发表机构
Sichuan University; Nuclear Power Institute of China（四川大学; 中国核动力研究设计院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出加性约化基预条件子，通过修正项正则化奇异近似，集成于FGMRES框架，加速大规模参数化PDE求解，数值实验验证其高效收敛。
AI 中文摘要
我们引入一类加性约化基预条件子，旨在加速由离散化参数化偏微分方程产生的大规模线性系统的迭代求解。主要思想是通过添加简单的修正项来正则化本质上奇异的约化阶近似：要么是缩放的单位修正，要么是基于基本预条件子的投影修正。这在显式且易于检查的条件下产生非奇异预条件子。预条件子的构造和应用通过离线策略集成到FGMRES框架中，该策略在每一步FGMRES迭代中通过本征正交分解动态构建约化基分量。我们建立了预条件子非奇异性的充分条件，并推导了预条件Richardson迭代的误差界。提供了对流扩散、各向异性涡旋、Stokes和Helmholtz问题的数值结果，以验证所提出的ARB预条件子的效率和收敛性。该方法始终在少数迭代内收敛，并显著减少在线求解时间，支持其在多查询工程场景中的效率。
英文摘要
We introduce a class of additive reduced basis preconditioners designed to accelerate the iterative solution of large-scale linear systems arising from discretized parametrized PDEs. The main idea is to regularize the inherently singular reduced-order approximation by adding simple correction terms: either a scaled identity correction or a projected correction based on a basic preconditioner. This yields nonsingular preconditioners under explicit and easily checked conditions. The construction and application of the preconditioners are integrated into an FGMRES framework through an offline strategy that dynamically builds the reduced-basis component by proper orthogonal decomposition at each FGMRES step. We establish sufficient conditions for the nonsingularity of the preconditioners and derive error bounds for the preconditioned Richardson iteration. Numerical results for convection-diffusion, anisotropic vortex, Stokes, and Helmholtz problems are provided to verify the efficiency and convergence of the proposed ARB preconditioners. The method consistently converges in a few iterations and substantially reduces online solve time, supporting its efficiency for multi-query engineering scenarios.
Comments23 pages, 8 figures
