Reduced Krylov Basis Methods for Parametric Partial Differential Equations
<|reference_start|>Reduced Krylov Basis Methods for Parametric Partial Differential Equations: 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.<|reference_end|>
@article{li2024reduced,
    title={Reduced Krylov Basis Methods for Parametric Partial Differential
  Equations},
    author={Yuwen Li, Ludmil T. Zikatanov, Cheng Zuo},
    journal={arXiv preprint arXiv:2405.07139},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.07139},
    primaryClass={math.NA cs.NA}
}