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Fast Karhunen-Loève Expansions via FFT-Accelerated Toeplitz Operators — arXiv:2609.26459

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.26459
License (as archived): CC0-1.0
Archived: 2026-10-08T15:55:32.456Z
SHA-256 of the archived text: c97d0a9ddf3748adb3a3a18a18656abe396277fa7eeaad43eafb6e7b47519020
Stable URL of this page: https://metraipolis.com/documento/dc_2cefaa0bbbdf

How to cite

Fast Karhunen-Loève Expansions via FFT-Accelerated Toeplitz Operators — arXiv:2609.26459 — arXiv (en), https://arxiv.org/abs/2609.26459 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_2cefaa0bbbdf (Metraipolis archive dc_2cefaa0bbbdf, sha256 c97d0a9ddf3748adb3a3a18a18656abe396277fa7eeaad43eafb6e7b47519020)

BibTeX

@misc{dc2cefaa0bbbdf,
  title = {Fast Karhunen-Loève Expansions via FFT-Accelerated Toeplitz Operators — arXiv:2609.26459},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.26459},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_2cefaa0bbbdf; sha256 c97d0a9ddf3748adb3a3a18a18656abe396277fa7eeaad43eafb6e7b47519020},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.26459. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\mathcal{O}(2^{d}N)$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-23 in math.NA).
Categorie arXiv: Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2609.26459 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T15:15:47.519Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-23?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).