{
  "abstract": "We study estimation of linear functionals $\\int fψ$ of the conductivity $f$ in the Darcy problem $\\nabla\\cdot(f\\nabla u)=g$ from observations of $u$ in Gaussian white noise of level $N^{-1/2}$. When the potential has a single nondegenerate minimum, we determine the minimax rates over Sobolev balls in the linearised experiment. Unless a compatibility condition along the gradient flow holds, the rate is governed by an effective dimension $T$ of the Hessian at the minimum, not by the ambient dimension: it is $N^{-(2s+T)/(4s+8)}$ for $T<4$, has an elbow at $T=4$ with an extra logarithmic factor, and is parametric for $T>4$. In the nonlinear experiment we show that these lower bounds persist when compatibility fails at the reference and $T\\le4$, and we prove the minimax rate on classes with isotropic critical Hessians, and the parametric rate near references with $T>4$. In dimension at most four, no nonzero linear functional is uniformly $\\sqrt N$-estimable on a Sobolev neighbourhood of the constant conductivity, even when it is regular there.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01290",
  "author_total": 3,
  "authors": [
    {
      "name": "Yi Liu"
    },
    {
      "name": "Xiaotian Chang"
    },
    {
      "name": "Qirui Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dt2d7k28n0rpp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01290",
  "fulltext_url": null,
  "id": "dt2d7k28n0rpp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01290",
  "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",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01290",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dt2d7k28n0rpp.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01290",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01290"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Minimax estimation of linear functionals in the Darcy inverse problem",
  "title_text": "Minimax estimation of linear functionals in the Darcy inverse problem",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/dt2d7k28n0rpp",
  "venue": "arXiv",
  "versions": []
}