全局Krylov子空间方法研究及其应用

发布时间：2018-01-06 10:24

本文关键词：全局Krylov子空间方法研究及其应用　出处：《电子科技大学》2015年硕士论文　论文类型：学位论文

【摘要】：本文主要研究求解大规模稀疏多右端向量线性方程组的全局Krylov子空间方法。由于许多应用领域,如求解偏微分方程问题,流体动力学,电路仿真,电磁场计算,线性控制论等,均需要求解多右端向量线性系统。因此建立多右端向量线性方程组的高效稳健的数值方法具有十分重要的意义。本文主要由下面两部分构成:首先,全面系统地阐述了基于Arnoldi过程的Krylov子空间方法,包括完全正交化方法(FOM)和广义最小残量方法(GMRES),数值实验表明GMRES方法要优于FOM方法。其次,按照这两种方法推导思路,分析了基于全局Arnoldi过程的Krylov子空间方法,即全局完全正交化方法(GL-FOM)和全局广义最小残量方法(GL-GMRES)。在此基础上,运用Givens旋转变换,给出了迭代解与相应的残量矩阵范数的表达式,并且使它们的表达式分别与FOM和GMRES的迭代解与相应的残量矩阵范数的表达式结构上保持类似。数值实验表明GL-GMRES方法要优于GL-FOM方法。基于全局Arnoldi过程,结合加权技术,提出了一种重启的加权全局广义最小残量方法用于求解多右端向量线性方程组。给出了两个定理和两个命题,目的在于:1.保证了D内积和D范数定义的合理性,并用拉直技术和克罗内克积表达D内积;2.使得所提的重启的加权全局广义最小残量方法的迭代解和相应的残量矩阵范数能够进一步计算;3.确保所提的重启的加权全局广义最小残量方法求出的迭代解和相应的残量矩阵范数的表达式与全局广义最小残量方法和广义最小残量方法求出的迭代解和相应的残量矩阵范数的表达式结构上保持一致;4.保证通过加权全局Arnoldi过程构造了块Krylov子空间的一组D正交基;5.旨在说明重启的加权全局广义最小残量算法具有尺度不变性。最后通过数值实验对收敛曲线、迭代次数、CPU消耗总时间方面进行比较,验证了所提重启的加权全局广义最小残量算法的有效性。
[Abstract]:This paper mainly studies how the right vectors for solving large sparse linear equations of global Krylov subspace methods. Because of many application areas, such as the problem of solving partial differential equations of fluid dynamics, circuit simulation, electromagnetic field calculation, linear control theory, requires solving multi right vector linear system. Therefore the establishment of a robust and efficient numerical method the right end of the vector of linear equations is very important. This paper consists of two parts following components: first, comprehensively and systematically elaborated the Krylov subspace method based on Arnoldi process, including complete orthogonal method (FOM) and generalized minimal residual method (GMRES), the numerical experiments show that the GMRES method is better than FOM according to this method. Secondly, the two methods are ideas, analysis of the Krylov subspace method based on global Arnoldi process, namely the global full orthogonalization method (GL-FOM) and the global generalized minimal Residual method (GL-GMRES). On this basis, using Givens transformation, an iteration solution and the corresponding residual matrix norm, and make them the expression of FOM and GMRES respectively and the iterative solution structure and residual matrix norm on the corresponding remain similar. Numerical experiments show the method to GL-GMRES better than the GL-FOM method. The global Arnoldi process based on combination weighting technique, a restart of the weighted global generalized minimal residual method for solving multi right vector linear equations is proposed. Two theorems are given and two propositions: 1., aims to ensure the rationality of D product and D norm is defined, and straighten Kronecker product technology and the expression of D 2. makes the inner product; iterative weighted global generalized minimal residual method for the restart of the solution and the corresponding residual matrix norm can be further calculated; 3. to ensure the resumption of the Consistent expression structure iteration expression and global GMRES iterative weighted global generalized minimal residual method to find solutions and the corresponding residual matrix norm and generalized minimal residual method to find solutions and the corresponding residual matrix norm; 4. is guaranteed by the weighted global Arnoldi process to construct a group D orthogonal basis block Krylov subspace; 5. to illustrate the weighted global generalized minimal residual algorithm restart with scale invariance. Finally, numerical experiments on the convergence curve, the number of iterations, the total time consumption of CPU were compared. The test validity of proposed weighted global generalized minimal residual algorithm restart.

【学位授予单位】：电子科技大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O177

【引证文献】