Convergence conditions for a restarted GMRES method augmented with eigenspaces
Convergence conditions for a restarted GMRES method augmented with eigenspaces
复制标题
使用特征空间增强的重新启动 GMRES 方法的收敛条件
DOI：
10.1002/nla.421
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发表时间：
2005
影响因子：
4.3
通讯作者：
J. Zítko
中科院分区：
文献类型：
--
作者：
J. Zítko
我们考虑使用 GMRES(m,k) 方法来求解线性系统 Ax=b，即重新启动 m 的 GMRES，其中向 m 维的标准 Krylov 子空间添加 k 维的另一个子空间，从而产生增广 Krylov 子空间。这个附加子空间通常近似于 A 不变子空间。通常使用与最接近零的特征值相关的特征空间，因为这些特征空间被认为最阻碍收敛。残差边界的行为是针对 GMRES(m,k) 过程中可能出现的各种情况进行描述的。所获得的残差向量范数估计表明了 GMRES(m,k) 收敛的充分条件，并说明这些增强技术可以在许多情况下消除 GMRES(m) 的停滞。所有估计均独立于初始近似值的选择。本文的结论和评论对所提出的边界的质量进行了数值评估。版权所有 © 2004 约翰·威利父子有限公司
We consider the GMRES(m,k) method for the solution of linear systems Ax=b, i.e. the restarted GMRES with restart m where to the standard Krylov subspace of dimension m the other subspace of dimension k is added, resulting in an augmented Krylov subspace. This additional subspace approximates usually an A‐invariant subspace. The eigenspaces associated with the eigenvalues closest to zero are commonly used, as those are thought to hinder convergence the most. The behaviour of residual bounds is described for various situations which can arise during the GMRES(m,k) process. The obtained estimates for the norm of the residual vector suggest sufficient conditions for convergence of GMRES(m,k) and illustrate that these augmentation techniques can remove stagnation of GMRES(m) in many cases. All estimates are independent of the choice of an initial approximation. Conclusions and remarks assessing numerically the quality of proposed bounds conclude the paper. Copyright © 2004 John Wiley & Sons, Ltd.