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arXiv · 2405.07139

Reduced Krylov Basis Methods for Parametric Partial Differential Equations

Yuwen Li·Ludmil T. Zikatanov·Cheng Zuo
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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.

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Yuwen Li, Ludmil T. Zikatanov, Cheng Zuo. 2024-05-12. Reduced Krylov Basis Methods for Parametric Partial Differential Equations. https://doi.org/10.1137/24m1661236

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