江苏省应用数学（中国矿业大学）中心系列学术报告
报告题目：Mixed Precision General Alternating-Direction Implicit Method for Solving Large Sparse Linear Systems
报 告 人：张娟 教授 单位：湘潭大学
报告时间：2025年12月20日（周六）上午 10:00--11:00
报告地点：数学学院A321
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数学学院
报告人简介：
张娟，教授，博士生导师，湘潭大学数学与计算科学学院副院长，“智能计算与信息处理”教育部重点实验室常务副主任。入选湖南省湖湘青年英才，湖南省青年骨干教师培养对象。多次赴新加坡国立大学、澳门大学访问。主持国自科面上、青年项目，博士后基金面上项目一等资助，湖南省教育厅重点项目等国家级省部级项目10余项。作为子课题负责人承担国家重点研发计划、军科委GF项目、工业软件内核研发及应用验证产业基础共性技术中心项目。主要从事数值代数、控制理论、矩阵计算等方面的研究。近五年在计算数学、控制领域权威期刊SIAM J. Numer. Anal.、SIAM J. Sci. Comput.、Automatica、J. Comput. Phys.、J. Sci. Comput.、CSIAM Tran. Appl. Math.发表和接收发表SCI论文20余篇。
Abstract:In this work, we introduce a three-precision formulation of the General Alternating-Direction Implicit method (GADI) designed to accelerate the solution of large-scale sparse linear systems Ax=b. GADI is a framework that can represent many existing Alternating-Direction Implicit (ADI) methods. With our proposed mixed precision layout for GADI, we aim to solve the structured subsystems in low precision to reduce the overall execution time while computing the residual and solution update in high precision to enable the solution to converge to high accuracy. We develop a rounding error analysis of mixed precision GADI that establishes the rates of convergence of the forward and normwise backward errors to certain limiting accuracies. Our analysis also highlights the conditions on the splitting matrices under which mixed precision GADI is guaranteed to converge for a given set of precisions. We then discuss a systematic and robust strategy for selecting the GADI regularization parameter α, whose adjustment is critical for performance. Specifically, our proposed strategy makes use of a Gaussian Process Regression (GPR) model trained on a dataset of low-dimensional problems to initialize α. Numerical experiments validate our approach, demonstrating that mixed precision GADI achieves a significant speedup of approximately 2.59x on a two-dimensional convection-diffusion problem when using Bfloat16 precision, compared to a full double-precision implementation.