{
  "id": "2401.00228",
  "version": "v1",
  "published": "2023-12-30T13:31:33.000Z",
  "updated": "2023-12-30T13:31:33.000Z",
  "title": "Parallel-in-time Multilevel Krylov Methods: A Prototype",
  "authors": [
    "Yogi A. Erlangga"
  ],
  "comment": "23 pages",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "This paper presents a parallel-in-time multilevel iterative method for solving differential algebraic equation, arising from a discretization of linear time-dependent partial differential equation. The core of the method is the multilevel Krylov method, introduced by Erlangga and Nabben~{\\it [SIAM J. Sci. Comput., 30(2008), pp. 1572--1595]}. In the method, special time restriction and interpolation operators are proposed to coarsen the time grid and to map functions between fine and coarse time grids. The resulting Galerkin coarse-grid system can be interpreted as time integration of an equivalent differential algebraic equation associated with a larger time step and a modified $\\theta$-scheme. A perturbed coarse time-grid matrix is used on the coarsest level to decouple the coarsest-level system, allowing full parallelization of the method. Within this framework, spatial coarsening can be included in a natural way, reducing further the size of the coarsest grid problem to solve. Numerical results are presented for the 1- and 2-dimensional heat equation using {\\it simulated} parallel implementation, suggesting the potential computational speed-up of up to 9 relative to the single-processor implementation and the speed-up of about 3 compared to the sequential $\\theta$-scheme.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-30T13:31:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "65F10",
      "65M55",
      "65Y05",
      "68W10"
    ],
    "keywords": [
      "parallel-in-time multilevel krylov methods",
      "differential algebraic equation",
      "linear time-dependent partial differential equation",
      "time grid"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}