题目：Accurate Numerical Solution For Shifted M-Matrix Algebraic Riccati Equation
报告人：薛军工 教授 单位：复旦大学数学学院
时间：2022年4月12日（周二）晚上7:30-8:30
腾讯会议：977-479-306
报告人及报告内容摘要：
薛军工, 1996年于复旦大学获理学博士学位,专业为计算数学, 目前为复旦大学数学科学学院教授, 博士生导师, 副院长. 主要研究方向为计算金融、数值代数和排队论. 1998年获德国洪堡基金, 2004年入选教育部新世纪优秀人才支持计划, 2006年入选上海市浦江计划. 在计算数学、金融数学和运筹学等领域国际上有影响的杂志, 如 Numer. Math., Math. Comp., SIAM Finan. Math, SIAM J. Matrix Anal. Appl. , IMA Numer. Anal., INFORMS J. Computing, Queueing System, J. Appl. Prob.等发表多篇论文.
Abstract: An algebraic Riccati (ARE) equation is called a shifted $M$-matrix algebraic Riccati equation (MARE) if it can be turned into an MARE after its matrix variable is partially shifted by a diagonal matrix. Such an ARE can arise from computing the invariant density of a Markov modulated Brownian motion. Sufficient and necessary conditions for an ARE to be a shifted MARE are obtained. Based on the condition, a highly accurate implementation of the alternating directional doubling algorithm (ADDA) is established to compute the extremal solution of a shifted \MARE, as well as a quantity needed for computing the invariant density in the application, with high entrywise relative accuracy. Numerical examples are presented to demonstrate the theory and algorithms.