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<p>In recent years, the Kaczmarz method has garnered significant attention for solving ultra-large-scale consistent linear systems due to its minimal storage requirements. To effectively enhance the efficiency of the Kaczmarz method, we employ <i>k</i>-means clustering method based on cosine distance to partition all equations (or equivalently, hyperplanes) into <i>k</i> clusters. Subsequently, in each iteration, the approximate solution is updated using the equation with the maximum residual within a randomly selected cluster, thereby constructing a <i>k</i>-means clustering based randomized maximum residual Kaczmarz (abbreviated as RMRK(<i>k</i>)) method. To further enhance the computational efficiency, a maximum residual block Kaczmarz method with <i>k</i>-means clustering (abbreviated as MRBK(<i>k</i>)) is constructed by simultaneously updating the approximate solution using the equations with the maximum residuals within each cluster. The convergence of both methods is rigorously proved, demonstrating that for any chosen initial value in the column space of coefficient matrix, the iterative sequences generated by both methods converge unconditionally to the minimum-norm solution of the linear system. Numerical experimental results validate the feasibility and effectiveness of these two methods.</p>

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K-means clustering based maximal residual (block) Kaczmarz methods for solving large scale system of linear equations

  • 期刊论文

  • 发表日期：

    2025年6月3日

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  • Chen Huan-Li,
  • Yang Ai-Li,
  • Liu Rui,
  • Cao Yang

摘要

In recent years, the Kaczmarz method has garnered significant attention for solving ultra-large-scale consistent linear systems due to its minimal storage requirements. To effectively enhance the efficiency of the Kaczmarz method, we employ k-means clustering method based on cosine distance to partition all equations (or equivalently, hyperplanes) into k clusters. Subsequently, in each iteration, the approximate solution is updated using the equation with the maximum residual within a randomly selected cluster, thereby constructing a k-means clustering based randomized maximum residual Kaczmarz (abbreviated as RMRK(k)) method. To further enhance the computational efficiency, a maximum residual block Kaczmarz method with k-means clustering (abbreviated as MRBK(k)) is constructed by simultaneously updating the approximate solution using the equations with the maximum residuals within each cluster. The convergence of both methods is rigorously proved, demonstrating that for any chosen initial value in the column space of coefficient matrix, the iterative sequences generated by both methods converge unconditionally to the minimum-norm solution of the linear system. Numerical experimental results validate the feasibility and effectiveness of these two methods.