学术报告

题      目： Numerical methods for the algebraic Riccati equation arising in transport theory in the critical case

报  告  人：郭晓霞   教授  (邀请人：黎稳 )

                                 中国海洋大学

时      间：2022-10-12 16:00-17:00

腾 讯 会 议：575 725 997

报告人简介:

       郭晓霞，中国海洋大学数学科学学院教授，2005年于中国科学院研究院计算数学与工程计算研究所毕业并获博士学位，一直从事矩阵计算的理论与方法研究，在非线性矩阵方程和图像处理的理论及数值方法方面取得了一些有意义的成果。曾主持并完成了国家自然科学基金项目两项和山东省自然科学基金项目一项,参与国家自然科学基金项目两项,在2008年入选了教育部的“新世纪优秀人才支持计划”，并在2018年以第一位次获得山东自然科学奖二等奖，在Numer.Math., SIAM J.Sci. Comput., Numer.Linear Algebra Appl., J.Comput.Math.等学术期刊上发表学术论文20余篇。

摘      要：

       In this paper, we are interested in computing the minimal positive solution  of an algebraic Riccati equation arising in transport theory. The coefficient matrices of this equation have two parameters $\alpha$ and $c$. When $0<\alpha< 1,0<c<1$, the existing numerical algorithms can solve its minimal  positive solution very quickly and efficiently. However, these algorithms don't converge or converge slowly to the minimal positive solution for $\alpha=0$ and $c=1$.  In this case, we propose two methods to improve the effectiveness of these algorithms. The first method is a modified double shift technique to transform the original algebraic Riccati equation into a new one, two equations have the same minimal positive solution. The second method is a deflating technique to transform the original equation into a new low-order algebraic Riccati equation, we  give the relationship between the minimal positive solutions of these two equations. The minimal positive solutions of two new equations both can be effectively solved by the existing algorithms. Numerical experiments are given to illustrate the results.