Large-scale linear systems, $Ax=b$, frequently arise in practice and demand effective iterative solvers. Often, these systems are noisy due to operational errors or faulty data-collection processes. In the past decade, the randomized Kaczmarz (RK) algorithm has been studied extensively as an efficient iterative solver for such systems. However, the convergence study of RK in the noisy regime is limited and considers measurement noise in the right-hand side vector, $b$. Unfortunately, in practice, that is not always the case; the coefficient matrix $A$ can also be noisy. In this paper, we analyze the convergence of RK for noisy linear systems when the coefficient matrix, $A$, is corrupted with both additive and multiplicative noise, along with the noisy vector, $b$. In our analyses, the quantity $\tilde R=\| \tilde A^{\dagger} \|_2^2 \|\tilde A \|_F^2$ influences the convergence of RK, where $\tilde A$ represents a noisy version of $A$. We claim that our analysis is robust and realistically applicable, as we do not require information about the noiseless coefficient matrix, $A$, and considering different conditions on noise, we can control the convergence of RK. We substantiate our theoretical findings by performing comprehensive numerical experiments.

翻译：大规模线性系统$Ax=b$在实践中频繁出现，需要高效的迭代求解器。由于操作错误或数据采集过程有缺陷，这些系统通常含有噪声。在过去十年中，随机Kaczmarz（RK）算法作为此类系统的高效迭代求解器得到了广泛研究。然而，RK在含噪情况下的收敛性研究十分有限，且仅考虑了右端向量$b$中的测量噪声。遗憾的是，实际情况并非总是如此；系数矩阵$A$也可能存在噪声。本文分析了当系数矩阵$A$同时被加性噪声和乘性噪声污染，且向量$b$也含有噪声时，RK算法对于含噪线性系统的收敛性。在我们的分析中，量$\tilde R=\| \tilde A^{\dagger} \|_2^2 \|\tilde A \|_F^2$影响RK的收敛性，其中$\tilde A$表示$A$的含噪版本。我们声称我们的分析是稳健且具有现实适用性的，因为我们不需要关于无噪声系数矩阵$A$的信息，并且通过考虑不同的噪声条件，可以控制RK的收敛性。我们通过进行全面的数值实验来证实我们的理论发现。