Iterative Lavrentiev regularization method under a heuristic rule for nonlinear ill-posed operator equations

Abstract

In this paper, we consider the iterative Lavrentiev regularization method for obtaining a stable approximate solution for a nonlinear ill-posed operator equation F (x) = y, where F : D(F ) ⊂ X → X is a nonlinear monotone operator on the Hilbert spaces X . In order to obtain a stable approximate solution using iterative regularization methods, it is important to use a suitable stopping rule to terminate the iterations at the appropriate step. Recently, Qinian Jin and Wei Wang (2018) have proposed a heuristic rule to stop the iterations for the iteratively regularized Gauss-Newton method. The advantage of a heuristic rule over the existing a priori and a posteriori rules is that it does not require accurate information on the noise level, which may not be available or reliable in practical applications. In this paper, we propose a heuristic stopping rule for an iterated Lavrentiev regularization method. We derive error estimates under suitable nonlinearity conditions on the operator F .