A general convergence analysis on inexact Newton method for nonlinear inverse problems

TL;DR

This paper provides a comprehensive convergence analysis of inexact Newton methods for solving nonlinear ill-posed inverse problems, establishing conditions for convergence and optimal rates under noise and source conditions.

Contribution

It offers the first general convergence proof for a broad class of spectral filter functions in inexact Newton methods for inverse problems.

Findings

Convergence of the method as noise level approaches zero.

Order optimal convergence rates under Hölder source conditions.

Numerical experiments confirming theoretical results.

Abstract

We consider the inexact Newton methods $$ x_{n+1}^\d=x_n^\d-g_{\a_n}(F'(x_n^\d)^* F'(x_n^\d)) F'(x_n^\d)^* (F(x_n^\d)-y^\d) $$ for solving nonlinear ill-posed inverse problems $F(x)=y$ using the only available noise data $y^\d$ satisfying $\|y^\d-y\|\le \d$ with a given small noise level $\d>0$. We terminate the iteration by the discrepancy principle $$ \|F(x_{n_\d}^\d)-y^\d\|\le \tau \d<\|F(x_n^\d)-y^\d\|, \qquad 0\le n<n_\d $$ with a given number $\tau>1$. Under certain conditions on $\{\a_n\}$ and $F$, we prove for a large class of spectral filter functions $\{g_\a\}$ the convergence of $x_{n_\d}^\d$ to a true solution as $\d\rightarrow 0$. Moreover, we derive the order optimal rates of convergence when certain H\"{o}lder source conditions hold. Numerical examples are given to test the theoretical results.