Right preconditioned MINRES for singular systems †
Right preconditioned MINRES for singular systems †
复制标题
正确预处理奇异系统的 MINRES †
DOI：
10.1002/nla.2277
复制
发表时间：
2020
影响因子：
4.3
通讯作者：
Zheng Ning
中科院分区：
文献类型：
--
作者：
Sugihara Kota;Hayami Ken;Zheng Ning
我们考虑求解大型稀疏对称奇异线性系统。我们首先引入右预处理最小残差（MINRES）算法，并证明即使线性系统是奇异且不一致的，它的迭代也收敛于预处理器加权最小二乘解，而不会破坏任意右侧向量和任意初始向量。对于系统一致的特殊情况，我们证明如果初始向量位于右预处理系数矩阵的范围空间中，则迭代收敛到关于预处理器的最小范数解。此外，我们提出了使用对称连续过度松弛（SSOR）和 Eisenstat 技巧的正确预处理 MINRES。电磁分析等半定系统的数值实验表明该方法高效、鲁棒。最后，我们表明可以通过重新启动迭代来进一步减少残差范数。
We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min‐norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over‐relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.