Abstract For solving large scale linear systems, a fast convergent randomized Kaczmarz-type method was constructed in Bai and Wu (2018) [4] . In this paper, we propose a geometric probability randomized Kaczmarz (GPRK) method by introducing a new index set J k and three supervised probability criteria defined on J k from a geometric point of view. Linear convergence of GPRK is proved, and the way of argument for the analysis of GPRK also leads to new sharper upper bounds for the randomized Kaczmarz (RK) method and the greedy randomized Kaczmarz (GRK) method. In practice, GPRK is implemented with a simple geometric probability criterion, i.e., the most efficient one of the aforementioned three supervised probability criteria defined on J k . The numerical results demonstrate that GPRK is robust and efficient, and it is faster than GRK in most of the tests in the sense of computing time.

中文翻译：

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一种用于大规模线性系统的几何概率随机 Kaczmarz 方法

摘要 为了求解大规模线性系统，Bai and Wu (2018) [4] 构建了一种快速收敛的随机 Kaczmarz 型方法。在本文中，我们通过从几何的角度引入新的索引集 J k 和定义在 J k 上的三个监督概率标准，提出了一种几何概率随机化 Kaczmarz (GPRK) 方法。证明了 GPRK 的线性收敛性，并且 GPRK 分析的论证方式也为随机 Kaczmarz (RK) 方法和贪婪随机 Kaczmarz (GRK) 方法带来了新的更清晰的上限。在实践中，GPRK 是用一个简单的几何概率准则来实现的，即在 J k 上定义的上述三个监督概率准则中最有效的一个。数值结果表明 GPRK 是稳健有效的，