Krylov子空间方法简介

摘要：

目前，许多研究领域都需要处理大型稀疏数据矩阵，而求这些矩阵的逆矩阵是最重要的问题。值得庆幸的是，Krylov子空间方法可以用来解决这些类型的问题。然而，很难理解这些方法背后的数学原理。在文章的第一部分，Krylov方法进行了详细的讨论。因此，具有线性代数基本知识的读者应该能够理解这些方法。在这一部分中，Krylov方法的知识被放入一些例子中，用于简单地实现一个众所周知的Krylov方法GMRES。在第二部分中，文章讨论了CG迭代，这是一种非常类似于Krylov方法的广为人知的方法。通过CG迭代与Krylov方法的比较，使读者对基于CG迭代的Krylov方法有一个更好的理解。在文章的第三部分，为了提高Krylov方法的效率，我们讨论了预条件子。此外，为了减少Krylov方法的空间消耗，本文还简要介绍了重新启动GMRES。

英文摘要：

Nowadays, many fields of study are have to deal with large and sparse data matrixes, but the most important issue is finding the inverse of these matrixes. Thankfully, Krylov subspace methods can be used in solving these types of problem. However, it is difficult to understand mathematical principles behind these methods. In the first part of the article, Krylov methods are discussed in detail. Thus, readers equipped with a basic knowledge of linear algebra should be able to understand these methods. In this part, the knowledge of Krylov methods are put into some examples for simple implementations of a commonly known Krylov method GMRES. In the second part, the article talks about CG iteration, a wildly known method which is very similar to Krylov methods. By comparison between CG iteration and Krylov methods, readers can get a better comprehension of Krylov methods based on CG iteration. In the third part of the article, aiming to improve the efficiency of Krylov methods, preconditioners are discussed. In addition, the restarting GMRES is briefly introduced to reduce the space consumption of Krylov methods in this part.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。