANM-based STAP method,the influence of the array amplitude-phase error is not considered and restrained,which inevitably causes performance deterioration.To solve this problem,the array amplitude-phase error is firstly estimated.Then,by pre-estimating the array amplitude-phase error information,a modified ANM model is built,in which the array amplitude-phase error factor is separated from the clutter response and the clutter covariance matrix(CCM)to improve the estimation accuracy of the CCM.To prove that the atomic norm theory is applicable in the presence of the array amplitude-phase error,the clutter sparsity is analyzed in this paper.Meanwhile,simulation results demonstrate that the proposed method is superior to the state-of-the-art STAP method.Moreover,the measured data is used to verify the effectiveness of the proposed method.

Generalization of the Global Error Minimization for Constructing Analytical Solutions to Nonlinear Evolution Equations

The global error minimization is a variational method for obtaining approximate analytical solutions to nonlinear oscillator equations which works as follows. Given an ordinary differential equation, a trial solution containing unknowns is selected. The method then converts the problem to an equivalent minimization problem by averaging the squared residual of the differential equation for the selected trial solution. Clearly, the method fails if the integral which defines the average is undefined or infinite for the selected trial. This is precisely the case for such non-periodic solutions as heteroclinic (front or kink) and some homoclinic (dark-solitons) solutions. Based on the fact that these types of solutions have vanishing velocity at infinity, we propose to remedy to this shortcoming of the method by averaging the product of the residual and the derivative of the trial solution. In this way, the method can apply for the approximation of all relevant type of solutions of nonlinear evolution equations. The approach is simple, straightforward and accurate as its original formulation. Its effectiveness is demonstrated using a Helmholtz-Duffing oscillator.

A multiaxial elastic potential with error-minimizing approximation to rubberlike elasticity

This study is concerned with a new,explicit approach by means of which forms of the large strain elastic potential for multiaxial rubberlike elasticity may be obtained based on data for a single deformation mode.As a departure from usual studies,here for the first time errors may be estimated and rendered minimal for all possible deformation modes and,furthermore,failure behavior may be incorporated.Numerical examples presented are in accurate agreement with Treloar＇s well-known data.

Kinematic Calibration and Forecast Error Compensation of a 2-DOF Planar Parallel Manipulator

Due to large workspace,heavy-duty and over-constrained mechanism,a small deformation is caused and the precision of the 2-DOF planar parallel manipulator is affected.The kinematic calibration cannot compensate the end-effector errors caused by the small deformation.This paper presents a method combined step kinematic calibration and linear forecast real-time error compensation in order to enhance the precision of a two degree-of-freedom(DOF) planar parallel manipulator of a hybrid machine tool.In the step kinematic calibration phase of the method,the end-effector errors caused by the errors of major constant geometrical parameters is compensated.The step kinematic calibration is based on the minimal linear combinations(MLCs) of the error parameters.All simple and feasible measurements in practice are given,and identification analysis of the set of the MLCs for each measurement is carried out.According to identification analysis results,both measurement costs and observability are considered,and a step calibration including step measurement,step identification and step error compensation is determined.The linear forecast real-time error compensation is used to compensate the end-effector errors caused by other parameters after the step kinematic calibration.Taking the advantages of the step kinematic calibration and the linear forecast real-time error compensation,a method for improving the precision of the 2-DOF planar parallel manipulator is developed.Experiment results show that the proposed method is robust and effective,so that the position errors are kept to the same order of the measurement noise.The presented method is attractive for the 2-DOF planar parallel manipulator and can be also applied to other parallel manipulators with fewer than six DOFs.

A Gauss-Newton Approach for Nonlinear Optimal Control Problem with Model-Reality Differences

Output measurement for nonlinear optimal control problems is an interesting issue. Because the structure of the real plant is complex, the output channel could give a significant response corresponding to the real plant. In this paper, a least squares scheme, which is based on the Gauss-Newton algorithm, is proposed. The aim is to approximate the output that is measured from the real plant. In doing so, an appropriate output measurement from the model used is suggested. During the computation procedure, the control trajectory is updated iteratively by using the Gauss-Newton recursion scheme. Consequently, the output residual between the original output and the suggested output is minimized. Here, the linear model-based optimal control model is considered, so as the optimal control law is constructed. By feed backing the updated control trajectory into the dynamic system, the iterative solution of the model used could approximate to the correct optimal solution of the original optimal control problem, in spite of model-reality differences. For illustration, current converted and isothermal reaction rector problems are studied and the results are demonstrated. In conclusion, the efficiency of the approach proposed is highly presented.

Improved modal truncation error in the directly analytical method for damage identification of frame structures

The damage identification is made by the numerical simulation analysis of a five-storey-and-two-span RC frame structure, using improved and unimproved direct analytical method respectively; and the fundamental equations were solved by the minimal least square method （viz. general inverse method）. It demonstrates that the feasibility and the accuracy of the present approach were impoved significantly, compared with the result of unimproved damage identification.

基于误差最小化的铸造不锈钢零件高速切削工艺方法研究

铸造不锈钢零件在切削过程中容易产生变形和振动,导致表面粗糙、切削尺寸偏差大。为了解决这一问题,提出基于误差最小化的铸造不锈钢零件高速切削工艺方法研究。在数控机床的支持下,采用在线测量的方式测量铸造不锈钢零件数据,将数控机床测量的坐标值转换为零件表面实测点,获取合适的测量点完成对加工路径的规划,在此基础上通过粗加工和精加工结合的形式对铸造不锈钢零件进行高速切削处理,并以误差最小为目的对加工误差进行补偿处理。实验结果表明:提出的基于误差最小化的高速切削工艺方法加工误差小、精度高,零件表面质量高,该工艺方法的实用性得到了增强。

基于相干性框架的部分支集已知的信号重建

文章使用l_(2)-αl_(2)(0<α≤1)最小化模型利用信号自身的先验支撑信息来重建高维稀疏信号。这是首篇基于相干性框架的部分支集已知的信号重建,重点讨论3种噪声(l_(2)有界噪声、Dantzig Selector噪声和脉冲噪声)情形下信号鲁棒恢复的充分条件和误差估计。

Tile selection method based on error minimization for photomosaic image creation

Photomosaic images are composite images composed of many small images called tiles.In its overall visual effect,a photomosaic image is similar to the target image,and photomosaics are also called“montage art”.Noisy blocks and the loss of local information are the major obstacles in most methods or programs that create photomosaic images.To solve these problems and generate a photomosaic image in this study,we propose a tile selection method based on error minimization.A photomosaic image can be generated by partitioning the target image in a rectangular pattern,selecting appropriate tile images,and then adding them with a weight coefficient.Based on the principles of montage art,the quality of the generated photomosaic image can be evaluated by both global and local error.Under the proposed framework,via an error function analysis,the results show that selecting a tile image using a global minimum distance minimizes both the global error and the local error simultaneously.Moreover,the weight coefficient of the image superposition can be used to adjust the ratio of the global and local errors.Finally,to verify the proposed method,we built a new photomosaic creation dataset during this study.The experimental results show that the proposed method achieves a low mean absolute error and that the generated photomosaic images have a more artistic effect than do the existing approaches.

面积误差最小约束下矢量数据向栅格数据转换的优化算法

在总结现有的矢量数据向栅格数据转换算法的基础上,提出一种面积误差最小约束下的优化算法。该算法保证了多边形矢量数据在栅格化后的面积误差尽可能最小,从而提高所得栅格数据在面积分析相关领域中应用的可信度。

以相对偏离度评估穿刺准确度对微创术治疗脑出血疗效的影响

目的介绍一种评估微创术穿刺准确度的方法——相对偏离度,并评估穿刺准确度对血肿清除效果的影响。方法回顾性分析2012年8月~2014年1月应用徒手穿刺法实施微创术联合重组组织型纤溶酶原激活剂治疗自发性幕上脑出血患者的临床资料,利用相对偏离度评估穿刺准确度,并分析穿刺准确度与抽吸、液化引流及血肿清除效果间的关系。结果共纳入53例患者,平均年龄（53.0±12.2）岁,术前血肿量为（38.3±12.1）mL。相对偏离度平均为（0.54±0.27）,相关分析显示相对偏离度与血肿清除率呈负相关（r=-0.591,P〈0.01）。相对偏离度≤0.6可能是较理想的准确度,53例中达到该准确度的比例为58.4%。结论相对偏离度是一种简便易行的评估穿刺准确度的方法。有必要进一步提高徒手穿刺法的穿刺准确度,以增强血肿清除效果。

基于UKF的四元数载体姿态确定

对于低精度高噪声的传感器组成的低成本姿态测量系统,本文引入U nscen ted K a lm an filtering(UKF)用于姿态确定,设计了有陀螺测量和四元数差分法的无陀螺测量两种UKF滤波器;应用四元数避免了欧拉角法的奇异问题;用高斯-牛顿误差最小法将六维参考向量转化为四元数,作为观测量的一部分,使九维非线性观测方程转化为七维线性方程进行滤波,减少了计算量;应用仿真数据进行算法验证,成功得到姿态估计;对两种算法在低速和高速状态下进行验证,仿真结果表明了该方法的有效性。

三角形定位算法的误差分析

为了研究三角形内定位误差,首先对三角形定位进行建模;其次在对该模型分析证明的基础上得出三角形内最小定位误差出现的条件;最后基于该条件抽象出了三角形内定位误差的变化规律,并通过实际测试验证该规律的正确性.得出的结论均来自于推导计算,为定位算法研究提供了有力理论依据与支持.

基于误差最小原理的微震震源参数反演

为减小矿山地质、岩性、施工等因素对震源参数测量精度的影响,提出一种基于误差最小原理的震源参数反演算法(均匀速度模型),在目标函数取最小值条件下拟合出最优震源参数。研究结果表明:无论是对内场震源点还是对外场震源点进行定位分析,提出的反演方法比传统的定位算法具有更高的定位精度。克服传统方法由于现场实测波速与真实波速之间存在误差而影响到拟合精度的缺点,可满足实际工程监测需要,研究结果可为相关工程的微震准确定位提供参考和借鉴。

估计GM(1,1)模型中参数的线性规划方法

估计GM(1,1)模型中的参数通常采用最小二乘准则,而在模型精度检验时又常采用平均相对误差。在平均相对误差达到最小准则或最大相对误差达到最小准则时,分别给出了估计GM(1,1)模型中参数的线性规划方法,并通过实例给出了不同极小化准则下数值结果的对比。数值结果表明,采用平均相对误差达到最小准则和最大相对误差达到最小准则比通常采用的最小二乘准则更合理,效果更好。

最小区域法评定圆度误差的计算机实现方法

针对在用计算机实现圆度测量的过程中,首先要解决的就是评定方法的算法,介绍了按最小区域法评定圆度误差的一种快速、简便、易于微机实现的数据处理方法,并给出了用C语言实现其算法后的结果。

太赫兹逆合成孔径雷达相位误差分析和补偿方法

通过对太赫兹逆合成孔径雷达(THz-ISAR)成像原理进行分析,构造目标回波距离维相位误差的多项式表示形式,提出了基于启发式最小熵搜索的二次相位误差校正方法和基于距离维自聚焦的三次及三次以上相位误差校正方法,校正了THz-ISAR回波信号距离维的相位误差,进而消除THz-ISAR一维距离像和二维成像结果中的散焦和展宽现象。理论分析和仿真实验结果都表明该方法能够有效补偿目标回波距离维相位误差,提高一维距离像和二维成像的聚焦效果。

锚喷巷道通风阻力系数的计算式研究

通过对平煤集团大量的锚喷巷道的回归分析 ,建立了巷道壁面绝对粗糙度和巷道断面积对摩擦阻力系数影响程度的回归方程 ,并对回归方程作了显著性检验 ,最后对巷道粗糙程度和断面积对锚喷巷道摩擦阻力系数的影响程度进行了讨论和总结 ,对同类巷道摩擦阻力系数的选取具有实践指导意义。

齿轮传动误差的数字仿真算法研究

齿轮副传动误差分析的传统方法为解析法,其过程十分复杂。根据最小有向距离原理,提出一种齿轮传动误差的数字仿真算法,具有数学模型简单、计算方便的特点,是分析齿轮传动误差的一种新方法。