Research On The Ill-posed And Solving Methods Of Nonlinear Least Squares Problem

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.