Xiaoqing Jin

Teng-Teng Yao, Zheng-Jian Bai, Xiao-Qing Jin et al.

This paper is concerned with the least squares inverse eigenvalue problem of reconstructing a linear parameterized real symmetric matrix from the prescribed partial eigenvalues in the sense of least squares, which was originally proposed by Chen and Chu [SIAM J. Numer. Anal., 33 (1996), pp. 2417--2430]. We provide a Riemannian inexact Gausss-Newton method for solving the least squares inverse eigenvalue problem. The global and local convergence analysis of the proposed method is discussed. Also, a preconditioned conjugate gradient method with an efficient preconditioner is proposed for solving the Riemannian Gauss-Newton equation. Finally, some numerical tests, including an application in the inverse Sturm-Liouville problem, are reported to illustrate the efficiency of the proposed method.

Chun-Yueh Chiang, Matthew M. Lin, Xiao-Qing Jin

Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This result by Weyl-Horn then leads to an interesting inverse problem, i.e., how to construct a matrix with desired eigenvalues and singular values. In this work, we do that and more. We propose an eclectic mix of techniques from differential geometry and the inexact Newton method for solving inverse eigenvalue and singular value problems as well as additional desired characteristics such as nonnegative entries, prescribed diagonal entries, and even predetermined entries. We show theoretically that our method converges globally and quadratically, and we provide numerical examples to demonstrate the robustness and accuracy of our proposed method. {Having theoretical interest, we provide in the appendix a necessary and sufficient condition for the existence of a $2\times 2$ real matrix, or even a nonnegative matrix, with prescribed eigenvalues, singular values, and main diagonal entries.

Zhi Zhao, Zheng-Jian Bai, Xiao-Qing Jin

This paper is concerned with the nonnegative inverse eigenvalue problem of finding a nonnegative matrix such that its spectrum is the prescribed self-conjugate set of complex numbers. We first reformulate the nonnegative inverse eigenvalue problem as an under-determined constrained nonlinear matrix equation over several matrix manifolds. Then we propose a Riemannian inexact Newton-CG method for solving the nonlinear matrix equation. The global and quadratic convergence of the proposed method is established under some mild conditions. We also extend the proposed method to the case of prescribed entries. Finally, numerical experiments are reported to illustrate the efficiency of the proposed method.

Marcell Vazquez-Chanlatte, Jyotirmoy V. Deshmukh, Xiaoqing Jin et al.

To effectively analyze and design cyberphysical systems (CPS), designers today have to combat the data deluge problem, i.e., the burden of processing intractably large amounts of data produced by complex models and experiments. In this work, we utilize monotonic Parametric Signal Temporal Logic (PSTL) to design features for unsupervised classification of time series data. This enables using off-the-shelf machine learning tools to automatically cluster similar traces with respect to a given PSTL formula. We demonstrate how this technique produces interpretable formulas that are amenable to analysis and understanding using a few representative examples. We illustrate this with case studies related to automotive engine testing, highway traffic analysis, and auto-grading massively open online courses.

Jyotirmoy V. Deshmukh, Alexandre Donzé, Shromona Ghosh et al.

Signal Temporal Logic (STL) is a formalism used to rigorously specify requirements of cyberphysical systems (CPS), i.e., systems mixing digital or discrete components in interaction with a continuous environment or analog com- ponents. STL is naturally equipped with a quantitative semantics which can be used for various purposes: from assessing the robustness of a specification to guiding searches over the input and parameter space with the goal of falsifying the given property over system behaviors. Algorithms have been proposed and implemented for offline computation of such quantitative semantics, but only few methods exist for an online setting, where one would want to monitor the satisfaction of a formula during simulation. In this paper, we formalize a semantics for robust online monitoring of partial traces, i.e., traces for which there might not be enough data to decide the Boolean satisfaction (and to compute its quantitative counterpart). We propose an efficient algorithm to compute it and demonstrate its usage on two large scale real-world case studies coming from the automotive domain and from CPS education in a Massively Open Online Course (MOOC) setting. We show that savings in computationally expensive simulations far outweigh any overheads incurred by an online approach.