Methods for solving nonlinear ill-posed problems based on the Tikhonov-Lavrentiev regularization and iterative approximation

A problem of constructing a stable approximate solution for a nonlinear irregular operator equation is investigated. For approximating solutions of the equations regularized by the Tikhonov-Lavrentiev methods, the Levenberg-Marquardt and Newton type processes are used, and the linear convergence rate and the Fej ́er property are proved. An asymptotic stopping rule of iterations is formulated that guarantees the regularizing properties of iterations and error estimate. Analogous questions are briefly discussed for the gradient methods.