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 xdg (a locally \emph{potential} operator), and only noisy data yδ with unknown noise norm δ:=|y−yδ| is available. For the nonstationary iterated Lavrentiev regularization xk+1δ=xkδ−(F'(xkδ)+αkI)−1(F(xkδ)−yδ+αk(xkδ−x0)), all classical stopping rules require δ. We analyze a \emph{noise-level-free} parameter-choice rule of retrospective type, selecting a minimizer k* of θ(k,yδ)=|F(xkδ)−yδ|2/αk2 over a finite computed window. Under a standard noise condition (necessary in view of the Bakushinskii veto) the rule is well defined and αk*2≥ c δ/|x0−xdg|. If moreover x0−xdg∈R(F'(xdg)) and k*≥1, we prove |xk*δ−xdg|=O(√(δ*)+√δ) with the computable δ*=|F(xk*δ)−yδ| satisfying κδ≤δ*≤ C√δ, yielding the baseline rate O(δ1/4) along admissible growing-horizon families, and O(√δ) whenever δ*=O(δ); no range-invariance or tangential cone conditions are imposed. Rates under Hölder source conditions with p≥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.
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BibTeX
@misc{singh2026noiselevelfree,
title = {{A noise-level-free parameter-choice rule for iterated Lavrentiev regularization of nonlinear potential monotone ill-posed problems}},
author = {Ankit singh and Hemant Sharma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26966},
eprint = {2610.26966},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26966},
}
RIS
TY - UNPB
TI - A noise-level-free parameter-choice rule for iterated Lavrentiev regularization of nonlinear potential monotone ill-posed problems
AU - Ankit singh
AU - Hemant Sharma
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.26966
UR - https://arxiv.org/abs/2610.26966
AB - 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^δ with unknown noise norm δ:=|y−y^δ| is available. For the nonstationary iterated Lavrentiev regularization x_k+1^δ=x_k^δ−(F'(x_k^δ)+α_kI)^−1(F(x_k^δ)−y^δ+α_k(x_k^δ−x_0)), all classical stopping rules require δ. We analyze a \emph{noise-level-free} parameter-choice rule of retrospective type, selecting a minimizer k_* of θ(k,y^δ)=|F(x_k^δ)−y^δ|^2/α_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≥ c δ/|x_0−x^dg|. If moreover x_0−x^dg∈R(F'(x^dg)) and k_*≥1, we prove |x_k_*^δ−x^dg|=O(√(δ_*)+√δ) with the computable δ_*=|F(x_k_*^δ)−y^δ| satisfying κδ≤δ_*≤ C√δ, yielding the baseline rate O(δ^1/4) along admissible growing-horizon families, and O(√δ) whenever δ_*=O(δ); no range-invariance or tangential cone conditions are imposed. Rates under Hölder source conditions with p≥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.
ER -