On the order optimality of the regularization via inexact Newton iterations
For researchers working on regularization methods for inverse problems, this provides a theoretical improvement in convergence guarantees.
The paper improves the convergence rate analysis of inexact Newton regularization methods for nonlinear ill-posed inverse problems, achieving order optimal convergence rates under certain conditions, which improves upon previous suboptimal results.
Inexact Newton regularization methods have been proposed by Hanke and Rieder for solving nonlinear ill-posed inverse problems. Every such a method consists of two components: an outer Newton iteration and an inner scheme providing increments by regularizing local linearized equations. The method is terminated by a discrepancy principle. In this paper we consider the inexact Newton regularization methods with the inner scheme defined by Landweber iteration, the implicit iteration, the asymptotic regularization and Tikhonov regularization. Under certain conditions we obtain the order optimal convergence rate result which improves the suboptimal one of Rieder. We in fact obtain a more general order optimality result by considering these inexact Newton methods in Hilbert scales.