四川大学学报（自然科学版）2026，Vol.63Issue(3)：559-565,7.DOI:10.19907/j.0490-6756.250055
参数化泊松方程的模型降阶预处理
Model order reduction preconditioning of parameterized Poisson equations
摘要
Abstract
This paper aims at the fast numerical solution of parameterized Poisson equation.A preconditioned conjugate gradient(PCG)method based on the model order reduction method is proposed to speed up the so-lution of the obtained discrete linear system.First,the general formulation of preconditioning matrix based on model order reduction is designed,and the symmetry and positive definition of the matrix are proved.In the off-line stage of the method,by using very few solution data,the PCG algorithm combined with the proper orthogonal decomposition(POD)method are adopted to generate a set of dynamic preconditioning matrices.In the on-line stage,the MPCG algorithm is proposed by combining the PCG algorithm and these dynamic preconditioning matrices.To verify the performance of the method,parameterized Poisson equations on the unit rectangular and L-shaped domains are numerically solved.It is shown that,in comparison with the stan-dard CG algorithm,the average computation time is speeded up by more than 43 times with the same calcula-tion accuracy.关键词
参数化泊松方程/模型降阶/预处理矩阵/共轭梯度Key words
parameterized Poisson equation/model order reduction/precondition matrix/conjugate gradient分类
数理科学引用本文复制引用
胡奇晓,徐友才,张世全..参数化泊松方程的模型降阶预处理[J].四川大学学报（自然科学版）,2026,63(3):559-565,7.基金项目
四川省自然科学基金(2023NSFSC0075) （2023NSFSC0075）
