Levenberg–Marquardt method and its modified versions for solving nonlinear equations with application to the inverse gravimetry problem

The Levenberg–Marquardt method and its modified versions are studied. Under some local conditions on the operator (in a neighborhood of a solution), strong and weak convergence of iterations is established with the solution error monotonically decreasing. The conditions are shown to be true for one class of nonlinear integral equations, in particular, for the structural gravimetry problem. Results of model numerical experiments for the inverse nonlinear gravimetry problem are presented.