摘要
贪婪随机增广Kaczmarz(GRAK)方法和加速GRAK(AGRAK)方法是求解不相容线性方程组转化为相容增广线性方程组后的有效迭代方法,但在处理大规模问题时仍面临较高的计算开销.为提升AGRAK方法的计算效率,提出了一种基于最大距离行选取策略的新型随机增长Kaczmarz(NRAK)方法.该方法通过优化当前迭代点的行选择机制,提升了方法的收敛速度.理论分析表明,该算法在均方上指数收敛于最小二乘解,并在合理的假设条件下,NRAK方法的收敛速度优于AGRAK方法.数值实验进一步表明,与当前典型的方法相比,NRAK方法所需的迭代步数和计算时间均有减少,即基于最大距离行选取策略的NRAK方法能够有效提升求解大规模不相容线性方程组的计算效率.
The greedy randomized augmented Kaczmarz(GRAK)method and its accelerated variant(AGRAK)method were effective iterative approaches for solving inconsistent linear systems transformed into consistent augmented linear systems.However,they still suffered from relatively high computational costs when handling large⁃scale problems.To improve the computational efficiency of the AGRAK method,a novel randomized augmexted Kaczmarz method based on a maximum⁃distance row selection strategy,termed NRAK,was proposed.By optimizing the row selection mechanism at the current iterate,the proposed method accelerated the convergence of the method.Theoretical analysis showed that the proposed method converged exponentially in the mean square to the least squares solution,and under reasonable assumptions,the NRAK method achieved superior convergence rates compared to the AGRAK method.Numerical experiments further demonstrated that,compared with state⁃of⁃the⁃art methods,the NRAK method required fewer iterations and less computational time,which indicated that the NRAK method based on the maximum⁃distance row selection strategy effectively improved the computational efficiency for solving large⁃scale inconsistent linear systems.
作者
吕玉鑫
张建华
LYV Yuxin;ZHANG Jianhua(School of Science,East China University of Technology,Nanchang 330000,China)
出处
《哈尔滨商业大学学报(自然科学版)》
2026年第1期122-128,共7页
Journal of Harbin University of Commerce(Natural Sciences Edition)
基金
国家自然科学基金(12061009)
江西省自然科学基金面上项目(2020BAB201002)。
