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. Numerical experiments are given to illustrate the theoretical results.
翻译:我们调查了随机的Kaczmarz方法,该方法利用随时可用的信息对步骤进行更新,以解决不一致的线性系统。我们提供了一种新的几何解释,它表明拟议的方法可以被视为某种意义上的正对投影方法。我们证明,这一方法线性趋同于线性系统独特的欧洲-clidean规范最低方位的最小标准解决方案。用数字实验来说明理论结果。