A noise-level-free parameter-choice rule for iterated Lavrentiev regularization of nonlinear potential monotone ill-posed problems

Abstract

We consider nonlinear ill-posed operator equations $F(x)=y$ in a Hilbert space, where $F$ is monotone with a Lipschitz continuous Fréchet derivative that is self-adjoint near the solution $x^\dg$ (a locally \emph{potential} operator), and only noisy data $y^{\delta }$ with unknown noise norm $\delta :=\Vert y-y^{\delta }\Vert$ is available. For the nonstationary iterated Lavrentiev regularization \[ x_{k+1}^δ=x_k^δ-\bigl(F'(x_k^δ)+α_kI\bigr)^{-1}\bigl(F(x_k^δ)-y^δ+α_k(x_k^δ-x_0)\bigr), \] all classical stopping rules require $\delta$. We analyze a \emph{noise-level-free} parameter-choice rule of retrospective type, selecting a minimizer $k_{\ast }$ of $\theta (k,y^{\delta })=\Vert F(x_k^{\delta })-y^{\delta }{\Vert }^2/{\alpha }_k^2$ over a finite computed window. Under a standard noise condition (necessary in view of the Bakushinskii veto) the rule is well defined and $α_{k_*}^2\ge c\,δ/\|x_0-x^\dg\|$. If moreover $x_0-x^\dg\in\R(F'(x^\dg))$ and $k_{\ast }\ge 1$, we prove $\|x_{k_*}^δ-x^\dg\|=O(\sqrt{δ_*}+\sqrtδ)$ with the computable ${\delta }_{\ast }=\Vert F(x_{k_{\ast }}^{\delta })-y^{\delta }\Vert$ satisfying $\kappa \delta \le {\delta }_{\ast }\le C\sqrt{\delta }$, yielding the baseline rate $O({\delta }^{1/4})$ along admissible growing-horizon families, and $O(\sqrt{\delta })$ whenever ${\delta }_{\ast }=O(\delta )$; no range-invariance or tangential cone conditions are imposed. Rates under Hölder source conditions with $p\ge 1$ (including saturation) follow; the case $p<1$, the removal of the self-adjointness assumption, and a full oracle inequality are identified as open problems.