Downward Continuation Iterative Regularization Solution Based on Quasi Optimal Regularization Factor Set

Objectives: Iterative Tikhonov regularization and iterative Landweber regularization are currently the two most commonly used iterative regularization algorithms, which can effectively solve the ill-posed problem during the continuation process. However, their optimal combination of iteration times and regularization parameters has not yet been determined, which limits their practicality. Methods: This paper constructs the concept of quasi-optimal regularization factor set and analyzes the distribution of iteration numbers and regularization parameters in the quasi-optimal regularization factor set. Based on this, a selection criterion for the optimal combination of iteration number and regularization parameter in iterative regularization algorithms is proposed, and a formula for using the L-curve method to obtain the optimal regularization parameter corresponding to the number of iterations is provided. Results: The study on the relationship between the extension error of iterative regularization algorithms and the variation of regularization factor sets shows that there is a significant correlation between the optimal regularization parameters and the number of iterations in the quasi optimal regularization factor set. When the number of iterations exceeds 10, the extension solutions corresponding to each quasi optimal regularization factor set are basically identical. When the number of iterations is large enough, any quasi optimal regularization factor set can be considered as the optimal regularization factor set. The iterative regularization algorithm based on the selected strategy in this article has a smoother extension solution and smaller error compared to the traditional Tikhonov regularization method. However, in areas with significant data changes, some high-frequency signals are filtered out as noise, and the extension effect is not significantly improved. Conclusions: The iterative regularization algorithm based on the selected strategy in this article has a smoother and smaller error extension solution compared to the traditional Tikhonov regularization method, which shows reliability and practicality.