Numerical Approximation of Joint Spectral Radius by Lower Rank Matrix Sets

该研究计划源于广泛的跨学科研究，结合数学和计算机器来解决计算科学领域的一个长期存在的问题，即，以有效的方式近似联合谱半径。与目前的方法不同，在这个项目中，PI提出了一个创新的方法，通过矩阵秩分解。其主要思想是通过秩分解的方法来降低计算复杂度，以达到理想的精度。本文的工作将为联合谱半径的理论研究和计算近似提供急需的共同基础，研究结果将有效地改进文献中的现有方法，从而对工程中的系统设计和控制等相关领域的应用科学产生有价值的影响，并有助于更好地理解一些长期存在的数学开放问题。该项目将为研究生和本科生提供指导和培训。学生的专业发展将通过辅导活动得到加强，这些活动包括数学和数值分析方面的广泛培训，口头和书面交流方面的指导，职业发展方面的建议，在国家一级传播项目成果本研究的目标是建立误差界，并开发有效的算法，通过使用联合谱半径的计算和近似低秩矩阵集PI将特别注意奇异值分解，尽管也将探索其他类型的分解，例如从高斯消除获得的行梯队形式的秩分解。利用对称规范函数及其次微分的概念，通过秩分解近似联合谱半径，并利用一般向量范数估计误差界。所提出的寻求一个理想的规范函数的程序是基于秩近似，允许其值在每一步的最小化。数值算法将被建立，并有望提供有效的计算工具，以应用于工程和相关领域。

This research program originates from a broad interdisciplinary study, combining mathematical and computational machineries to address a long-standing problem in the field of computational science, i.e., to approximate the joint spectral radius in an efficient way. Different from current approaches, in this project, the PIs propose an innovative approach via matrix rank decomposition. The main idea is to reduce the computational complexity through rank decomposition methodology so that desirable precision can be achieved. The proposed work will provide much needed common ground between theoretical study and computational approximation of joint spectral radius, and the research findings would effectively improve the existing approaches in literature, which in turn will lead to a valuable impact on applied sciences, such as system design and control in engineering and other related fields, as well as to better understandings of some long-standing open problems in mathematics. This project will provide mentorship and training for graduate and undergraduate students. Professional development of the students will be enhanced through mentoring activities which include broad training in mathematics and numerical analysis, guidance in oral and written communication, advice on career development, and dissemination of project results at national/international conferences.The goal of this research project is to establish error bounds and to develop efficient algorithms for the computation and approximation of joint spectral radius by using lower rank matrix sets. The PIs will pay particular attention to the singular value decomposition, although other types of decompositions will also be explored, such as the rank decomposition from row echelon form obtained from Gaussian elimination. By using the concept of symmetric gauge function and its subdifferential, the joint spectral radius will be approximated through rank decompositions and the error bounds will be estimated by using generic vector norms. The proposed procedure for seeking a desirable gauge function is based on the rank approximation, allowing the minimization of its value at each step. Numerical algorithms will be established and are expected to provide effective computational tools to be applied to engineering and related fields.