On maximum residual block Kaczmarz method for solving large consistent linear systems

For solving large consistent linear systems by iteration methods, inspired by the maximum residual Kaczmarz method and the randomized block Kaczmarz method, we propose the maximum residual block Kaczmarz method, which is designed to preferentially eliminate the largest block in the residual vector $r_{k}$ at each iteration. At the same time, in order to further improve the convergence rate, we construct the maximum residual average block Kaczmarz method to avoid the calculation of pseudo-inverse in block iteration, which completes the iteration by projecting the iteration vector $x_{k}$ to each row of the constrained subset of $A$ and applying different extrapolation step sizes to average them. We prove the convergence of these two methods and give the upper bounds on their convergence rates, respectively. Numerical experiments validate our theory and show that our proposed methods are superior to some other block Kaczmarz methods.