On the adaptive-momentum variant for maximal weighted residual row-action methods

For solving a consistent linear system, classical row-action methods like the maximal weighted residual Kaczmarz (denoted by MWRK) method are straightforward yet highly effective iterative solvers. Expanding on a greedy index selection strategy and Polyak's heavy ball momentum technique, we present an adaptive momentum-accelerated MWRK (denoted by AmMWRK) method. At each iteration, the including step-size and momentum parameter are updated according to the Petrov-Galerkin conditions. Significantly, the geometric interpretation of the AmMWRK method is also given. By using this geometric property, we prove that our method achieves a linear convergence to the least-squares solution with minimum Euclidean norm in a concise way and attains a tighter upper convergence bound than the classical MWRK method.Furthermore, we extend this approach to solving the inconsistent linear systems and provide corresponding convergence analysis.Several numerical studies are presented to corroborate our theoretical findings. The real-world applications are also presented for illustration purposes. AMS subject classifications. 65F10, 65F25