数学讲坛
发布于:2017-07-07 15:10:32   |   作者:[学院] 数学学院   |   浏览次数:472

报告题目(一):Ubiquitous Doubling Algorithms, General Theory, and Applications

报告时间:7月17日(星期一)14:30-15:30

    人:Ren-Cang Li

报告地点:清水河校区主楼A1-512 (数学学院会议室)

 

Abstact:

Iterative methods are widely and indispensably used in numerical approximations. Basically, any iterative method is a rule that produces a sequence of approximations and with a reasonable expectation that newer approximations in the sequence are better. The goal of a doubling algorithm is to significantly speed up the approximation process by seeking ways to  skip computing most of the approximations in the sequence but sporadically few, in fact, extremely very few: only the $2^i$-th approximations in the sequence, kind of like computing $\alpha^{2^i}$ via repeatedly squaring. However, this idea is only worthwhile if there is a much cheaper way to directly obtain the $2^i$-th approximation from the $2^{i-1}$-st one  than simply following the rule to generate every approximation between the $2^{i-1}$-st and $2^i$-th approximations in order to obtain the $2^i$-th approximation. Anderson (1978) had sought the idea to speed up the simple fixed point iteration for solving the discrete-time algebraic Riccati equation via repeatedly compositions of the fixed point iterative function. As can be imagined, under repeatedly compositions, even a simple function can usually and quickly turn into nonetheless a complicated and unworkable one. In the last 20 years or so in large part due to an extremely elegant way of formulation and analysis, the researches in doubling algorithms thrived and continues to be very active, leading to numerical effective and robust algorithms not only for the continuous-time and discrete-time algebraic Riccati equations from optimal control that motivated the researches  in the first place but also for $M$-matrix algebraic Riccati equations (MARE), structured eigenvalue problems, and other nonlinear matrix equations. But the resulting theory is somewhat fragmented and sometimes ad hoc. In this talk, we will seek to provide a general and coherent theory, discuss new highly accurate doubling algorithm for MARE, and look at several important applications.

 

报告题目(二):A block term decomposition of third order tensors

报告时间:7月17日(星期一)15:30-16:30

    人:Ren-Cang Li

报告地点:清水河校区主楼A1-512 (数学学院会议室)

 

Abstract:

Matrix joint block diagonalization problem is a particular block term decomposition of third order tensors, which  has many important applications in blind source separation and independent component analysis. In this talk, I will talk about two classes of algebraic methods for JBD problem: one is based on matrix polynomial, the other is based on matrix commutation. Existence and uniqueness conditions for JBD problem are discussed. Numerical methods are developed and validated by simulating examples.

 

报告人简介:Li Rencang现任美国德克萨斯大学惠灵顿分校教授,于1985年在厦门大学计算数学专业本科毕业,1988年中科院计算数学所获得硕士学位,1995年在美国加州大学伯克利分校获得应用数学博士学位。主要研究兴趣包含高性能计算、常微分方程数值求解、数值代数和模型降阶。曾担任知名期刊SIAM Journal on Matrix Analysis and Applications 的编委,现任Mathematical Communications和Numerical Algebra, Control and Optimization期刊的副主编,在SIAM Journal on Matrix Analysis and Applications、Linear Algebra and its Applications等重要学术期刊上发表论文八十多篇。