{
  "abstract": "This paper introduces a novel method for approximating the inverse of discrete Fourier integral operators (FIOs). Given an $N \\times N$ matrix representation $K$ of an FIO, the proposed algorithm consists of two stages. In the offline stage, we first construct a butterfly factorization (BF) $\\tilde{K}$ of $K$, which enables fast forward matrix-vector multiplication. We then construct a hierarchically semiseparable (HSS) approximation $\\tilde{G} \\approx G$, where $G = K^{*} K$, using fast applications of $\\tilde{K}$ and $\\tilde{K}^{*}$ to random matrices. Finally, we apply the ULV factorization to the HSS matrix $\\tilde{G}$ to obtain an approximation $\\tilde{F} \\approx G^{-1}$. Combining these approximations yields an approximate inverse $K^{-1} \\approx \\tilde{F} \\tilde{K}^{*}$. The offline stage has complexity $O(N \\log^{2} N)$ for 1D problems and $O(N^{1.5} \\log N)$ for 2D problems. In the online stage, the proposed method approximates $K^{-1} u$ for a given input vector $u$ with complexity $O(N \\log N)$ for both 1D and 2D problems. The proposed method can be used either as a direct solver or as a preconditioner for iterative methods. Numerical results for 1D and 2D FIOs demonstrate the effectiveness of the proposed method.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11520",
  "author_total": 2,
  "authors": [
    {
      "name": "Yingzhou Li"
    },
    {
      "name": "Jingyu Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bv2ww3ca5prt8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11520",
  "fulltext_url": null,
  "id": "bv2ww3ca5prt8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11520",
  "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": "W7212296957",
  "pdf_url": "https://arxiv.org/pdf/2609.11520",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bv2ww3ca5prt8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11520",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11520"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Approximate Inversion of Discrete Fourier Integral Operators via Hierarchically Semiseparable Matrices",
  "title_text": "Approximate Inversion of Discrete Fourier Integral Operators via Hierarchically Semiseparable Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/bv2ww3ca5prt8",
  "venue": "arXiv",
  "versions": []
}