Iterative solution methods
Iterative solution methods
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DOI：
10.1016/j.apnum.2004.06.003
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发表时间：
2004-12-01
影响因子：
2.8
通讯作者：
Beauwens, R
中科院分区：
文献类型：
--
作者：
Beauwens, R
本报告旨在审查国家的最先进的迭代方法解决大型稀疏线性系统，如在有限差分和有限元近似的边值问题。然而，为了保持这一审查在合理的范围内，我们只审查这些方法的代数分析已经实现。我们首先审查的基本原则和迭代求解方法的组成部分，并更详细地描述了用于设计preconditioners的主要设备，显示了如何建立当今复杂的preconditioners通过简单的加法和/或乘法组成。我们还注意到，加速方法有时可能会被视为，因此使用，作为preconditioners.Next，使用近似分解作为基本框架，我们展示了他们的发展如何导致所谓的修改方法的研究，以及为什么注意力转移到特定的排序，多级类型。最后，我们将展示如何成功开发的多重网格和层次的基础方法，促使引进等效的代数技术：除了递归排序，一个额外的步骤称为稳定多项式预处理，在这里发挥作用的W-循环的多重网格方法和代数版本的V-循环平滑。(C)2004年IMACS。Elsevier B. V.出版，保留所有权利。
This presentation is intended to review the state-of-the-art of iterative methods for solving large sparse linear systems such as arising in finite difference and finite element approximations of boundary value problems. However, in order to keep this review within reasonable bounds, we only review those methods for which an algebraic analysis has been achieved.We first review the basic principles and components of iterative solution methods and describe in more detail the main devices used to design preconditioners, showing how the present day complex preconditioners are built through additive and/or multiplicative composition of simpler ones. We also note that acceleration methods may sometimes be viewed, and thus used, as preconditioners.Next, using approximate factorizations as basic framework, we show how their development led to the study of so-called modified methods and why attention then shifted to specific orderings, of multilevel type. Finally we show how the successful development of multigrid and hierarchical basis methods prompted the introduction of equivalent algebraic techniques: besides recursive orderings, an additional step called stabilization by polynomial preconditioning that plays here the role of the W-cycles of the multigrid method and an algebraic version of V-cycles with smoothing. (C) 2004 IMACS. Published by Elsevier B.V. All rights reserved.