{
  "abstract": "Learned multigrid components are often evaluated at a fixed iteration budget, but such comparisons do not distinguish local approximation gains from practical solver efficiency. We study this distinction for high-contrast elliptic problems on unstructured finite element meshes. A learned V-cycle uses operator-dependent transfers, Galerkin coarse operators, near-kernel deflation and an energy-norm safeguard; only its smoothers are trained. The architecture represents a specified class of fixed polynomial smoothers and is positively homogeneous in the load. On fresh conductive-channel problems, one learned pass reduces the energy error by factors of 5.7-6.1 relative to the same cycle with four-step Jacobi-preconditioned conjugate-gradient smoothing; the corresponding factors are 1.38-1.45 for insulating barriers. In a separate fixed-input test on channel problems, the fine-level learned smoother attains a smaller energy-norm error than the exact minimum over a five-dimensional Krylov space of its own input on 97-100 of 100 fresh inputs for each position and training initialization; this does not establish superiority over all polynomial-smoothed cycles. When the classical smoothing budget is increased, the paired geometric-mean accuracy crosses that of the learned cycle between 8 and 14 steps per smoothing call on channels and between 5 and 8 on barriers. The measured CPU implementation offers no computational advantage over the tested classical solvers, and load changes and floating islands further restrict its accuracy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30692",
  "author_total": 1,
  "authors": [
    {
      "name": "Kumar Kaushik Ranjan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/24rxn14vq0vkt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30692",
  "fulltext_url": null,
  "id": "24rxn14vq0vkt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30692",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7223316845",
  "pdf_url": "https://arxiv.org/pdf/2610.30692",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/24rxn14vq0vkt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30692",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30692"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Learned smoothing in operator-adapted multigrid: controlled accuracy gains and computational limits",
  "title_text": "Learned smoothing in operator-adapted multigrid: controlled accuracy gains and computational limits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/24rxn14vq0vkt",
  "venue": "arXiv",
  "versions": []
}