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Mathematical analysis of linear/nonlinear iterative algorithms including GMRES, SOR, etc.

线性/非线性迭代算法的数学分析，包括GMRES、SOR等。

基本信息

批准号：
09640277
负责人：
YAMAMOTO Tetsuro
金额：
$ 0.96万
依托单位：
Ehime University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640277/
关键词：
Iterative methods for salring equations GMRES method SOR method SSOR method Global convergence theorem Uzawa法 Jacobi-Davidson法非線形SOR型解法非線形SSOR法 SOR型D-K法一般局所収束定理

项目摘要

The purpose of this research was to give mathematical foundations for linear/nonlinear iterative methods including GMRES, SOR, etc. Concerning SOR, we obtained the following results :1. Unified treatment of known convergence theorems for linear SOR methods : The Ostrowski-Reich theorem is the most famous result for convergence of the SOR method applied to the linear system Ax = b which asserts that if A is hermitian with positive diagonals, then the SOR method converges for 0 < omega < 2 if and only if A is positive definite. Some results by Householder-John, Ortega-Plemmons, etc. are also known, which are closely related to the Ostrowski-Reich theorem. We found out that. these results can uniformly be derived from the Stein theorem, which asserts that a matrix H is a convergent matrix if and only if there exists a positive definite matrix B such that BETA - H^*BH is positive definite. This simplifies and unifies the convergence proofs by Ostrowski and others.2. Global convergence of nonlinear SOR-like methods. Although Brewster-Kannan's result is known for convergence of SOR-Newton's method, it only asserts that for any initial vector there exists a sequence of parameter {omega_k}, 0< omega_k < 2 such that the process with wk at each step converges to the solution. however, the explicit choice of wk is not known. We obtained a global convergence theorem for SOR-Newton's method applied to a discretized equation for semilinear PDE, which guarantees the convergence for 0 < omega< omega^<**>_h = 2 - OMICRON(h^2) where Ii denotes the mesh size.

本研究的目的是为线性/非线性迭代方法（包括GMRES、SOR 等）提供数学基础。关于SOR，我们获得了以下结果： 1．线性 SOR 方法已知收敛定理的统一处理：Ostrowski-Reich 定理是应用于线性系统 Ax = b 的 SOR 方法收敛的最著名的结果，它断言如果 A 是具有正对角线的埃尔米特式，则当且仅当 A 是正定时，SOR 方法收敛于 0 < omega < 2。 Householder-John、Ortega-Plemmons等人的一些结果也是已知的，它们与Ostrowski-Reich定理密切相关。我们发现了这一点。这些结果可以统一从 Stein 定理导出，该定理断言矩阵 H 是收敛矩阵当且仅当存在正定矩阵 B 使得 BETA - H^*BH 是正定的。这简化并统一了Ostrowski等人的收敛证明。2．非线性 SOR 类方法的全局收敛。尽管 Brewster-Kannan 的结果因 SOR-Newton 方法的收敛而闻名，但它仅断言对于任何初始向量，都存在参数序列 {omega_k}，0< omega_k < 2，使得每一步使用 wk 的过程收敛到解。然而，wk 的明确选择尚不清楚。我们获得了应用于半线性 PDE 离散方程的 SOR-Newton 方法的全局收敛定理，该定理保证了 0 < omega< omega^<**>_h = 2 - OMICRON(h^2) 的收敛性，其中 Ii 表示网格大小。