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<p>This paper considers the distributed solution of large-scale sparse overdetermined linear systems and proposes an event-triggered adaptive accelerated parallel randomized Kaczmarz (ET-AAPRK) algorithm. The method is designed to reduce the communication bottleneck in parallel solvers and to improve the projection efficiency of conventional parallel randomized Kaczmarz methods with fixed relaxation parameters. Within a parallel randomized projection framework, ET-AAPRK combines greedy sampling with an adaptive relaxation strategy. The relaxation parameter is computed online by approximating residual energy minimization over a small set of probe rows, which improves the quality of local projections and reduces numerical oscillations. To avoid unnecessary synchronization, an event-triggered communication rule based on the residual variation rate is introduced. Global communication is performed only when the effectiveness of local projections becomes limited. After synchronization, a residual-driven weighted aggregation strategy is used to combine local iterates. Theoretical analysis shows that, for consistent overdetermined systems with full-column-rank coefficient matrices, ET-AAPRK achieves stable error reduction and local linear convergence in expectation. Numerical experiments were carried out on the Shanhe Supercomputer for four sparse overdetermined systems using 1 to 32 processes. Compared with the parallel randomized Kaczmarz projection (PRKP) algorithm and the parallel conjugate gradient (PCG) method, ET-AAPRK achieves the shortest runtime on the tested problems, with average speedups of 4.15 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\times\)</EquationSource> <EquationSource Format="MATHML"><math> <mo>×</mo> </math></EquationSource> </InlineEquation> and 156.06 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\times\)</EquationSource> <EquationSource Format="MATHML"><math> <mo>×</mo> </math></EquationSource> </InlineEquation>, respectively. The results indicate that the proposed event-triggered and adaptive projection strategies can reduce communication overhead and improve parallel efficiency for the tested large-scale sparse systems.</p>

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ET-AAPRK: An event-triggered adaptive accelerated parallel randomized Kaczmarz algorithm for large-scale overdetermined systems

  • 期刊论文

  • 发表日期：

    2026年7月31日

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  • Yulong Jiang,
  • Tao Liu,
  • Yongli Wang,
  • Guoping He

摘要

This paper considers the distributed solution of large-scale sparse overdetermined linear systems and proposes an event-triggered adaptive accelerated parallel randomized Kaczmarz (ET-AAPRK) algorithm. The method is designed to reduce the communication bottleneck in parallel solvers and to improve the projection efficiency of conventional parallel randomized Kaczmarz methods with fixed relaxation parameters. Within a parallel randomized projection framework, ET-AAPRK combines greedy sampling with an adaptive relaxation strategy. The relaxation parameter is computed online by approximating residual energy minimization over a small set of probe rows, which improves the quality of local projections and reduces numerical oscillations. To avoid unnecessary synchronization, an event-triggered communication rule based on the residual variation rate is introduced. Global communication is performed only when the effectiveness of local projections becomes limited. After synchronization, a residual-driven weighted aggregation strategy is used to combine local iterates. Theoretical analysis shows that, for consistent overdetermined systems with full-column-rank coefficient matrices, ET-AAPRK achieves stable error reduction and local linear convergence in expectation. Numerical experiments were carried out on the Shanhe Supercomputer for four sparse overdetermined systems using 1 to 32 processes. Compared with the parallel randomized Kaczmarz projection (PRKP) algorithm and the parallel conjugate gradient (PCG) method, ET-AAPRK achieves the shortest runtime on the tested problems, with average speedups of 4.15 \(\times\) × and 156.06 \(\times\) × , respectively. The results indicate that the proposed event-triggered and adaptive projection strategies can reduce communication overhead and improve parallel efficiency for the tested large-scale sparse systems.