{
  "abstract": "We propose a localized reduced basis method for parametrized parabolic partial differential equations based on an adaptive partitioning of the parameter domain using hierarchical binary trees. A POD-Greedy procedure is employed to construct reduced spaces associated with each parameter subdomain. A certified \\emph{a posteriori} error estimator is derived from an inf-sup stability analysis and shown to be equivalent to the underlying approximation error. This equivalence, together with suitable regularity assumptions on the parameter-to-solution map, enables a rigorous convergence analysis and is used to derive a strategy for selecting the number of POD modes to retain in each local reduced space. Numerical experiments for a parametrized convection--diffusion problem demonstrate the effectiveness of the proposed approach and confirm the predicted convergence behavior.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36141",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohamed Barakat"
    },
    {
      "name": "Diane Guignard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3nty1hpt08q90.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36141",
  "fulltext_url": null,
  "id": "3nty1hpt08q90",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36141",
  "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",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7214894524",
  "pdf_url": "https://arxiv.org/pdf/2609.36141",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3nty1hpt08q90.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36141",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36141"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs",
  "title_text": "A Tree-Based Localized Reduced Basis Method for Parametrized Parabolic PDEs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/3nty1hpt08q90",
  "venue": "arXiv",
  "versions": []
}