A method for solving ill-conditioned separable nonlinear least squares problems and its application

For ill-conditioned separable nonlinear least squares problems, the LM (Levenberg-Marquardt) iteration method based on the VP (Variable Projection) algorithm and SVD (Singular Value Decomposition) for solving nonlinear parameters is explored in this paper, and a ridge estimation is proposed based on the TSVD (Truncated Singular Value Decomposition) and MSVD (Modified Singular Value Decomposition) methods to solve linear parameters. It is proved that the LM method based on SVD and the improved algorithm based on TSVD and MSVD can overcome the ill-conditioning of the matrix to a certain extent and enhance the reliability of parameter estimation through Mackey-Glass time series simulation and height anomaly fitting experiments. TSVD, MSVD and improved algorithm are compared and analyzed in terms of the accuracy and stability of parameter estimation and the curve fitting effect in the Mackey-Glass time series simulation experiment. The height anomaly fitting experiment verifies the feasibility and applicability of the algorithm in such practical problems. The experimental results show that the improved algorithm based on TSVD and MSVD enables the parameter estimation to be more stable, and the parameter values calculated by the algorithm bring about a better goodness of fit and fitting effect of the model.