Heuristic rule for inexact Newton-Landweber iteration with convex penalty terms of nonlinear: ill-posed problems

Abstract

Inexact Newton regularization methods are a family of prominent regularization methods for solving nonlinear ill-posed problems, which consist of an outer Newton iteration and an inner scheme providing increments by applying the regularization technique to the local linearized equations. In this paper, we propose a heuristic stopping rule for the inexact Newton regularization method, where the inner scheme is defined by Landweber iteration and the strong convex function is incorporated as the penalty term. In contrast to a prior and a posteriori stopping rules, our heuristic rule is purely data driven and does not require the information on noise level, which renders the method feasible when the noise level is unknown or unreliable. Under certain assumptions on the random noise, we establish a new convergence analysis for the inexact Newton-Landweber method under the heuristic rule. The numerical simulations are provided to demonstrate the performance of our heuristic rule.