Seminars and Colloquia by Series

Semi-Infinite Relaxations for a Dynamic Knapsack Problem with Stochastic Item Sizes

Series
ACO Student Seminar
Time
Friday, October 30, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alejandro Toriello – Georgia Tech
We consider a version of the knapsack problem in which an item size is random and revealed only when the decision maker attempts to insert it. After every successful insertion the decision maker can choose the next item dynamically based on the remaining capacity and available items, while an unsuccessful insertion terminates the process. We propose a new semi-infinite relaxation based on an affine value function approximation, and show that an existing pseudo-polynomial relaxation corresponds to a non-parametric value function approximation. We compare both theoretically to other relaxations from the literature and also perform a computational study. Our new relaxation provides tight bounds over a variety of different instances and surprisingly becomes tighter as the number of items increases. Joint work with Daniel Blado (ACO) and Weihong Hu (ISyE).

Thermostated Kac Models

Series
Dissertation Defense
Time
Friday, October 30, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ranjini Vaidyanathan – School of Mathematics, Georgia Tech

Please Note: Advisor: Dr. Federico Bonetto

We consider a model of N particles interacting through a Kac-style collision process, with m particles among them interacting, in addition, with a thermostat. When m = N, we show exponential approach to the equilibrium canonical distribution in terms of the L2 norm, in relative entropy, and in the Gabetta-Toscani-Wennberg (GTW) metric, at a rate independent of N. When m < N , the exponential rate of approach to equilibrium in L2 is shown to behave as m/N for N large, while the relative entropy and the GTW distance from equilibrium exhibit (at least) an "eventually exponential” decay, with a rate scaling as m/N^2 for large N. As an allied project, we obtain a rigorous microscopic description of the thermostat used, based on a model of a tagged particle colliding with an infinite gas in equilibrium at the thermostat temperature. These results are based on joint work with Federico Bonetto, Michael Loss and Hagop Tossounian.

Construction of whiskered invariant tori for fibered holomorphic dynamics II

Series
Dynamical Systems Working Seminar
Time
Thursday, October 29, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mikel Viana – Georgia Tech (Math)
We consider fibered holomorphic dynamics, generated by a skew product over an irrational translation of the torus. The invariant object that organizes the dynamics is an invariant torus. Often one can find an approximately invariant torus K_0, and we construct an invariant torus, starting from K_0. The main technique is a KAM iteration in a-posteriori format. In this talk we give the details of the iterative procedure using the geometric and number-theoretic conditions presented last time.

Recent Berry-Esseen bounds obtained with Stein's method and Poincare inequalities, with Geometric applications

Series
Stochastics Seminar
Time
Thursday, October 29, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Raphael Lachieze-Rey – University of Southern California
Recently, new general bounds for the distance to the normal of a non-linear functional have been obtained, both with Poisson input and with IID points input. In the Poisson case, the results have been obtained by combining Stein's method with Malliavin calculus and a 'second-order Poincare inequality', itself obtained through a coupling involving Glauber's dynamics. In the case where the input consists in IID points, Stein's method is again involved, and combined with a particular inequality obtained by Chatterjee in 2008, similar to the second-order Poincar? inequality. Many new results and optimal speeds have been obtained for some Euclidean geometric functionals, such as the minimal spanning tree, the Boolean model, or the Voronoi approximation of sets.

Some algebraic techniques in the numerical analysis of ordinary differential equations

Series
Applied and Computational Mathematics Seminar
Time
Thursday, October 29, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Philippe Chartier – INRIA Rennes, Université de Rennes I, ENS Rennes –

Please Note: Joint with School of Math Colloquium. Special time (colloquium time).

In this talk, I will introduce B-series, which are formal series indexed by trees, and briefly expose the two laws operating on them. The presentation of algebraic aspects will here be focused on applications to numerical analysis. I will then show how B-series can be used on two examples: modified vector fields techniques, which allow for the construction of arbitrarly high-order schemes, and averaging methods, which lie at the core of many numerical schemes highly-oscillatory evolution equations. Ultimately and if time permits, I will illustrate how these concepts lead to the accelerated simulation of the rigid body and the (nonlinear) Schrödinger equations. A significant part of the talk will remain expository and aimed at a general mathematical audience.

Reflectionless Measures for Singular Integral Operators

Series
Analysis Seminar
Time
Wednesday, October 28, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Benjamin Jaye – Kent State University –
We shall describe how the study of certain measures called reflectionless measures can be used to understand the behaviour of oscillatory singular integral operators in terms of non-oscillatory quantities. The results described are joint work with Fedor Nazarov, Maria Carmen Reguera, and Xavier Tolsa

Mechanisms of Chaos

Series
Research Horizons Seminar
Time
Wednesday, October 28, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Leonid Bunimovich – School of Mathematics, Georgia Institute of Technology –

Please Note: Food and Drinks will be provided before the seminar

In this seminar,we will explain why and how unpredictable (chaotic) dynamics arises in deterministic systems. Some open problems in dynamical systems, probability, statistical mechanics, optics, (differential) geometry and number theory will be formulated.

Generalized Dantzig Selector: Application to the k-support norm

Series
High-Dimensional Phenomena in Statistics and Machine Learning Seminar
Time
Tuesday, October 27, 2015 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 249
Speaker
Changong Li – Georgia Inst. of Technology, School of Mathematics

Please Note: Review of a recent paper by Chatterjee et al. (Arxiv 1406.5291)

We propose a Generalized Dantzig Selector (GDS) for linear models, in which any norm encoding the parameter structure can be leveraged for estimation. We investigate both computational and statistical aspects of the GDS. Based on conjugate proximal operator, a flexible inexact ADMM framework is designed for solving GDS, and non-asymptotic high-probability bounds are established on the estimation error, which rely on Gaussian width of unit norm ball and suitable set encompassing estimation error. Further, we consider a non-trivial example of the GDS using k-support norm. We derive an efficient method to compute the proximal operator for k-support norm since existing methods are inapplicable in this setting. For statistical analysis, we provide upper bounds for the Gaussian widths needed in the GDS analysis, yielding the first statistical recovery guarantee for estimation with the k-support norm. The experimental results confirm our theoretical analysis.

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