A greedy randomized augmented Kaczmarz (GRAK) method was proposed in [Z.-Z. Bai and W.-T. WU, SIAM J. Sci. Comput., 43 (2021), pp. A3892-A3911] for large and sparse inconsistent linear systems. However, one has to construct two new index sets via computing residual vector with respect to the augmented linear system in each iteration. Thus, the computational overhead of this method is large for extremely large-scale problems. Moreover, there is no reliable stopping criterion for this method. In this work, we are interested in solving large-scale sparse or dense inconsistent linear systems, and try to enhance the numerical performance of the GRAK method. First, we propose an accelerated greedy randomized augmented Kaczmarz method. Theoretical analysis indicates that it converges faster than the GRAK method under very weak assumptions. Second, in order to further release the overhead, we propose a semi-randomized augmented Kaczmarz method with simple random sampling. Third, to the best of our knowledge, there are no practical stopping criteria for all the randomized Kaczmarz-type methods till now. To fill-in this gap, we introduce a practical stopping criterion for Kaczmarz-type methods, and show its rationality from a theoretical point of view. Numerical experiments are performed on both real-world and synthetic data sets, which demonstrate the efficiency of the proposed methods and the effectiveness of our stopping criterion.

翻译：针对大规模稀疏不相容线性系统，文献[Z.-Z. Bai and W.-T. WU, SIAM J. Sci. Comput., 43 (2021), pp. A3892-A3911]提出了贪婪随机化增广Kaczmarz（GRAK）方法。然而，该方法每次迭代需通过计算增广线性系统的残差向量构建两个新的指标集，导致处理超大规模问题时计算开销较大，且缺乏可靠的停机准则。本文致力于求解大规模稀疏或稠密不相容线性系统，并尝试提升GRAK方法的数值性能。首先，我们提出一种加速贪婪随机化增广Kaczmarz方法，理论分析表明其在非常弱的假设条件下收敛速度快于GRAK方法。其次，为进一步降低计算开销，我们提出一种结合简单随机采样的半随机化增广Kaczmarz方法。第三，据我们所知，目前所有随机化Kaczmarz类方法均缺乏实用的停机准则。为填补这一空白，我们引入一种适用于Kaczmarz类方法的实用停机准则，并从理论角度论证其合理性。在真实数据集与合成数据集上的数值实验验证了所提方法的有效性及停机准则的实用性。