• 论文 •

### 求解稀疏连续线性系统的自适应SGCRO-DR 算法

1. 1. 数学科学学院/计算科学研究所, 电子科技大学, 成都 611731;
2. 北京应用物理与计算数学研究所, 计算物理重点实验室, 北京 100094;
3. 中物院高性能数值模拟软件中心, 北京 100088
• 收稿日期:2021-10-12 发布日期:2022-06-10
• 通讯作者: 荆燕飞,Email:yanfeijing@uestc.edu.cn or 00jyfvictory@163.com.
• 基金资助:
科学挑战专题(TZ2016002),国家自然科学基金项目(12071062,61772003),电子科技大学理科实力提升计划资助

Xu Daqiang, Jing Yanfei, Hu Shaoliang, Xu Xiaowen. ADAPTIVE SGCRO-DR ALGORITHM FOR SOLVING SPARSE SEQUENCES OF LINEAR SYSTEMS[J]. Journal on Numerica Methods and Computer Applications, 2022, 43(2): 125-141.

### ADAPTIVE SGCRO-DR ALGORITHM FOR SOLVING SPARSE SEQUENCES OF LINEAR SYSTEMS

Xu Daqiang1, Jing Yanfei1, Hu Shaoliang2,3, Xu Xiaowen2,3

1. School of Mathematical Sciences/Institute of Computational Science, University of Electronic Science and Technology of China, Chengdu 611731, China;
2. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China;
3. CAEP Software Center for High Performance Numerical Simulation, Beijing, 100088, China
• Received:2021-10-12 Published:2022-06-10
GCRO-DR方法是求解一系列连续线性系统常用迭代方法.本文首先提出了~simpler GCRO-DR方法,在单个循环中它的计算成本比GCRO-DR更少.本文为了避免算法的不稳定性问题和内存溢出问题,并提高~simpler GGRO-DR算法的收敛性,引入了重启参数自适应策略.另外,在用该方法求解大型连续线性系统需要多次重启次数情形以及相邻系数矩阵之间谱信息相关情形,本文利用重启参数自适应策略提供的学习样本,通过强化学习来选取一个比较好的重启参数.最后,数值实验证明了所提三类算法的有效性.
The GCRO-DR method is considered as a frequently-used iterative solver for solving a series of linear systems given in sequence. In this paper, a simpler variant of GCRO-DR is firstly proposed, which takes less computational cost within one cycle than GCRO-DR. Furthermore, an adaptive strategy for setting restart parameters is introduced in order to overcome the non-stability problem and out-of-memory problem of the proposed simpler GGRO-DR method, as well as to improve its convergence. Moreover, in both circumstances that many restart numbers are required by such method for solving large linear systems given in sequence, and neighbouring coefficient matrices share relevant spectral information, a better restart parameter is determined by reinforcement learning based on the learning samples provided by the adaptive strategy for setting restart parameters. Finally, numerical experiments demonstrate the efficiency of the proposed three types of algorithms.

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