谢家新 副教授 (北京航空航天大学 )

陈亮 工程师

Connecting randomized iterative methods with Krylov subspaces

9月23日（星期二）15:30-16:30, N202

Randomized iterative methods, such as the randomized Kaczmarz method, have gained significant attention for solving large-scale linear systems due to their simplicity and efficiency. Meanwhile, Krylov subspace methods have emerged as a powerful class of algorithms, known for their robust theoretical foundations and rapid convergence properties. Despite the individual successes of these two paradigms, their underlying connection has remained largely unexplored. In this talk, we develop a unified framework that bridges randomized iterative methods and Krylov subspace techniques, supported by both rigorous theoretical analysis and practical implementation. The core idea is to formulate each iteration as an adaptively weighted linear combination of the sketched normal vector and previous iterates, with the weights optimally determined via a projection-based mechanism. This formulation not only reveals how subspace techniques can enhance the efficiency of randomized iterative methods, but also enables the design of a new class of iterative-sketching-based Krylov subspace algorithms. We  prove that our method converges linearly in expectation and validate our findings with numerical experiments. The arXiv link: https://arxiv.org/abs/2505.20602.

报告人简介：谢家新，北京航空航天大学数学科学学院副教授，博士生导师，2017年获湖南大学计算数学博士学位。研究兴趣为数据科学中的数学问题，特别是随机迭代法及其加速技术等问题。已在SIOPT, SIMAX, JCM, COAP, IJM等期刊发表论文多篇。主持国家重点研发计划子课题、国家自然科学基金面上项目等项目。现为中国运筹学会算法软件及应用分会理事，中国运筹学会数学规划分会青年理事。