最小二乘法的反射块 Kaczmarz 算法

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-10-10 DOI:10.1016/j.laa.2024.10.009

Changpeng Shao

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引用次数: 0

摘要

在 Steinerberger (2021) [23] 和 Shao (2023) [21]中，针对一致线性系统提出了两种新型 Kaczmarz 算法，它们有一些相似之处。这两种算法不仅能与之前的许多 Kaczmarz 算法相媲美，更重要的是，它们揭示了线性系统解的一些有趣的新几何性质，而这些性质从 Kaczmarz 算法的标准观点来看并不明显。在本文中，我们将全面研究这两种算法。首先，我们从理论上分析了求解最小二乘法的算法，这在实践中更为常见。其次，我们将这两种算法扩展为块版本，并对其收敛率进行了严格估计。第三，作为理论补充，我们提供了更多关于收敛率属性的结果。所有这些结果都有助于更透彻地理解这些算法。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Reflective block Kaczmarz algorithms for least squares

In Steinerberger (2021) [23] and Shao (2023) [21], two new types of Kaczmarz algorithms, which share some similarities, for consistent linear systems were proposed. These two algorithms not only compete with many previous Kaczmarz algorithms but, more importantly, reveal some interesting new geometric properties of solutions to linear systems that are not obvious from the standard viewpoint of the Kaczmarz algorithm. In this paper, we comprehensively study these two algorithms. First, we theoretically analyse the algorithms for solving least squares, which is more common in practice. Second, we extend the two algorithms to block versions and provide their rigorous estimate on the convergence rates. Third, as a theoretical complement, we provide more results on properties of the convergence rate. All these results contribute to a more thorough understanding of these algorithms.

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.