{
  "id": "2502.03881",
  "version": "v1",
  "published": "2025-02-06T08:54:52.000Z",
  "updated": "2025-02-06T08:54:52.000Z",
  "title": "Numerical Aspects of the Tensor Product Multilevel Method for High-dimensional, Kernel-based Reconstruction on Sparse Grids",
  "authors": [
    "Markus Büttner",
    "Rüdiger Kempf",
    "Holger Wendland"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "This paper investigates the approximation of functions with finite smoothness defined on domains with a Cartesian product structure. The recently proposed tensor product multilevel method (TPML) combines Smolyak's sparse grid method with a kernel-based residual correction technique. The contributions of this paper are twofold. First, we present two improvements on the TPML that reduce the computational cost of point evaluations compared to a naive implementation. Second, we provide numerical examples that demonstrate the effectiveness and innovation of the TPML.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-06T08:54:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65D12",
      "65D15",
      "65D40"
    ],
    "keywords": [
      "tensor product multilevel method",
      "numerical aspects",
      "kernel-based reconstruction",
      "high-dimensional",
      "smolyaks sparse grid method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}