Robust Kaczmarz methods for nearly singular linear systems

Abstract

The Kaczmarz method is an efficient iterative algorithm for large-scale linear systems. However, its linear convergence rate suffers from ill-conditioned problems and is highly sensitive to the smallest nonzero singular value. In this work, we aim to extend the classical Kaczmarz to nearly singular linear systems that are row rank-deficient. We introduce a new concept of nearly singular property by treating the row space as an unstable subspace in the Grassman manifold. We then define a related important space called the approximate kernel, based on which a robust kernel-augmented Kaczmarz (KaK) is introduced via the subspace correction framework and analyzed by the well-known Xu--Zikatanov identity. To get an implementable version, we further introduce the approximate dual kernel and transform KaK into an equivalent kernel-augmented coordinate descent. Furthermore, we develop an accelerated variant and establish the improved rate of convergence matching the optimal complexity of first-order methods. Compared with existing methods, ours achieve uniform convergence rates for nearly singular linear systems, and the robustness has been confirmed by some numerical tests.