一类快速免伪逆贪婪块KACZMARZ方法  

A CLASS OF FASTER FREE PSEUDOINVERSE GREEDY BLOCK KACZMARZ METHOD

作　　者：张建华 郭静慧 Zhang Jianhua;Guo Jinghui(School of Science,East China University of Technology,Nanchang 330013)

机构地区：[1]东华理工大学理学院,南昌330013

出　　处：《高等学校计算数学学报》2022年第3期285-298,共14页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金(12061009);江西省自然科学基金面上项目(20202BAB201002)。

摘　　要：1引言大规模层析图像重建[1]、分类问题[2]、图像重构[3]、数字信号处理[4]和数据挖掘[5]等科学与工程应用中,大规模相容线性方程组Ax=b,A∈R^(m×n),b∈R^(m),(1)的高效求解具有重要的理论意义和实际应用价值.基于矩阵分裂的直接法(例如LU分解和QR分解等)和基于全矩阵-向量积的迭代法(例如CG方法和Lanczos方法等)通常不能适用于大规模线性方程组的求解.近年来,以Kaczmarz方法为代表的行作用方法受到人们的广泛关注并成功应用于CT成像技术等实际问题中.Recently,block Kaczmarz method and randomized block Kaczmarz method have been successfully provided to solve large-scale consistent linear systems.Even more recently,Niu and Zheng proposed a greedy block Kaczmarz(GBK)using a greedy strategy,which is a projection-based block method.However,at each iteration step,the GBK method contains pseudoinverse computation and it is also not adequate for distributed implementations.In this paper,we present a pseudoinverse-free GBK(FGBK)method and derive its convergence analyses.Moreover,in order to accelerate convergence rate of the FGBK method,we establish a faster FGBK method using matrix sketching techniques and study its convergence property.Finally,numerical examples are tested to illustrate the effectiveness of our new methods.

关 键 词：LU分解 LANCZOS方法 向量积 QR分解 迭代法 矩阵分裂 数字信号处理 图像重构 

分 类 号：O242.28[理学—计算数学]