{
  "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 $y^δ$ with unknown noise norm $δ:=\\|y-y^δ\\|$ 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 $δ$. 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 $α_{k_*}^2\\ge c\\,δ/\\|x_0-x^\\dg\\|$. If moreover $x_0-x^\\dg\\in\\R(F'(x^\\dg))$ and $k_*\\ge1$, we prove $\\|x_{k_*}^δ-x^\\dg\\|=O(\\sqrt{δ_*}+\\sqrtδ)$ with the computable $δ_*=\\|F(x_{k_*}^δ)-y^δ\\|$ satisfying $κδ\\leδ_*\\le C\\sqrtδ$, yielding the baseline rate $O(δ^{1/4})$ along admissible growing-horizon families, and $O(\\sqrtδ)$ whenever $δ_*=O(δ)$; no range-invariance or tangential cone conditions are imposed. Rates under Hölder source conditions with $p\\ge1$ (including saturation) follow; the case $p<1$, the removal of the self-adjointness assumption, and a full oracle inequality are identified as open problems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26966",
  "author_total": 2,
  "authors": [
    {
      "name": "Ankit singh"
    },
    {
      "name": "Hemant Sharma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9b26cvvrrbe6c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26966",
  "fulltext_url": null,
  "id": "9b26cvvrrbe6c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26966",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2610|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7223118526",
  "pdf_url": "https://arxiv.org/pdf/2610.26966",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9b26cvvrrbe6c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.26966",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26966"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A noise-level-free parameter-choice rule for iterated Lavrentiev regularization of nonlinear potential monotone ill-posed problems",
  "title_text": "A noise-level-free parameter-choice rule for iterated Lavrentiev regularization of nonlinear potential monotone ill-posed problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/9b26cvvrrbe6c",
  "venue": "arXiv",
  "versions": []
}