{
  "abstract": "In this work, we develop low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations and incorporate them into a basis-update \\& Galerkin mass-lumped finite element method (\\mbox{BUG-MLFEM}). By incorporating mass lumping via Gauss-Lobatto quadrature, the spatial discretization yields a matrix differential equation equipped with diagonal positive-definite mass matrices. We establish the weighted dynamical low-rank approximation (WDLRA) within the Galerkin framework of the MLFEM system, proving the theoretical equivalence between the BUG integrator and MLFEM on low-rank subspaces, which guarantees the discrete well-posedness of the implicit BUG steps. To overcome the severe computational bottleneck of standard $O(m^3+n^3)$ Sylvester solvers, we develop tailored, low-cost non-iterative solvers that exploit the inherent sparse-by-small algebraic structure. Under Dirichlet and Neumann boundary conditions (BCs), the large coefficient matrices are banded, enabling efficient diagonalization of the small $r \\times r$ factor matrix and reducing the per-step computational complexity to $O((m+n)r^2)$ and storage to $O((m+n)r)$. For periodic BCs, we combine this diagonalization strategy with a rank-2 Sherman-Morrison-Woodbury update to handle wrap-around corner entries at negligible additional cost. Numerical experiments demonstrate first-order temporal and $(k+1)$th-order spatial convergence of the proposed BUG-MLFEM in the discrete $L^2$ norm, where $k$ denotes the polynomial degree of the finite element space, together with significant computational speedups over existing low-rank integrators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18419",
  "author_total": 3,
  "authors": [
    {
      "name": "Joy Saha"
    },
    {
      "name": "Jun Yang"
    },
    {
      "name": "Peimeng Yin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6271bq9tc66pa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18419",
  "fulltext_url": null,
  "id": "6271bq9tc66pa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18419",
  "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",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7221793861",
  "pdf_url": "https://arxiv.org/pdf/2610.18419",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6271bq9tc66pa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18419",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18419"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations",
  "title_text": "Low-cost non-iterative Sylvester solvers for dynamical low-rank approximation of reaction-diffusion equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/6271bq9tc66pa",
  "venue": "arXiv",
  "versions": []
}