{
  "abstract": "Approximating linear operators on high-dimensional spaces using only forward operator-vector product queries is a common but challenging task in scientific computing and machine learning arising when access to the adjoint action is experimentally unavailable or difficult to construct computationally. Existing query-efficient methods with polylogarithmic query complexity in the error tolerance rely on self-adjointness or access to the adjoint operator, while existing adjoint-free methods based on global low-order regularity have polynomial query complexity. We show that a priori local row-space information can be exploited for query-efficient, adjoint-free operator approximation, and that this information can be supplied by local high-order regularity estimates. Our approach first extends the fixed-sparsity matrix approximation algorithm of (Amsel et al., SIMAX, 2026) to approximating a matrix by one whose rows lie in prescribed subspaces, providing a general template for exploiting local row-space structure from forward queries alone. We then apply this framework to Hilbert-Schmidt integral operators with asymptotically smooth kernels. We establish asymptotic smoothness for Green's functions of canonical second-order elliptic PDEs in $d\\leq 3$ dimensions including Poisson's equation in balls and constant coefficient equations on tori. In $d$ spatial dimensions, our algorithm uses $\\mathcal{O}\\big(\\log(\\varepsilon^{-1})\\log^{d}(\\varepsilon^{-1}δ^{-1/2})\\big)$ non-adaptive randomized queries to produce an $\\varepsilon$-accurate approximation in Hilbert-Schmidt norm with probability $1-δ$. This implies the error decreases as $\\mathcal{O}\\big( \\exp(- c q^{1/(d+1)}) \\big)$ in the query count $q$, which we observe in numerical experiments using tens to hundreds of queries to approximate solution operators of one-dimensional variable-coefficient second-order elliptic equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07513",
  "author_total": 2,
  "authors": [
    {
      "name": "Esther Gallmeier"
    },
    {
      "name": "Samuel E. Otto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ahbcg23hk10hy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07513",
  "fulltext_url": null,
  "id": "ahbcg23hk10hy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07513",
  "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.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7220899050",
  "pdf_url": "https://arxiv.org/pdf/2610.07513",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ahbcg23hk10hy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07513",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07513"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Efficient Approximation of Structured Linear Operators using Randomized One-Sided Queries",
  "title_text": "Efficient Approximation of Structured Linear Operators using Randomized One-Sided Queries",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/ahbcg23hk10hy",
  "venue": "arXiv",
  "versions": []
}