{
  "abstract": "The efficacy of hierarchical B-spline bases in solving high-dimensional partial differential equations has been demonstrated in several studies. However, the process of assembling isogeometric Galerkin matrices remains computationally challenging with standard methods, particularly as the dimension and the spline degree of the basis increase. In this work, we demonstrate the possibility of exploiting the structure of the system matrices to derive a fast assembly method. First, we introduce the Hadamard format, a compact representation of the Galerkin matrices based on low-rank approximation of the integral kernels and univariate integration. Second, we present an efficient algorithm for evaluating the Hadamard format and constructing the full system matrix. The algorithm relies on a dedicated data structure, denoted retraction tree, for representing the index set of the active basis functions, and exploits the restricted deep-first search to quickly compute the sparsity pattern of the full matrix. The complexity analysis shows that the proposed method achieves quasi-optimal complexity with respect to the matrix size. Our experiments, conducted using the C++ library G+Smo, validate the derived complexity and indicate that the proposed method significantly enhances the assembly process.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "name": "Carlotta Giannelli"
    },
    {
      "name": "Angelos Mantzaflaris"
    },
    {
      "name": "Mattia Matucci"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3qmnffv2ymsqv.bib",
  "cited_by": 0,
  "doi": "10.1016/j.cma.2026.119285",
  "fulltext_url": null,
  "id": "3qmnffv2ymsqv",
  "issns": [
    "0045-7825",
    "1879-2138"
  ],
  "landing_url": "https://hal.science/hal-05805454v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
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      "openalex",
      "2206|Advanced Numerical Analysis Techniques"
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  "openalex": "W7202171956",
  "pdf_url": "https://hal.science/hal-05805454/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": "Elsevier",
  "ris": "https://research.draconitic.com/paper/3qmnffv2ymsqv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:59Z",
      "native_id": "hal-05805454",
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      "url": "https://hal.science/hal-05805454v1"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-na",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "On the Hadamard form of hierarchical spline Galerkin matrices and their efficient assembly in isogeometric analysis",
  "title_text": "On the Hadamard form of hierarchical spline Galerkin matrices and their efficient assembly in isogeometric analysis",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:59Z",
  "url": "https://research.draconitic.com/paper/3qmnffv2ymsqv",
  "venue": "Computer Methods in Applied Mechanics and Engineering",
  "versions": []
}