{
  "abstract": "A high-order hierarchical direct solver for variable-media scattering in two dimensions is presented. Standard discretizations for these problems produce dense linear systems that are commonly solved iteratively, requiring a new global solve for each incident field. The method proposed here instead constructs a reusable scattering operator for the inhomogeneous medium, based on a hierarchical subdivision of the computational domain. First, high-accuracy leaf scattering operators are built from discretized local integral equations. Up the hierarchy, neighboring boxes are merged to build the scattering operators for successively larger boxes. A key feature of the method is that each scattering operator is represented as a map of incoming traces to outgoing traces. Consequently, volume unknowns appear only in the leaf computations; all higher levels of the hierarchy operate entirely on boundary traces. The hierarchical solver is presented first at the continuum level and discretized only afterward using high-order local volume quadrature, polynomial trace representations, and boundary integral techniques. For a discretization involving $N$ degrees of freedom, the method requires $O(N^{3/2})$ work and $O(N\\log N)$ storage to build all scattering operators. Once these are available, any incoming field can be processed at cost $O(N)$ if only the outgoing field is required, and cost $O(N\\log N)$ if the full field is required. Numerical experiments demonstrate high-order accuracy on a range of challenging scattering problems including cavities, back-scattering, limited regularity, and high wavenumber, that pose significant challenges to iterative methods.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00494",
  "author_total": 4,
  "authors": [
    {
      "name": "Yunhui Cai"
    },
    {
      "name": "Simon Dirckx"
    },
    {
      "name": "Joar Bagge"
    },
    {
      "name": "Per-Gunnar Martinsson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6070vybeys4py.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00494",
  "fulltext_url": null,
  "id": "6070vybeys4py",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00494",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00494",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6070vybeys4py.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:59Z",
      "native_id": "2611.00494",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00494"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Hierarchical direct solver for Lippmann-Schwinger equations based on boundary traces",
  "title_text": "A Hierarchical direct solver for Lippmann-Schwinger equations based on boundary traces",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:05Z",
  "url": "https://research.draconitic.com/paper/6070vybeys4py",
  "venue": "arXiv",
  "versions": []
}