An efficient Levenberg–Marquardt method with a new LM parameter for systems of nonlinear equations

A modified Levenberg–Marquardt method for solving singular systems of nonlinear equations was proposed by Fan [J Comput Appl Math. 2003;21;625–636]. Using trust region techniques, the global and quadratic convergence of the method were proved. In this paper, to improve this method, we decide to introduce a new Levenberg–Marquardt parameter while also incorporate a new nonmonotone technique to this method. The global and quadratic convergence of the new method is proved under the local error bound condition. Numerical results show the new algorithm is efficient and promising.     

双边永磁同步直线电机随机模型系统辨识

为解决双边永磁同步直线电机离散传递函数模型参数动态变化、传统方法辨识精度低的问题,提出一种基于随机模型的系统辨识方法,分析运动过程中的系统特性,结合具有随机扰动项的Box-Jenkins模型,辨识动态系统传递函数模型参数。通过注入不同频率逆M序列电流信号,充分激发系统响应,比较分析不同实验条件下的系统辨识结果。选择预报误差法定义平方差代价函数,运用Levenberg Marquardt算法不断迭代优化以获得传递函数模型最优参数。通过对不同采样频率、不同注入电流幅值和不同电机运动速度条件下的系统辨识对比分析,获得最佳辨识模型。结果表明,当采样频率为1000 Hz,注入电流幅值为2.0 A和电机运动速度为50 mm?s-1时,获得最优辨识仿真输出匹配度为94.81%,平方差为7.84×10-4。     

局域定位系统拟应用于工程测量系统自定位法

提出了一种系统自定位方法.该方法采用控制点坐标进行约束,按照非线性最小二乘的Levenberg-Marquardt算法求解系统自定位参数.以双发射站为例,给出了新的系统自定位参数的解算模型.通过试验证明此方法是可行的,解算模型是正确有效的,其系统自定位精度完全能够满足工程测量的要求.     

BP网络光度法测定去痛片中的四组分

应用Levenberg-Marquardt优化算法的BP网络,采用紫外分光光度法测定去痛片中四组分的含量。测定低含量组分咖啡因和苯巴比妥时,结合吸光度减法技术。氨基比林、非那西丁、咖啡因和苯巴比妥的平均回收率和RSD分别为99.3%,1.2%：;100.0%,1.1%;100.3%,1.5%;100.1%,1.3%。     

Analytical solution of stochastic differential equation by multilayer perceptron neural network approximation of Fokker–Planck equation

The Fokker–Planck equation is a useful tool to analyze the transient probability density function of the states of a stochastic differential equation. In this paper, a multilayer perceptron neural network is utilized to approximate the solution of the Fokker–Planck equation. To use unconstrained optimization in neural network training, a special form of the trial solution is considered to satisfy the initial and boundary conditions. The weights of the neural network are calculated by Levenberg–Marquardt training algorithm with Bayesian regularization. Three practical examples demonstrate the efficiency of the proposed method.     

高CO2含量天然气偏差因子影响研究

天然气偏差因子是天然气物性参数和地质储量计算的基础,其不仅受温度压力的影响,也是天然气组分与组成的函数。目前,对低含量CO₂的天然气偏差因子计算方法精度较高,但对以DF气田典型代表的高含CO₂天然气偏差因子,其计算精度难以得到保障。针对不同CO₂含量天然气采用PVT实验方法,分析了CO₂含量对天然气偏差因子的影响规律,并以实验结果为基础,用麦夸特法及通用全局优化法修正了DAK计算天然气偏差因子经验公式。结果表明,与重烃（乙烷、丙烷）对天然气偏差因子的影响规律相似,CO₂的存在明显降低了天然气的偏差因子,且影响幅度随CO₂含量增加而逐渐增大;当CO₂含量较低时,DAK经验公式能够较准确计算天然气偏差因子,而当CO₂含量大于50%时,应用修正的DAK经验公式能够获得满意的效果。     

The Convergence of a Levenberg–Marquardt Method for Nonlinear Inequalities

In this paper, we consider the least l ₂-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg–Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l ₂-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions.     

纯距离法激光跟踪坐标测量系统的布局与仿真

一般激光跟踪测量系统通过测量距离变化量和跟踪镜转角来确定目标的位置坐标。而纯距离法激光跟踪测量系统只测量距离和距离变化量 ,而不用测量角度量 ,即可进行坐标测量 ,因此可得到较高的测量精度。其系统几何参数可以不借助外界实物标准尺进行自标定 ,并建立虚拟坐标测量参考系。多站测量系统的布局、测量点选择、计算的收敛性与误差等必须进行分析。从激光跟踪测量系统自标定的数学模型出发 ,分析其方程组的雅可比矩阵 ,推导出在自标定时测量系统中各站点和测量点的布局限制 ,并用计算机进行了仿真验证。     

解Duffing方程周期解的Levenberg-Marquardt方法

先把求解微分方程的周期解问题转化为无约束最优化问题, 再利用无约束最优化问题的最优性条件及Levenberg-Marquardt方法求解了满足限制共
振条件下的一类Duffing方程的周期解. 数值计算结果表明了方法的有效性.     

自适应光学系统变形镜控制电压预测

在校正大气湍流畸变波前相差的自适应光学系统中,利用基于Levenberg-Marquardt学习算法的非线性反向传播神经网络技术（LMBP）对变形镜控制电压进行预测。以对受横向风影响的大气湍流畸变波前斜率数据为研究对象,通过数值仿真方法,研究了基于LMBP算法的自适应光学系统变形镜电压非线性预测控制算法。通过实验发现,预测电压和变形镜实际控制电压拟合效果良好。讨论了回溯帧数对预测效果的影响,并与基于递推最小二乘（RLS）算法的线性预测算法进行比较。对比结果表明,基于LMBP算法的非线性电压预测方法比基于递推最小二乘法的线性电压预测方法能更有效地降低系统由伺服延迟引起的误差。