2024/05/12 by Yuwen Li, Ludmil Zikatanov, Li, Yuwen +3
Computer Science · Mathematics · #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2405.07139
openalex publication_date 2024/05/12 · openalex created_date 2024/05/15 · openalex updated_date 2026/07/28
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.