Georgia Institute of Technology College of Sciences

Mathematical imaging is not only a multidisciplinary research area but also a major cross-disciplinesubject within mathematical sciences as image analysis techniques involve analysis, optimization, differential geometry and nonlinear partial differential equations, computational algorithms and numerical analysis.In this talk I first review various models and techniques in the variational frameworkthat are used for restoration of images. Then I discuss more recent work on i) choice of optimal coupling parameters for the TV model,ii) the blind deconvolution and iii) high order regularization models.This talk covers joint work with various collaborators in imaging including J. P. Zhang, T.F. Chan, R. H. Chan, B. Yu, L. Sun, F. L. Yang (China), C. Brito (Mexico), N. Chumchob (Thailand), M. Hintermuller (Germany), Y. Q. Dong (Denmark), X. C. Tai (Norway) etc.

Initially homogeneous vehicular traffic flow can become inhomogeneous even in the absence of obstacles. Such phantom traffic jams'' can be explained as instabilities of a wide class of second-order'' macroscopic traffic models. In this unstable regime, small perturbations amplify and grow into nonlinear traveling waves. These traffic waves, called jamitons'', are observed in reality and have been reproduced experimentally. We show that jamitons are analogs of detonation waves in reacting gas dynamics, thus creating an interesting link between traffic flow, combustion, water roll waves, and black holes. This analogy enables us to employ the Zel'dovich-von Neumann-Doering theory to predict the shape and travel velocity of the jamitons. We furthermore demonstrate that the existence of jamiton solutions can serve as an explanation for multi-valued parts that fundamental diagrams of traffic flow are observed to exhibit.

The symmetric configurations for the equilibrium shape of a fluid interfaceare given by the geometric differential equation mean curvature isproportional to height. The equations are explored numerically tohighlight the differences in classically treated capillary tubes andsessile drops, and what has recently emerged as annular capillary surfaces. Asymptotic results are presented.

Dynamical processes, such as information diffusion in social networks, gene regulation in biological systems and functional collaborations between brain regions, generate a large volume of high dimensional “asynchronous” and “interdependent” time-stamped event data. This type of timing information is rather different from traditional iid. data and discrete-time temporal data, which calls for new models and scalable algorithms for learning, analyzing and utilizing them. In this talk, I will present methods based on multivariate point processes, high dimensional sparse recovery, and randomized algorithms for addressing a sequence of problems arising from this context. As a concrete example, I will also present experimental results on learning and optimizing information cascades in web logs, including estimating hidden diffusion networks and influence maximization with the learned networks. With both careful model and algorithm design, the framework is able to handle millions of events and millions of networked entities.