{
  "id": "2405.07139",
  "version": "v1",
  "published": "2024-05-12T02:29:03.000Z",
  "updated": "2024-05-12T02:29:03.000Z",
  "title": "Reduced Krylov Basis Methods for Parametric Partial Differential Equations",
  "authors": [
    "Yuwen Li",
    "Ludmil T. Zikatanov",
    "Cheng Zuo"
  ],
  "comment": "23 pages, 6 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "This work is on a user-friendly reduced basis method for solving a family of parametric PDEs by preconditioned Krylov subspace methods including the conjugate gradient method, generalized minimum residual method, and bi-conjugate gradient method. The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. As shown in the theory and experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale systems in the reduced basis subspace. This reduces the computational cost dramatically because (1) to construct the reduced basis vectors, we only solve one large-scale problem in the high-fidelity level; and (2) the family of large-scale problems restricted to the reduced basis subspace have much smaller sizes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-12T02:29:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parametric partial differential equations",
      "reduced krylov basis methods",
      "reduced basis subspace",
      "preconditioned krylov subspace method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}