求解大型线性系统的K-均值最大残差块方法
K-Means Maximum Residual Block Method for Solving Large Linear Systems
摘要: 求解大型线性系统，带K-均值聚类的贪婪随机块Kaczmarz方法是近几年被广受关注的一类方法。本文在该方法基础上做了进一步的研究即在每一次迭代中优先消除残差向量中的最大块，构建了最大残差块Kaczmarz方法及其加速版本并进行了收敛性分析。数值实验证实了本文算法的有效性。
Abstract: The greedy random block Kaczmarz method with K-means clustering for solving large linear systems has been widely studied in recent years. This article conducted further research on this method by prioritizing the elimination of the largest block in the residual vector in each iteration, constructing the Kaczmarz method for the maximum residual block and its accelerated version, and conducting convergence analysis. Numerical experiments have confirmed the effectiveness of the algorithm proposed in this paper.
文章引用：黄柏顺. 求解大型线性系统的K-均值最大残差块方法[J]. 应用数学进展, 2024, 13(11): 5063-5072. https://doi.org/10.12677/aam.2024.1311489

参考文献

[1] Strohmer, T. and Vershynin, R. (2008) A Randomized Kaczmarz Algorithm with Exponential Convergence. Journal of Fourier Analysis and Applications, 15, 262-278. [Google Scholar] [CrossRef] 
[2] Jiang, X., Zhang, K. and Yin, J. (2022) Randomized Block Kaczmarz Methods with K-Means Clustering for Solving Large Linear Systems. Journal of Computational and Applied Mathematics, 403, Article 113828. [Google Scholar] [CrossRef] 
[3] 李雪芳. 基于自动控制步长的病态线性方程组算法研究[D]: [硕士学位论文]. 太原: 太原科技大学, 2013.
[4] 段云岭. 非线性方程组的解法: 局部弧长法[J]. 力学学报, 1997, 29(1): 116-122.
[5] 张广求, 余青, 洪果媛. 线性方程组分光光度法测定痤疮搽剂中甲硝唑和氯霉素的含量[J]. 中国药房, 1995, 6(1): 39-40.
[6] 蔡建新, 包尚联, 王卫东. 核医学影像中单光子定位的最小二乘方法[J]. 核电子学与探测技术, 1996, 16(3): 189-193.
[7] 陶本藻, 张勤. GPS 非线性数据处理的同伦最小二乘模型[J]. 武汉大学学报(信息科学版), 2003, 28(S1): 115-118.
[8] 胡圣荣, 戴纳新. 病态线性方程组新解法: 增广方程组法[J]. 华南农业大学学报, 2009, 30(1): 119-121.
[9] 胡川, 陈义. 非线性整体最小平差迭代算法[J]. 测绘学报, 2014, 43(7): 668-674+760.
[10] 刘经南, 曾文宪, 徐培亮. 整体最小二乘估计的研究进展[J]. 武汉大学学报(信息科学版), 2013, 38(5): 505-512.
[11] Heath, M.T. (1984) Numerical Methods for Large Sparse Linear Least Squares Problems. SIAM Journal on Scientific and Statistical Computing, 5, 497-513. [Google Scholar] [CrossRef] 
[12] Farebrother, R.W. (2018) Linear Least Squares Computations. Routledge.
[13] Bai, Z. and Wu, W. (2019) On Greedy Randomized Coordinate Descent Methods for Solving Large Linear Least-Squares Problems. Numerical Linear Algebra with Applications, 26, e2237. [Google Scholar] [CrossRef] 
[14] Paige, C.C. and Saunders, M.A. (1982) LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares. ACM Transactions on Mathematical Software, 8, 43-71. [Google Scholar] [CrossRef] 
[15] Yongjie, Z. and Qin, S. (2007) A New ICCG Method of Large-Scale Sparse Linear Equation Group. 2007 International Symposium on Microwave, Antenna, Propagation and EMC Technologies for Wireless Communications, Hangzhou, 16-17 August 2007, 848-851. [Google Scholar] [CrossRef] 
[16] Bai, Z. and Wu, W. (2018) On Greedy Randomized Kaczmarz Method for Solving Large Sparse Linear Systems. SIAM Journal on Scientific Computing, 40, A592-A606. [Google Scholar] [CrossRef] 
[17] Zhang, K., Liu, C.T., Jiang, X.L. (2024) A Greedy Randomized Block Coordinate Descent Algorithm with K-Means Clustering for Solving Large Linear Least-Squares Problems. International Journal of Computer Science, 51, 511-518.
[18] Davis, T.A. and Hu, Y. (2011) The University of Florida Sparse Matrix Collection. ACM Transactions on Mathematical Software, 38, 1-25. [Google Scholar] [CrossRef] 

为你推荐