Seminars and Colloquia by Series

Monday, April 10, 2017 - 14:00 , Location: Skiles 005 , Elisabetta Matsumoto , GT Physics , Organizer: Sung Ha Kang

The properties of euclidean space seem natural and obvious to us, to thepoint that it took mathematicians over two thousand years to see analternative to Euclid’s parallel postulate. The eventual discovery ofhyperbolic geometry in the 19th century shook our assumptions, revealingjust how strongly our native experience of the world blinded us fromconsistent alternatives, even in a field that many see as purelytheoretical. Non-euclidean spaces are still seen as unintuitive and exotic,but with direct immersive experiences we can get a better intuitive feel forthem. The latest wave of virtual reality hardware, in particular the HTCVive, tracks both the orientation and the position of the headset within aroom-sized volume, allowing for such an experience. We use this nacenttechnology to explore the three-dimensional geometries of theThurston/Perelman geometrization theorem. This talk focuses on oursimulations of H³ and H²×E.

Monday, April 3, 2017 - 14:00 , Location: Skiles 005 , Prof. Michael Muskulus , NTNU: Norwegian University of Science and Technology , michael.muskulus@ntnu.no , Organizer:

This talk addresses an important problem in arctic engineering due to interesting dynamic phenomena in a forced linear system. The nonautonomous system considered is representative of a whole class of engineering problems that are not approachable by standard techniques from dynamical system theory.The background are ice-induced vibrations of structures (e.g. wind turbines or measurement masts) in regions with active sea ice. Ice is a complex material and the mechanism for ice-induced vibrations is not fully clear at present. In particular, the conditions under which the observed, qualitatively different vibration regimes are active cannot be predicted accurately so far. A recent mathematical model developed by Delft University of Technology assumes that a number of parallel ice strips are pushing with a constant velocity against a flexible structure. The structure is modelled as a single degree of freedom harmonic oscillator. The contact force acts on the structure, but at the same time slows down the advancement of the ice, thereby introducing a dynamic nonlinearity in the otherwise linear system. When the local contact force becomes large enough, the ice crushes and the corresponding strip is reset to a random offset in front of the structure.This is the first mathematical model that exhibits all three different dynamic regimes that are observed in reality: for slow ice velocities the structure undergoes quasi-static sawtooth responses where all ice strips fail at the same time (a kind of synchronization phenomenon), for large ice velocities the structure response appears random, and for intermediate ice velocities the system exhibits vibrations at the structure eigenfrequency, commonly called frequency lock-in behavior. The latter type of vibrations causes a lot of damage to the structure and poses a safety and economic risk, so its occurrence needs to be predicted accurately.As I will show in this talk, the descriptive terms for the three vibration regimes are slightly misleading, as the mechanisms behind the observed behaviors are somewhat different than intuition suggests. I will present first results in analyzing the system and offer some explanations of the observed behaviors, as well as some simple criteria for the switch between the different vibration regimes.

Monday, March 13, 2017 - 14:00 , Location: Skiles 005 , Prof. Yao Li , University of Massachusetts Amherst , yaoli@math.umass.edu , Organizer: Molei Tao

In
this talk I will present my recent result about the ergodic properties
of nonequilibrium steady-state (NESS) for a stochastic energy exchange
model. The energy exchange model is numerically reduced from a
billiards-like deterministic particle system that models the microscopic
heat conduction in a 1D chain. By using a technique called the induced
chain method, I proved the existence, uniqueness, polynomial speed of
convergence to the NESS, and polynomial speed of mixing for the
stochastic energy exchange model. All of these are consistent with the
numerical simulation results of the original deterministic
billiards-like system.

Thursday, March 2, 2017 - 14:00 , Location: Skiles 006 , Professor Kui Ren , University of Texas, Austin , Organizer: Sung Ha Kang

Two-photon photoacoustic tomography (TP-PAT) is a non-invasive optical molecular imaging modality that aims at inferring two-photon absorption property of heterogeneous media from photoacoustic measurements. In this work, we analyze an inverse problem in quantitative TP-PAT where we intend to reconstruct optical coefficients in a semilinear elliptic PDE, the mathematical model for the propagation of near infra-red photons in tissue-like optical media, from the internal absorbed energy data. We derive uniqueness and stability results on the reconstructions of single and multiple coefficients, and perform numerical simulations based on synthetic data to validate the theoretical analysis.

Monday, February 27, 2017 - 14:00 , Location: Skiles 005 , Gunay Dogan , National Institute of Standards and Technology , Organizer: Sung Ha Kang

For many problems in science and engineering, one needs to quantitatively compare shapes of objects in images, e.g., anatomical structures in medical images, detected objects in images of natural scenes. One might have large databases of such shapes, and may want to cluster, classify or compare such elements. To be able to perform such analyses, one needs the notion of shape distance quantifying dissimilarity of such entities. In this work, we focus on the elastic shape distance of Srivastava et al. [PAMI, 2011] for closed planar curves. This provides a flexible and intuitive geodesic distance measure between curve shapes in an appropriate shape space, invariant to translation, scaling, rotation and reparametrization. Computing this distance, however, is computationally expensive. The original algorithm proposed by Srivastava et al. using dynamic programming runs in cubic time with respect to the number of nodes per curve. In this work, we propose a new fast hybrid iterative algorithm to compute the elastic shape distance between shapes of closed planar curves. The asymptotic time complexity of our iterative algorithm is O(N log(N)) per iteration. However, in our experiments, we have observed almost a linear trend in the total running times depending on the type of curve data.

Saturday, February 25, 2017 - 09:00 , Location: University of Georgia, Paul D. Coverdell Center for Biomedical & Health Sciences, Athens, GA 30602 , Haomin Zhou , GT Math , Organizer: Sung Ha Kang

The Georgia Scientific Computing Symposium (GSCS) is a forum for
professors, postdocs, graduate students and other researchers in Georgia
to meet in an informal setting, to exchange ideas, and to highlight
local scientific computing research. The symposium has been held every
year since 2009 and is open to the entire research community.

The format of the day-long symposium is a set of invited
presentations, poster sessions and a poster blitz, and plenty of time to
network with other attendees. More information at http://euler.math.uga.edu/cms/GSCS-2017

Monday, November 28, 2016 - 14:05 , Location: Skiles 005 , Prof. Enlu Zhou , Georgia Tech ISyE , Organizer: Martin Short
Many real-life systems require simulation techniques to evaluate the system performance and facilitate decision making. Stochastic simulation is driven by input model — a collection of probability distributions that model the system stochasticity. The choice of the input model is crucial for successful modeling and analysis via simulation. When there are past observed data of the system stochasticity, we can utilize these data to construct an input model. However, there is only a finite amount of data in practice, so the input model based on data is always subject to uncertainty, which is the so-called input (model) uncertainty. Therefore, a typical stochastic simulation faces two types of uncertainties: one is the input (model) uncertainty, and the other is the intrinsic stochastic uncertainty. In this talk, I will discuss our recent work on how to assess the risk brought by the two types of uncertainties and how to make decisions under these uncertainties.
Monday, November 21, 2016 - 14:05 , Location: Skiles 005 , Dr. Christina Frederick , Georgia Tech Mathematics , Organizer: Martin Short
We present a multiscale approach for identifying features in ocean beds
by solving inverse problems in high frequency seafloor acoustics. The
setting is based on Sound Navigation And Ranging (SONAR) imaging used in
scientific, commercial, and military applications. The forward model
incorporates multiscale simulations, by coupling Helmholtz equations and
geometrical optics for a wide range of spatial scales in the seafloor
geometry. This allows for detailed recovery of seafloor parameters
including material type. Simulated backscattered data is generated using
numerical microlocal analysis techniques. In order to lower the
computational cost of the large-scale simulations in the inversion
process, we take advantage of a \r{pre-computed} library of
representative acoustic responses from various seafloor
parameterizations.
Monday, November 14, 2016 - 14:05 , Location: Skiles 005 , Dr. Maryam Yashtini , Georgia Tech Mathematics , Organizer: Martin Short
Many real-world problems reduce to optimization problems that are solved
by iterative methods. In this talk, I focus on recently developed
efficient algorithms for solving large-scale optimization problems that
arises in medical imaging and image
processing. In the first part of my talk, I will introduce the Bregman
Operator Splitting with Variable Stepsize (BOSVS) algorithm for solving
nonsmooth inverse problems. The proposed algorithm is designed to handle
applications where the matrix in the fidelity
term is large, dense, and ill-conditioned. Numerical results are provided
using test problems from parallel magnetic resonance imaging. In the
second part, I will focus on the Euler's Elastica-based model which is
non-smooth and non-convex, and involves high-order
derivatives. I introduce two efficient alternating minimization methods
based on operator splitting and alternating direction method of
multipliers, where subproblems can be solved efficiently by Fourier
transforms and shrinkage operators. I present the analytical
properties of each algorithm, as well as several numerical experiments
on image inpainting problems, including comparison with some existing
state-of-art methods to show the efficiency and the effectiveness of the
proposed methods.
Monday, November 7, 2016 - 14:05 , Location: Skiles 005 , JD Walsh , GA Tech Mathematics, doctoral candidate , Organizer: Martin Short
The boundary method is a new algorithm for solving semi-discrete
transport problems involving a variety of ground cost functions. By
reformulating a transport problem as an optimal coupling problem, one
can construct a partition of its continuous space whose boundaries allow
accurate determination of the transport map and its associated
Wasserstein distance. The boundary method approximates region boundaries
using the general auction algorithm, controlling problem size with a
multigrid discard approach. This talk describes numerical and
mathematical results obtained when the ground cost is a convex
combination of lp norms, and shares preliminary work involving other
ground cost functions.

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