A generalized adaptive Levenberg–Marquardt method for solving nonlinear ill-posed problems

Abstract

Abstract We investigate the method of regularization for the stable approximate solution to nonlinear ill-posed problems whose forward operators may not be Gâteaux differentiable. The method is designed by combining the classical Levenberg–Marquardt method with the two-point gradient iteration, and the adaptive stepsize which is related to the Tikhonov regularization parameter and the structure of the forward operator. In order to further enhance the acceleration effect, we employ a modified discrete backtracking search algorithm to determine the combination parameters involved. With the help of the concept of asymptotic stability and a generalized tangential cone condition, the convergence analysis of the proposed method is studied. Moreover, several numerical experiments are performed to illustrate the effectiveness and acceleration effect.