Accelerated greedy randomized augmented Kaczmarz algorithm for inconsistent linear systems

Abstract

For solving inconsistent linear systems of equations iteratively, we further generalize the greedy randomized augmented Kaczmarz (GRAK) algorithm by introducing a nonzero parameter in the involved augmented linear system of the above inconsistent linear system, obtaining a class of accelerated greedy randomized augmented Kaczmarz algorithms. These algorithms involve one iteration parameter whose special choice can recover the GRAK algorithm, as well as yield new ones. Theoretical analyses show that the new algorithms converge to the unique solution of the augmented linear system. Moreover, the optimal choice of the parameter involved and the corresponding convergence rates of the new algorithms are computed exactly. Numerical results show that our algorithms can be much more effective than the GRAK algorithm in terms of both iteration counts and computing times.