An acceleration scheme for row projection methods

Cimminoʹs and Kaczmarzʹs methods are two classes of row projection (RP) methods for solving structured linear systems efficiently on parallel computers. They have been extended for solving nonlinear systems. Since the convergence rates of the two methods are unsatisfactorily slow, several acceleration schemes for the Kaczmarz method have been proposed. In this paper, we refine and improve the existing acceleration techniques and develop a new acceleration scheme for both of the methods. The new scheme involves finding points closest to a solution x∗ along some lines. Since x∗ is not known a priori, recursion formulas are derived. Three new accelerated algorithms—two for the linear Kaczmarz method and the linear Cimmino method and the other one for the nonlinear Cimmino method—are presented. The convergence of the three proposed algorithms is proved. The linear accelerated Kaczmarz algorithm and Cimmino algorithm converge globally. The nonlinear accelerated Cimmino algorithm converges locally. An important characteristic is that the accelerated Cimmino method retains the same degree of parallelism as the unaccelerated one. Results of initial numerical experiments with new algorithms are presented.