AI 中文总结
本文针对不定最小二乘问题，提出块正则化分裂（BRS）及松弛BRS（RBRS）预条件迭代方法，经理论分析与数值实验验证，其性能优于现有方法，可高效求解大规模问题。
AI 中文摘要
本文提出一种新颖的块正则化分裂（BRS）框架，用于高效求解不定最小二乘（ILS）问题。基于融入矩阵分裂方案的逐块正则化策略，我们开发了BRS迭代方法及有效的BRS预条件子。对所提迭代方法的收敛性进行了严格分析，并建立了BRS预条件矩阵的谱界。为进一步提升计算性能，我们引入松弛BRS（RBRS）预条件子，其可提供更优的谱性质并显著加速Krylov子空间方法的收敛。对稠密与稀疏测试问题的大量数值实验表明，所提BRS和RBRS预条件子在迭代次数、计算时间及整体效率上均始终优于现有方法。这些结果凸显了所提块正则化分裂框架在求解大规模ILS问题时的有效性与鲁棒性。
英文摘要
This paper proposes a novel block-regularized splitting (BRS) framework for the efficient solution of indefinite least-squares (ILS) problems. Based on a block-wise regularization strategy incorporated into a matrix splitting scheme, we develop a BRS iterative method together with an effective BRS preconditioner. The convergence of the proposed iterative method is rigorously analyzed, and spectral bounds for the BRS-preconditioned matrix are established. To further enhance computational performance, we introduce a relaxed BRS (RBRS) preconditioner, which provides improved spectral properties and significantly accelerates the convergence of Krylov subspace methods. Extensive numerical experiments on both dense and sparse test problems demonstrate that the proposed BRS and RBRS preconditioners consistently outperform existing approaches in terms of iteration count, computational time, and overall efficiency. These results highlight the effectiveness and robustness of the proposed block-regularized splitting framework for solving large-scale ILS problems.
