Program
Computational & Applied Mathematics (CAM) Seminar
Student No.:40
Time:Tue 15:20-16:55, Oct.9-Jan.15
Instructor:Shi Zuoqiang, Jing Wenjia  
Place:Conference Room 3, Jin Chun Yuan West Bldg.
Starting Date:2018-10-9
Ending Date:2019-1-15

2018-12-20

Time: Tue 9:50-11:25, Dec.20

Speaker: Guodong Pang (Penn State University)

Title: Subexponential rate of convergence of a class of Levy-driven SDEs and applications in stochastic networks

Abstract: We study the ergodic properties of a class of multidimensional piecewise Ornstein--Uhlenbeck processes with jumps. They include the scaling limits arising from stochastic networks with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these models, the Ito equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Levy process, or (2) an anisotropic Levy process with independent one-dimensional symmetric \alpha-stable components, or (3) an anisotropic Levy process as in (2) and a pure-jump Levy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) \alpha-stable Levy process as a special case. We identify conditions on the parameters in the drift, the Levy measure and/or covariance function which result in subexponential and/or exponential ergodicity. We show that these assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. For the stochastic network models, we show that the rate of convergence is polynomial and provide a sharp quantitative characterization of the rate via matching upper and lower bounds.


2018-10-16

Speaker: Wen Huang (厦门大学)

Title: Riemannian optimization and averaging symmetric positive definte matrices

Abstract: Symmetric positive definite matrices have become fundamental computational objects in many areas. It is often of interest to average a collection of symmetric positive definite matrices. In this presentation, we investigate different averaging techniques for symmetric positive definite matrices. We use recent developments in Riemannian optimization to develop efficient and robust algorithms to handle this computational task. We provide methods to produce efficient numerical representations of geometric objects that are required for Riemannian optimization methods on the manifold of symmetric positive definite matrices. In addition, we offer theoretical and empirical suggestions on how to choose between various methods and parameters. In the end, we evaluate the performance of different averaging techniques in applications. This is joint work with Xinru Yuan, Pierre-Antoine Absil and Kyle A. Gallivan.



2018-10-9

Speaker: Xin Liu (中科院计算数学所)

Title: A Continuous Optimization Model for Clustering

Abstract: We study the problem of clustering a set of objects into groups according to a certain measure of similarity among the objects. This is one of the basic problems in data processing with various applications ranging from computer science to social analysis. We propose a new continuous model for this problem, the idea being to seek a balance between maximizing the number of clusters and minimizing the similarity among the objects from distinct clusters. Adopting the methodology of spectral clustering, our model quantifies the number of clusters via the rank of a graph Laplacian, and then relaxes rank minimization to trace minimization with orthogonal constraints. We analyze the properties of our model, propose a block coordinate descent algorithm for it, and establish the global convergence of the algorithm. We then demonstrate our model and algorithm by several numerical examples.