On greedy randomized average block Kaczmarz method for solving large linear systems

Inspired by the greedy randomized Kaczmarz method, we propose a probability criterion which can capture subvectors of the residual whose norms are relatively large. According to this probability criterion we select a submatrix randomly from the coefficient matrix, then average the projections of the current iteration vector onto each individual row of this chosen submatrix, constructing the greedy randomized average block Kaczmarz method for solving the consistent system of linear equations, which can be implemented in a distributed environment. When the size of each block is one, the probability criterion in the greedy randomized average block Kaczmarz method is a generalization of that in the greedy randomized Kaczmarz method. The greedy randomized Kaczmarz method is also a special case of the greedy randomized average block Kaczmarz method. Two kinds of extrapolated stepsizes for the greedy randomized average block Kaczmarz method are analyzed. The experimental results show the advantage of the greedy randomized average block Kaczmarz method over the greedy randomized Kaczmarz method and several existing randomized block Kaczmarz methods.