Wednesdays at 4:15 PM in 380C.
A natural optimization model that formulates many online resource allocation and revenue management problems is the online linear program (LP) where the constraint matrix, along with the objective coefficients, are revealed column by column. We discuss optimal algorithms for solving this surprisingly general class of online problems under the assumption of random order of arrival and some conditions on the data and size of the problem. These algorithms have a feature of "learning while doing" by dynamically updating a threshold price vector at certain time intervals, where the dual prices learned from revealed columns in the previous period are used to determine the sequential decisions in the current period. In particular, the algorithms don't assume any distribution information on the input itself, thus it is robust to data uncertainty and variations due to its dynamic learning capability. Applications include many online multi-resource allocation and multi-product revenue management problems such as online routing and packing, online combinatorial auctions, display advertizing, etc.
Understanding the coupling of physical models at different scales is important and quite challenging. In this talk, we focus on the issue of kinetic-fluid coupling, in particular, the half-space problems for kinetic equations coming from the boundary layer. We will present some recent progress in algorithm development and analysis for the linear half-space kinetic equations, and its application in coupling of neutron transport equations with diffusion equations. (joint work with Qin Li and Weiran Sun)
We will consider systems of stochastic delay differential equations driven by colored noise. The two important time scales characterizing such system are the delay time and the correlation time of the noise. We will study perturbatively the limiting behavior of the solutions as these two time scales become small, showing the emergence of a noise-induced drift in the effective equation describing the limit. The results are confirmed by a noisy electrical circuit experiment. This is a joint work with experimental physicists Giuseppe Pesce and Giovanni Volpe and with graduate students Scott Hottovy and Austin McDaniel.
The ability of animals to self-organize into remarkable patterns of movement is seen throughout nature from herds of large mammals, to flocks of birds, schools of fish, and swarms of insects. Remarkably, patterns of collective movement can be seen even in the simplest forms of life such as bacteria. M. xanthus are common soil bacteria that are among the most "social" bacteria in nature. In this talk clustering mechanism of swarming M. xanthus will be described using combination of experimental movies and stochastic model simulations. Continuous limits of discrete stochastic dynamical systems simulating cell aggregation will be described in the form of reaction-diffusion and nonlinear diffusion equations. Surface motility such as swarming is thought to precede biofilm formation during infection. Population of bacteria P. aeruginosa, major infection in hospitals, will be shown to efficiently propagate as high density waves that move symmetrically as rings within swarms towards the extending tendrils. Multi-scale model simulations suggest a mechanism of wave propagation as well as branched tendril formation at the edge of the population that depend upon competition between the changing viscosity of the bacterial liquid suspension and the liquid film boundary expansion caused by Marangoni forces. This collective mechanism of cell- cell coordination was recently shown to moderate swarming direction of individual bacteria to avoid a toxic environment.
Abstract: This talk compares local and global conditions for polynomial optimization problems. First, we review the classical local optimality conditions: constraint qualification, strict complementarity and second order sufficiency conditions. We show that they are always satisfied, except a zero measure set of input data. Second, we review global optimality conditions that are expressed by sum-of-squares type representations. We show that if the above classical local optimality conditions hold, then the sum-of-squares type global optimality conditions must be satisfied. Third, we review Lasserre's hierarchy for solving polynomial optimization, and show that it always has finite convergence, except a zero measure set of input data.
Linear response eigenvalue problems (LREPs) arise from the study of collective motion of many particle systems, such as excited states of electronic structures. In this talk, we will present recent theoretical results on variational principles of LREPs and discuss conjugate gradient-like algorithms. Numerical results of large-scale LREPs for computing multiple low-lying excitation energies of molecules by the time-dependent density functional theory will be presented. This is a joint work with Ren-cang Li, Dario Rocca and Giulia Galli.
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