项目名称: 电磁场涡流问题中结构化线性方程组的预处理方法
项目编号: No.11301521
项目类型: 青年科学基金项目
立项/批准年度: 2014
项目学科: 数理科学和化学
项目作者: 任志茹
作者单位: 中国科学院数学与系统科学研究院
项目金额: 22万元
中文摘要: 大规模结构化线性方程组的求解来源于科学与工程计算的众多领域。例如:流体力学中Navier-Stokes方程,电磁学中的时谐涡流模型和偏微分方程最优控制问题等等,经过数值离散后都转化为结构化线性方程组的求解。由此类问题离散所得到的矩阵通常是大型稀疏的,且具有一定的结构,因此对结构化线性方程组预处理迭代法的研究有着广泛的应用背景和重要的理论意义。本课题将对来源于电磁场涡流问题中的结构化线性方程组,利用其系数矩阵的特殊结构构造有效的预处理子,并结合分裂迭代法或Krylov子空间方法探索相应的预处理技术,从而为这些结构化线性方程组设计高效的数值算法。首先,基于这些系数矩阵的分块结构,我们对其对称性、正定性以及特征信息进行理论分析;然后利用这些矩阵的特殊结构和具体性质来构造有效的预处理子,设计相应的算法,并通过程序对算法进行实现。此外,期望从理论上分析预处理后矩阵的特征值范围以及算法的收敛性。
中文关键词: 结构线性方程组;预处理;收敛性;特征值;涡流模型
英文摘要: Large-scale structured systems of linear equations arise from many fields in scientific and engineering computing. For example, Navier-Stokes equation in fluid mechanics, time-harmonic eddy current model in electromagnetics, optimal control problems governed by partial differential equations and so on, can be transformed into solving the structured systems of linear equations after numerical discretizations. The coefficient matrices of these structured systems of linear equations are usually large sparse, and have certain structures. Therefore, it has a broad application background and significant theoretical value to study on preconditioned methods for these structured systems of linear equations. In order to slove the structured systems of linear equations arising from eddy current problems in electromagnetics, we consider to construct effective preconditioners for the coefficient matrices according to its special structures, explore preconditioning technique combined with splitting iteration methods and Krylov subspace methods, and design effective numerical algorithms further in this project. Firstly, we will analyze its symmetry, positive definite, and characteristic information theoretically based on the block structures of the coefficient matrices. Then, we will propose efficient preconditioners by making
英文关键词: structured linear systems;precondition;convergence;eigenvalue;eddy current model