【作者】 唐利民；

【导师】 朱建军；

【作者基本信息】 中南大学 ， 大地测量学与测量工程， 2011， 博士

【摘要】 非线性模型广泛存在于测量平差、变形监测及路面模量反算领域中。诸如导线测量中以待定点坐标为未知参数的角度观测方程和边长观测方程,它们都是待定点坐标的非线性函数；GPS伪距测量中,卫星至测站的几何距离的观测方程是测站点待定坐标的非线性函数；路基沉降变形分析与预测模型中,沉降量是关于时间的非线性函数；路面模量反算的模型也是非线性函数等等。一般地,这些模型的求解均基于非线性最小二乘原理。针对工程应用上的特点,研究非线性最小二乘问题的性质和解算方法,具有重要的理论意义和实践价值。求解非线性最小二乘问题可能会存在不适定现象。不合适的原始问题的函数模型或者解算方法的选择不当都有可能导致非线性最小二乘问题的求解失败,特别是数值迭代算法中涉及到迭代矩阵的求逆运算,而迭代矩阵的病态或奇异则会导致算法失败。有必要根据不适定理论,研究非线性最小二乘的不适定现象,为解决非线性最小二乘在工程实际问题应用中的局限性提供理论和方法支持。本文针对不适定非线性最小二乘问题的现状和存在的问题进行了研究,主要贡献有以下几点：1.论文分析了求解非线性最小二乘问题的经典数值迭代法,提出了求解非线性最小二乘问题的数值迭代法的统一模型,在此基础上对非线性最小二乘问题可能产生的两种不适定性进行了分析,给出了两种不适定性的定义。并明确指出统一模型中迭代矩阵的求逆运算是产生第一种不适定现象的重要原因。2.由于迭代矩阵求逆会导致非线性最小二乘问题产生第一种不适定性,研究迭代矩阵的病态判据就显得尤其重要。论文分析了非线性最小二乘数值迭代算法中迭代矩阵的特性,结合一般病态矩阵判据理论,给出了迭代矩阵的病态判据,并研究了一种新的替代矩阵,此替代矩阵可以有效的降低迭代矩阵的条件数,为后续研究相关算法提供了理论基础。3.由于Landweber迭代无需矩阵求逆,避免了一般非线性最小二乘问题解算方法中由于矩阵求逆而产生的不适定现象,论文基于Landweber迭代理论,建立了不适定非线性最小二乘的Landweber迭代算法,此算法无需迭代矩阵求逆即可有效收敛。4.论文结合同伦延拓和正则化方法,构造了一个稳定泛函,建立了非线性最小二乘问题正则同伦解算方法,对其迭代过程进行了详细的推导。在该泛函中,正则参数可同时起到同伦参数的作用。由于正则同伦参数的存在,可以根据迭代矩阵病态特点选取不同的正则同伦参数值来避免和改善迭代矩阵在迭代过程中的病态性或奇异性。5.基于Tikhonov正则化原理,通过添加稳定泛函项,结合修正高斯-牛顿法,论文还建立起Tikhonov正则化修正高斯-牛顿法,此方法可以有效改善迭代矩阵在迭代过程中的病态程度。6.通过对迭代矩阵添加一个正则化因子α来改善矩阵在迭代过程中的病态程度,论文建立了不适定非线性最小二乘问题的正则化牛顿迭代算法,给出了算法迭代步骤。基于路基沉降泊松预测模型的计算实例表明,此算法在改善迭代矩阵病态性或奇异性的同时,也较普通牛顿迭代法具有更好的拟合曲线。7.介绍了正则参数选择策略的研究进展,并对七种正则化因子选择方法作了简单的介绍。论文根据不适定非线性最小二乘问题的特点,提出了直接搜索法和区间划分法这两种确定正则化因子的方法,此方法非常便于计算机实现。8.结合计算机语言,开发出了相应的求解不适定非线性最小二乘问题算法的INLS工具箱,利用此工具箱,可以很方便的解算不适定的非线性最小二乘问题。由于不适定现象会导致非线性最小二乘问题解算失败,为了避免其产生不适定现象,有必要认真研究非线性最小二乘的不适定理论和方法。论文首先研究了非线性最小二乘的不适定现象,然后分析了迭代矩阵的病态判据,最后研究了适合不适定非线性最小二乘问题的多种解算算法。结合非线性平差实例及道路工程中非线性最小二乘问题的应用,论文对给出的算法与已有的算法进行了比较,并进行了数值分析和实例解算,从而验证了算法的有效性,使得算法能够很好地应用于工程计算。更多还原

【Abstract】 Nonlinear models exist in survey adjustment, deformation monitoring and pavement modulus back calculations fields widely. For example, in traverse survey, angle observations equation and side length equation which use fixed point coordinate as unknown parameters are a nonlinear function of fixed point coordinate. In GPS pseudo-range measurement, observation equation of geometric distance between satellite and observation station is also a nonlinear function of fixed point coordinate. In subgrade settlement Analysis and Prediction models, the settling volume is a nonlinear function of time. Models of modulus back calculations are nonlinear function etc. Solution of those models is based on principle of nonlinear least square general. Study the nature and calculation method of nonlinear least square problems according to the characteristics of engineering application has important theoretical and practical value.Nonlinear least squares problem solving may exit ill-posed phenomena. It may lead to failure to solve nonlinear least squares problem such as inappropriate function model of the original problem or inappropriate choice of solution method. Especially, value iteration algorithms which need iterative matrix inversion, the ill-posed of iterative matrix will lead algorithms to failure. It is necessary to study the ill-posed phenomenon of nonlinear least squares according to ill-posed theory, and provide theoretical and methodological support for solving the limitations of the nonlinear least squares using in engineering applications.In view of its current situation and problems, ill-posed nonlinear least squares problem is studies in this paper. Its main achievements are as follows:(1) The classical numerical iterative methods for solving nonlinear least squares problem are analyzed in this paper. A unified model of numerical iterative for solving nonlinear least squares problem is proposed. On this basis, two ill-posed phenomena of nonlinear least squares problem are analysis. Definitions of the two ill-posed are also given. And make clear that iterative matrix inversion in the unified model of numerical iterative is the important reason of produce the first phenomenon.(2) Study of pathological criteria of iterative matrix is particularly important when iterative matrix inversion lead to the first phenomenon of nonlinear least squares problem. The iteration matrix properties in numerical iterative algorithm of nonlinear least squares were analyzed in this paper. According to the criterion theory of general matrix, iteration matrix pathological criteria are given. A new alternative matrix is also given which can effectively reduce the condition number of iterative matrix. This provides a theoretical basis for the follow-up correlation algorithm study.(3) Landweber iteration method which avoids the ill-posed phenomena of general solution method for nonlinear least squares problem in matrix inversion for it without matrix inversion. Based on Landweber iteration theory, Landweber iteration algorithm is established for ill-posed nonlinear least squares problem in this paper. This algorithm can be effective iteration convergence without matrix inversion.(4) Combine with continuation homotopy and regularization method, a regular homotopy solution method for nonlinear least squares problem with constructing a stable functional is established this paper. The iterative process is derived in detail.Regularization parameter can also play the role of homotopy parameter in the stable functional. It can select a different regularization homotopy parameter values to reduce the matrix condition number and alleviates matrix ill-conditioning degree according to characteristics of the ill-posed iterative matrix.(5) Based on Tikhonov regularization theory, by adding a stable function and referring to the modified Gauss-Newton method, the regularization modified Gauss-Newton method to solve the nonlinear least square problem is constructed in this paper. The method can alleviates matrix ill-conditioning degree to the iterative matrix in iterative process.(6) Based on regularization theory, with an addition of a regular factor to iterative matrix to alleviate matrix ill-conditioning degree, regularized Newton iterative method for ill-posed nonlinear least square problem is constructed in this paper. Iterative procedure is also given. The analysis of settlement data using the Poisson prediction model show that the regularized Newton iterative method not only can alleviates matrix ill-conditioning degree to the iterative matrix, but also can get better fitting curve than general Newton iterative method.(7) The research progress of the regular parameters selection method is described this paper. Seven regular factor selection methods are described briefly. According to the characteristics of ill-posed nonlinear least square problem, two new regularization parameter selecting strategies are proposed, which called as direct search method and interval division method. The two methods are very convenient for computer implementation.(8) An INLS Toolbox to solve the ill-posed nonlinear least squares problem combined with computer language is developed in this paper. Solutions to the ill-posed nonlinear least squares problem can be easily getting when use this Toolbox.Since ill-posed phenomena will lead to nonlinear least squares problem solver failure, it is necessary to study the theory and methods of ill-posed nonlinear least squares problem to avoid the ill-posed phenomena. Ill-posed phenomena of nonlinear least squares problem is studied firstly in this paper. Then the pathological criteria of the iterative matrix are analyzed. Finally, variety of algorithms is researched to suitable for solving ill-posed nonlinear least squares problem. With examples of nonlinear surveying adjustment and road engineering application of nonlinear least squares problem, the algorithm presented in the paper is compared with existing algorithms and simulated with experimental data, thus verifying the validity of the algorithm and showing that the algorithm is applicable to engineering calculation.更多还原