Convergence rates for Kaczmarz-type algorithms
Provides theoretical convergence guarantees for practically used Kaczmarz-type algorithms, benefiting researchers and practitioners in iterative linear system solvers.
This paper proves at least linear convergence rates for Kaczmarz-Tanabe and Extended Kaczmarz methods, and establishes sublinear/linear rates for almost cyclic and remotest set control strategies, completing existing results for random selection.
In this paper we make a theoretical analysis of the convergence rates of Kaczmarz and Extended Kaczmarz projection algorithms for some of the most practically used control sequences. We first prove an at least linear convergence rate for the Kaczmarz-Tanabe and its Extended version methods (the one in which a complete set of projections using row/column index is performed in each iteration). Then we apply the main ideas of this analysis in establishing an at least sublinear, respectively linear convergence rate for the Kaczmarz algorithm with almost cyclic and the remotest set control strategies, and their extended versions, respectively. These results complete the existing ones related to the random selection procedures.