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.
