Stable iteratively regularized gradient method for nonlinear irregular equations under large noise

Abstract

We consider an iteratively regularized version of the method of gradient descent for solving nonlinear irregular equations F(x) = 0 in a Hilbert space. When studying regularization methods for such equations with noisy operators F, traditional conditions on available approximations amount to error estimates of the form , for x from a neighbourhood of a solution. Convergence of the methods is usually established on the assumption that the error level δ → 0, i.e. that noisy elements strongly converge to the exact value F(x). In this paper we analyse approximating properties of the regularized gradient method assuming that may converge to F(x) only weakly. We suggest an a priori stopping rule for the gradient iteration and give error estimates for obtained approximate solutions in terms of levels of strong and weak perturbations of the original operator. The main theorem generalizes recent results of Bakushinsky and Kokurin (2004 Iterative Methods for Approximate Solution of Inverse Problems (Dordrecht: Springer)) on the stopping of regularized gradient method under strong perturbations of F.