Separable Nonlinear Least Squares And Application In Solving For Parameter Of The Spatial Cartesian Coordinate Transformation

Nonlinear least squares is an important branch of optimization problems.With the development and application of electronic computer,the theory and method of nonlinear optimization have developed rapidly.Parameter estimation or data fitting are needed in the fields of measurement adjustment,deformation monitoring,neural network and so on.Generally,the mathematical models of these two kinds of problems have special forms according to their sources and practical engineering applications.If the form of the model is a linear combination of nonlinear functions,then this kind of problem is called separable nonlinear least square problem.This paper focuses on the improvement of variable projection algorithm(VP),the iterative algorithm of nonlinear parameter estimation and its application in the process of solving separable nonlinear least squares.The main research contents are as follows:(1)For the complexity of matrix calculation of nonlinear functions in the process of separable nonlinear least squares parameter separation,the variable projection algorithm is improved based on the matrix decomposition methods such as full rank decomposition,QR decomposition,singular value decomposition,Schmidt orthogonalization.The matrix operation in the process of parameter separation is simplified by matrix decomposition,and the calculation efficiency is improved;(2)For the iterative method in the process of solving nonlinear equations,this paper analyzes the basic principles of Trust-region method and Levenberg-Marquardt algorithm,and compares the advantages and disadvantages of the two iterative algorithms through Mackey-Glass time series simulation experiment and conversion model parameter solving experiment between Beijing 54 Coordinate System and WGS84 Coordinate System;(3)The improved variable projection algorithm based on Schmidt orthogonal decomposition is applied to solve the problem of space rectangular coordinate parameters.The traditional method of parameter non separation,the classical variable projection method(VP),the VP algorithm based on full rank decomposition,the VP algorithm based on QR decomposition,the VP algorithm based on singular value decomposition and the VP algorithm based on Schmidt orthogonalization are used to solve the problem respectively.The algorithms is compared in terms of the number of iterations,the number of function calculations and the sum of squares of residuals.This paper focuses on the improvement of the variable projection algorithm in the process of solving the separable nonlinear least squares,the iterative algorithm of parameter estimation to study the optimization problem of parameter estimation.Considering the special structure of the separable nonlinear least squares problem,a more effective solution method than the general nonlinear least squares algorithm was designed.Combined with the separable nonlinear least squares model in geodesy,the application of the solution method is studied.In this paper,combined with the structural characteristics of the separable nonlinear least squares problem,the linear parameters in the model are represented by nonlinear parameters by the improved variable projection algorithm based on matrix decomposition,and then transformed into the least squares problem with only nonlinear parameters.The Trust-region method and Levenberg-Marquardt algorithm are used to optimize the estimation of the nonlinear parameters.Through the simulation experiment and real experiment data,the improved variable projection algorithm is verified and analyzed.The experimental results show that under the condition of the same accuracy of the solution results,different improved variable projection algorithms have different degrees of improvement in the number of iterations,the number of function calculations and calculation efficiency.