Two-stage method for solving systems of nonlinear equations and its applications to the inverse atmospheric sounding problem

For an overdetermined system of nonlinear equations, a two-stage method is suggested for constructing an error-stable approximate solution. The first stage consists in constructing a regularized set of approximate solutions for finding normal quasi-solutions of the original system. At the second stage, the regularized quasi-solutions are approximated using an iterative process based on square approximation of the Tikhonov functional and a prox-method. For this Newton-type method, a convergence theorem is proved and the Fejér property of the iterations is established. Additionally, the two-stage method is applied to the inverse problem of reconstructing heavy water (HDO) in the atmosphere from infrared spectra of solar light transmission.