OBK-RCM: Accelerated Orthogonal Block Kaczmarz Algorithm via RCM Reordering and Dynamic Grouping for Sparse Linear Systems

TL;DR

This paper introduces OBK-RCM, an accelerated orthogonal block Kaczmarz algorithm that uses RCM reordering and dynamic grouping to efficiently solve large sparse linear systems with scattered nonzero patterns, achieving significant speedups.

Contribution

The paper proposes a novel OBK-RCM algorithm combining RCM reordering and orthogonal block partitioning, with extensions for non-square systems, improving efficiency and convergence in sparse linear algebra.

Findings

Achieves 10-50 times faster CPU time than existing methods.

Reduces iterations by 50-90% compared to state-of-the-art algorithms.

Demonstrates effectiveness on real-world and synthetic matrices.

Abstract

Existing block Kaczmarz methods face challenges in balancing computational efficiency and convergence for large sparse linear systems with scattered nonzero patterns, due to costly partitioning strategies and non-orthogonal projections. In this paper, we propose the orthogonal block Kaczmarz (OBK-RCM) algorithm with the Reverse Cuthill-McKee (RCM), which integrates the RCM reordering with a novel orthogonal block partitioning strategy. RCM transforms sparse matrices into banded structures to enhance inter-block orthogonality, while dynamic grouping of mutually orthogonal blocks based on angle cosine thresholds reduces iterative complexity. In addition, two extended versions (SOBK-RCM and UOBK-RCM) are proposed to deal with non-square systems by constructing extended matrices without sacrificing sparsity. This work offers a practical framework for efficient sparse linear algebra solvers. Experiments on 33 real-world and synthetic matrices show that OBK-RCM achieves 10-50 times faster CPU time (up to several hundred) and 50-90% fewer iterations than state-of-the-art methods (RBK,RBK(k),GREBK(k),aRBK), especially for scattered sparse structures in most cases. Theoretical analysis confirms linear convergence, driven by hyperplane orthogonality.