{
  "abstract": "This article introduces four parallel numerical techniques for computing entries of the inverse of large sparse symmetric systems, all grounded in incomplete $LDL^T$ (ILDL) factorizations: (1) the selected inversion method (SelInv), which applies the $LDL^T$ factorization to recover entries of the matrix inverse within the sparsity pattern of the computed factors; (2) an approximate inversion method based on a truncated Neumann series expansion applied to the inverse of the L factor (NInv), providing an alternative at the cost of reduced accuracy; (3) a Mix approximation that merges the best of both SelInv and NInv; and (4) Mix-SPAI, which applies sparse approximate inverse (SPAI) refinement on the output of the Mix method to improve entry-level accuracy. To further improve accuracy while maintaining a stable sparsity pattern, we additionally employ a low-rank correction based on eigenvector updates, providing an alternative to tightening the drop tolerance. We report the performance of the proposed numerical techniques on a comprehensive collection of sparse matrices from scientific and industrial applications.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11363",
  "author_total": 3,
  "authors": [
    {
      "name": "Tahamina Akter"
    },
    {
      "name": "Matthias Bollhöfer"
    },
    {
      "name": "Olaf Schenk"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cpnj67zy90p59.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11363",
  "fulltext_url": null,
  "id": "cpnj67zy90p59",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11363",
  "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": "W7212259871",
  "pdf_url": "https://arxiv.org/pdf/2609.11363",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cpnj67zy90p59.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11363",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11363"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Scalable Approximate Selected Inversion Based on Single-Level Incomplete $LDL^T$ Factorization and Spectral Corrections for Large Sparse Systems",
  "title_text": "Scalable Approximate Selected Inversion Based on Single-Level Incomplete LDL^T Factorization and Spectral Corrections for Large Sparse Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/cpnj67zy90p59",
  "venue": "arXiv",
  "versions": []
}