{
  "id": "2405.15573",
  "version": "v1",
  "published": "2024-05-24T14:03:42.000Z",
  "updated": "2024-05-24T14:03:42.000Z",
  "title": "Uniform $\\mathcal{H}$-matrix Compression with Applications to Boundary Integral Equations",
  "authors": [
    "Kobe Bruyninckx",
    "Daan Huybrechs",
    "Karl Meerbergen"
  ],
  "categories": [
    "math.NA",
    "cs.MS",
    "cs.NA"
  ],
  "abstract": "Boundary integral equation formulations of elliptic partial differential equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\\mathcal{O}(N\\log N)$ complexity for storage and multiplication by a vector. This is achieved purely algebraically, based on low-rank approximations of subblocks, and hence the format is also applicable to a wider range of problems. The $\\mathcal{H}^2$-matrix format improves the complexity to $\\mathcal{O}(N)$ by introducing a recursive structure onto subblocks on multiple levels. However, in practice this comes with a large proportionality constant, making the $\\mathcal{H}^2$-matrix format advantageous mostly for large problems. In this paper we investigate the usefulness of a matrix format that lies in between these two: Uniform $\\mathcal{H}$-matrices. An algebraic compression algorithm is introduced to transform a regular $\\mathcal{H}$-matrix into a uniform $\\mathcal{H}$-matrix, which maintains the asymptotic complexity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-24T14:03:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "matrix format",
      "matrix compression",
      "elliptic partial differential equations",
      "boundary integral equation formulations",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}