Numerical Approximation of Joint Spectral Radius by Lower Rank Matrix Sets

这一研究计划源于一项广泛的跨学科研究，将数学和计算机械结合起来，以解决计算科学领域中的一个长期存在的问题，即以一种有效的方式近似联合谱半径。与现有的方法不同，在本项目中，PI提出了一种通过矩阵秩分解的创新方法。其主要思想是通过秩分解的方法来降低计算复杂度，从而达到理想的精度。这项工作将在联合谱半径的理论研究和计算近似之间提供亟需的共同点，研究成果将有效地改进现有的文献方法，从而对应用科学，如工程和其他相关领域的系统设计和控制，以及对数学中一些长期悬而未决的问题有更好的理解。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，这些活动包括广泛的数学和数值分析培训、口头和书面交流指导、职业发展建议，以及在国内/国际会议上传播项目成果。这项研究项目的目标是建立误差界，并开发有效的算法，通过使用较低等级的矩阵集来计算和近似联合谱半径。PI将特别注意奇异值分解，尽管也将探索其他类型的分解，例如通过高斯消除获得的行梯形的等级分解。利用对称规范函数及其次微分的概念，通过秩分解逼近联合谱半径，并利用广义向量范数估计误差界。所提出的寻求理想规范函数的方法是基于等级近似，允许在每一步使其值最小化。将建立数值算法，并有望提供有效的计算工具，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.