On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

TL;DR

This paper analyzes the use of Newton type methods with the discrepancy principle for stable solutions to nonlinear ill-posed inverse problems, proving convergence and order optimality under certain conditions.

Contribution

It introduces convergence and order optimality results for Newton type methods with discrepancy principle under Lipschitz conditions on the derivative.

Findings

Convergence of the method as noise level decreases.

Order optimal convergence rates established.

Applicability under Lipschitz continuity of the derivative.

Abstract

We consider the computation of stable approximations to the exact solution $x^\dag$ of nonlinear ill-posed inverse problems $F(x)=y$ with nonlinear operators $F:X\to Y$ between two Hilbert spaces $X$ and $Y$ by the Newton type methods $$ x_{k+1}^\delta=x_0-g_{\alpha_k} (F'(x_k^\delta)^*F'(x_k^\delta)) F'(x_k^\delta)^* (F(x_k^\delta)-y^\delta-F'(x_k^\delta)(x_k^\delta-x_0)) $$ in the case that only available data is a noise $y^\delta$ of $y$ satisfying $\|y^\delta-y\|\le \delta$ with a given small noise level $\delta>0$. We terminate the iteration by the discrepancy principle in which the stopping index $k_\delta$ is determined as the first integer such that $$ \|F(x_{k_\delta}^\delta)-y^\delta\|\le \tau \delta <\|F(x_k^\delta)-y^\delta\|, \qquad 0\le k<k_\delta $$ with a given number $\tau>1$. Under certain conditions on $\{\alpha_k\}$, $\{g_\alpha\}$ and $F$, we prove that $x_{k_\delta}^\delta$ converges to $x^\dag$ as $\delta\to 0$ and establish various order optimal convergence rate results. It is remarkable that we even can show the order optimality under merely the Lipschitz condition on the Fr\'{e}chet derivative $F'$ of $F$ if $x_0-x^\dag$ is smooth enough.