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.

翻译：本文提出一种用户友好的简化基方法，用于通过预条件Krylov子空间方法（包括共轭梯度法、广义最小残差法和双共轭梯度法）求解参数化偏微分方程族。所提方法利用针对单一参数实例的高保真离散化预条件Krylov子空间迭代，生成简化基子空间的正交基向量，随后在低维Krylov子空间中近似求解大规模参数依赖型离散问题。理论和实验表明，仅需少量Krylov子空间迭代即可同时生成高保真大规模系统族在简化基子空间中的近似解。该方法显著降低计算成本，原因在于：(1) 构建简化基向量时仅需求解单个高保真大规模问题；(2) 限制在简化基子空间中的大规模问题族规模大幅减小。