{
  "abstract": "Gaussian random fields are a versatile tool used in the fields of stochastic PDEs, uncertainty quantification, and geostatistical simulation. One way to obtain them is to use a truncated Karhunen-Loève expansion (KLE). Computing the expansion requires the leading eigenpairs of an $N \\times N$ covariance matrix, where $N$ is the total number of grid cells. These are usually computed with a Krylov eigensolver, which relies on the covariance operator only within matrix-vector products. Stored densely, the matrix takes $\\mathcal{O}(N^{2})$ memory and each product $\\mathcal{O}(N^{2})$ time. For a stationary kernel on an equispaced grid, the covariance matrix becomes (block-) Toeplitz and the product evaluates in $\\mathcal{O}(N\\log N)$ time using FFT-based circulant embedding, without the need to assemble the dense matrix. In a matched single-threaded comparison, the median speedup of the eigensolve grows from $18 \\times$ at $N = 4096$ to $183 \\times$ at $N = 2^{15}$. This makes it possible to compute discretized fields that would otherwise be infeasible to compute in the standard formulation. We show that the same construction carries over to non-separable kernels as well as $d$ dimensions, using block-Toeplitz matrices. We extend it to piecewise-constant fields on arbitrary domains given as subsets of a tensor grid. Computational savings grow with problem size, and storage drops from $\\mathcal{O}(N^{2})$ to $\\mathcal{O}(2^{d}N)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26459",
  "author_total": 2,
  "authors": [
    {
      "name": "Nils Wildt"
    },
    {
      "name": "Wolfgang Nowak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ztffw9mf7mp2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26459",
  "fulltext_url": null,
  "id": "3ztffw9mf7mp2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26459",
  "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.NA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7214088479",
  "pdf_url": "https://arxiv.org/pdf/2609.26459",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ztffw9mf7mp2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.26459",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26459"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Fast Karhunen-Loève Expansions via FFT-Accelerated Toeplitz Operators",
  "title_text": "Fast Karhunen-Loève Expansions via FFT-Accelerated Toeplitz Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/3ztffw9mf7mp2",
  "venue": "arXiv",
  "versions": []
}