Convergence analyses based on frequency decomposition for the randomized row iterative method

Abstract

For solving the large-scale linear systems, a unified randomized row iterative (RRI) method was proposed in Gower and Richtárik (2015 SIAM J. Matrix Anal. Appl. 36 1660–1690), where its mean squared error is shown to decrease exponentially under some induced energy norm. In this work, for solving the perturbed system of linear equations, we give a new convergence analysis for the RRI method in the context of inverse problems. We divide the total error into two parts: the low- and high-frequency errors, which fully exploits the weighted singular value decomposition of the coefficient matrix. The upper bounds in the convergence rate of these two errors of RRI are analyzed for a noisy right-hand side, which can be specialized to the noise-free right-hand side case. Our estimates are compared with the upper bounds in Jiao et al (2017 Inverse Problems 33 125012) when RRI is reduced to the standard randomized Kaczmarz method. Finally we present numerical examples to confirm the analyses.