On the order optimality of the regularization via inexact Newton iterations

TL;DR

This paper analyzes inexact Newton regularization methods for nonlinear ill-posed inverse problems, demonstrating they can achieve order optimal convergence rates, especially when using various inner schemes like Landweber, Tikhonov, and others.

Contribution

It establishes the order optimality of inexact Newton methods with different inner schemes in Hilbert scales, improving upon previous suboptimal results.

Findings

Achieves order optimal convergence rates under certain conditions.

Extends results to various inner schemes including Landweber and Tikhonov.

Provides a more general framework in Hilbert scales.

Abstract

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.