标题： 关于交替随机块Kaczmarz方法 显示英文标题

标题： On the alternating randomized block Kaczmarz method

摘要： 块Kaczmar兹方法及其变体旨在解决超定线性系统。 它们涉及迭代地将当前点投影到一组约束的解空间上。 在这项工作中，通过交替处理两个子问题（即具有多个右端项的线性系统）使用块Kaczmar兹方法，我们提出了{\it 交替随机块Kaczmarz算法}（ARBK）方法来求解线性矩阵方程$AXB=F$，该方法包含了随机索引选择方案以确定约束的子集。 收敛性分析表明， ARBK方法具有一个由明确表达式限定的线性收敛率。 已经进行了若干数值研究来验证理论发现。 显示英文摘要

摘要： The block Kaczmarz method and its variants are designed for solving the over-determined linear system. They involve iteratively projecting the current point onto the solution space of a subset of constraints. In this work, by alternately dealing with two subproblems (i.e., linear system with multiple right-hand sides) using the block Kaczmarz method, we propose the {\it Alternating Randomized Block Kaczmarz} (ARBK) method to solve the linear matrix equation $AXB=F$, which incorporates a randomized index selection scheme to determine the subset of constraints. The convergence analysis reveals that the ARBK method has a linear convergence rate bounded by an explicit expression. Several numerical studies have been conducted to validate the theoretical findings.