Seminars and Colloquia Schedule

Polytopal Element Methods in Mathematics and Engineering

Series
Other Talks
Time
Monday, October 26, 2015 - 09:05 for 8 hours (full day)
Location
Student Center Theater, Georgia Tech
Speaker
Various speakers – Georgia Tech
The workshop will be held from Monday October 26 - Wednesday October 28, 2015. The purpose of this workshop is to promote communication among the many mathematical and engineering communities currently researching polytopal discretization methods for the numerical approximation of solutions of partial differential equations. A variety of distinct polytopal element methods (POEMs) have been designed to solve the same types of problems, but a workshop-type environment is required to foster a community-wide understanding of the comparative advantages of each technique and to develop a set of ‘best practices’ in regards to implementation. Registration is required.

Calculation of a Power Price Equilibrium under Risk Averse Trading

Series
Other Talks
Time
Monday, October 26, 2015 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Raphael Hauser – Mathematical Institute, University of Oxford
We propose a term structure power price model that, in contrast to widely accepted no-arbitrage based approaches, accounts for the non-storable nature of power. It belongs to a class of equilibrium game theoretic models with players divided into producers and consumers. The consumers' goal is to maximize a mean-variance utility function subject to satisfying an inelastic demand of their own clients (e.g households, businesses etc.) to whom they sell the power. The producers, who own a portfolio of power plants each defined by a running fuel (e.g. gas, coal, oil...) and physical characteristics (e.g. efficiency, capacity, ramp up/down times...), similarly, seek to maximize a mean-variance utility function consisting of power, fuel, and emission prices subject to production constraints. Our goal is to determine the term structure of the power price at which production matches consumption. We show that in such a setting the equilibrium price exists and discuss the conditions for its uniqueness. The model is then extended to account for transaction costs and liquidity considerations in actual trading. Our numerical simulations examine the properties of the term structure and its dependence on various model parameters. We then further extend the model to account for the startup costs of power plants. In contrast to other approaches presented in the literature, we incorporate the startup costs in a mathematically rigorous manner without relying on ad hoc heuristics. Through numerical simulations applied to the entire UK power grid, we demonstrate that the inclusion of startup costs is necessary for the modeling of electricity prices in realistic power systems. Numerical results show that startup costs make electricity prices very spiky. In a final refinement of the model, we include a grid operator responsible for managing the grid. Numerical simulations demonstrate that robust decision making of the grid operator can significantly decrease the number and severity of spikes in the electricity price and improve the reliability of the power grid.

Uniqueness of seismic inverse source problems modeling microseismicity and ruptures

Series
Applied and Computational Mathematics Seminar
Time
Monday, October 26, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Professor Maarten de Hoop – Rice University –
We consider an inverse problem for an inhomogeneous wave equation with discrete-in-time sources, modeling a seismic rupture. We assume that the sources occur along an unknown path with subsonic velocity, and that data is collected over time on some detection surface. We explore the question of uniqueness for these problems, and show how to recover the times and locations of sources microlocally first, and then the smooth part of the source assuming that it is the same at each source location. In case the sources (now all different) are (roughly speaking) non-negative and of limited oscillation in space, and sufficiently separated in space-time, which is a model for microseismicity, we present an explicit reconstruction, requiring sufficient local energy decay. (Joint research with L. Oksanen and J. Tittelfitz)

Triangulation independent Ptolemy varieties

Series
Geometry Topology Seminar
Time
Monday, October 26, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 270
Speaker
Christian Zickert – University of Maryland –
The Ptolemy variety is an invariant of a triangulated 3-manifoldM. It detects SL(2,C)-representations of pi_1(M) in the sense that everypoint in the Ptolemy variety canonically determines a representation (up toconjugation). It is closely related to Thurston's gluing equation varietyfor PSL(2,C)-representations. Unfortunately, both the gluing equationvariety and the Ptolemy variety depend on the triangulation and may missseveral components of representations. We discuss the basic properties ofthese varieties, how to compute invariants such as trace fields and complexvolume, and how to obtain a variety, which is independent of thetriangulation.

Repairing tropical curves by means of tropical modifications

Series
Algebra Seminar
Time
Monday, October 26, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Maria Angelica Cueto – The Ohio State University
Tropical geometry is sensitive to embeddings of algebraic varieties inside toric varieties. In this talk, I will advertise tropical modifications as a tool to locally repair bad embeddings of plane curves, allowing the re-embedded tropical curve to better reflect the geometry of the input one. Our motivating examples will be plane elliptic cubics and genus two hyperelliptic curves. Based on joint work with Hannah Markwig (arXiv:1409.7430) and ongoing work in progress with Hannah Markwig and Ralph Morrison.

Seismic inverse problems

Series
IMPACT Distinguished Lecture
Time
Tuesday, October 27, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Professor Maarten de Hoop – Rice University –
We give a brief analysis of the oscillations of the earth and then extract the system of equations describing acousto-elastic, seismic waves. Processes in Earth's interior are encoded in the coefficients of this system, which also parametrize its structure and material properties. We introduce the seismic inverse problem with its different aspects including a dual time-frequency point of view. Central in the analysis is the formulation as an inverse boundary value problem with the Dirichlet-to-Neumann map or Neumann-to-Dirichlet map as the data. We discuss various conditional Lipschitz stability estimates for this problem for coefficients containing discontinuities, and with partial boundary data, which involves the introduction of an unstructured tetrahedral mesh. Quantitative estimates of the stability constants play acritical role in analyzing convergence for iterative reconstruction schemes, making use of Hausdorff warping and leading to a multilevel approach requiring hierarchical, multi-scale compression. We present computational experiments on the regional and geophysical exploration scales. We conclude with some results pertaining to the high-frequency inverse boundary value or geometric inverse problems, again, in the presence of discontinuities.

Relative Entropy Relaxations for Signomial Optimization

Series
Applied and Computational Mathematics Seminar
Time
Tuesday, October 27, 2015 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Venkat Chandrasekaran – Cal Tech
Due to its favorable analytical properties, the relative entropy function plays a prominent role in a variety of contexts in information theory and in statistics. In this talk, I'll discuss some of the beneficial computational properties of this function by describing a class of relative-entropy-based convex relaxations for obtaining bounds on signomials programs (SPs), which arise commonly in many problems domains. SPs are non-convex in general, and families of NP-hard problems can be reduced to SPs. By appealing to representation theorems from real algebraic geometry, we show that sequences of bounds obtained by solving increasingly larger relative entropy programs converge to the global optima for broad classes of SPs. The central idea underlying our approach is a connection between the relative entropy function and efficient proofs of nonnegativity via the arithmetic-geometric-mean inequality. (Joint work with Parikshit Shah.)

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

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.

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 –

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.

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

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 –

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.

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.

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.

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

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.

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).