题目：Regularized randomized iterative algorithms for factorized linear systems
报告人：杜魁 教授 单位：厦门大学 数学科学学院
时间：2022年4月29日（周五）上午10:00-11:00
腾讯会议：383-738-106
报告人及报告内容摘要：
杜魁，厦门大学数学科学学院教授，2009年8月博士毕业于香港城市大学，2009年9月至2011年4月在芬兰阿尔托大学数学研究所做博士后，2011年6月至今在厦门大学数学科学学院工作；现任中国数学会计算数学分会第十届委员会理事，福建省运筹学会理事；主持和参与国家自然科学基金项目多项；目前主要研究兴趣为大规模问题随机算法，反问题的计算方法等。
Abstract: Randomized iterative algorithms for solving a factorized linear system, $\mathbf A\mathbf B\mathbf x=\mathbf b$ with $\mathbf A\in{\mathbb{R}}^{m\times \ell}$, $\mathbf B\in{\mathbb{R}}^{\ell\times n}$, and $\mathbf b\in{\mathbb{R}}^m$, have recently been proposed. They take advantage of the factorized form and avoid forming the matrix $\mathbf C=\mathbf A\mathbf B$ explicitly. However, they can only find the minimum norm (least squares) solution. In contrast, the regularized randomized Kaczmarz (RRK) algorithm can find solutions with certain structures from consistent linear systems. In this work, by combining the randomized Kaczmarz algorithm or the randomized Gauss--Seidel algorithm with the RRK algorithm, we propose two novel regularized randomized iterative algorithms to find (least squares) solutions with certain structures of $\mathbf A\mathbf B\mathbf x=\mathbf b$. We prove linear convergence of the new algorithms. Computed examples are given to illustrate that the new algorithms can find sparse (least squares) solutions of $\mathbf A\mathbf B\mathbf x=\mathbf b$ and can be better than the existing randomized iterative algorithms for the corresponding full linear system $\mathbf C\mathbf x=\mathbf b$ with $\mathbf C=\mathbf A\mathbf B$.