We investigate the randomized Kaczmarz method that adaptively updates the stepsize using readily available information for solving inconsistent linear systems. A novel geometric interpretation is provided which shows that the proposed method can be viewed as an orthogonal projection method in some sense. We prove that this method converges linearly in expectation to the unique minimum Euclidean norm least-squares solution of the linear system, and provide a tight upper bound for the convergence of the proposed method. Numerical experiments are also given to illustrate the theoretical results.

翻译：我们研究了利用易获取信息自适应更新步长的随机Kaczmarz方法，用于求解不相容线性系统。本文提供了一种新的几何解释，表明所提方法在某种意义上可视为正交投影方法。我们证明该方法期望线性收敛至线性系统的最小欧几里得范数最小二乘解，并给出了该方法收敛性的紧上界。最后通过数值实验验证了理论结果。