Seminars and Colloquia by Series

Ergodic theory: a statistical description of chaotic dynamical systems

Series
SIAM Student Seminar
Time
Friday, December 3, 2021 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Alex BlumenthalGeorgia Tech

Dynamical systems model the way that real-world systems evolve in time. While the time-asymptotic behavior of many systems can be characterized by “simple” dynamical features such as equilibria and periodic orbits, some systems evolve in a chaotic, seemingly random way. For such systems it is no longer meaningful to track one trajectory at a time individually- instead, a natural approach is to treat the initial condition as random and to observe how its probabilistic law evolves in time. This is the core idea of ergodic theory, the topic of this talk. I will not assume much beyond some basics of probability theory, e.g., random variables. 

About Coalescence of Eigenvalues for Matrices Depending on Several Parameters

Series
SIAM Student Seminar
Time
Friday, November 12, 2021 - 14:30 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Luca DieciGeorgia Institute of Technology

We review some theoretical and computational results on locating eigenvalues coalescence for matrices smoothly depending on parameters. Focus is on the symmetric 2 parameter case, and Hermitian 3 parameter case. Full and banded matrices are of interest.

Mathematical approaches to Imaging and data

Series
SIAM Student Seminar
Time
Friday, September 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 169
Speaker
Sung-Ha KangSchool of Math, Georgia Tech,

 

I will talk about introduction to mathematical image processing, and cover how numerical PDE can be used in data understanding.  This talk will present some of variational/PDE-based methods for image processing, such as denoising, inpainting, colorization.  If time permits, I will introduce identification of differential equation from given noisy data.   

A Self-Limiting Hawkes Process

Series
SIAM Student Seminar
Time
Monday, November 16, 2020 - 12:30 for 1 hour (actually 50 minutes)
Location
ONLINE at https://bluejeans.com/703668715
Speaker
John OlindeGT Math

Many real life processes that we would like to model have a self-exciting property, i.e. the occurrence of one event causes a temporary spike in the probability of other events occurring nearby in space and time.  Examples of processes that have this property are earthquakes, crime in a neighborhood, or emails within a company.  In 1971, Alan Hawkes first used what is now known as the Hawkes process to model such processes.  Since then much work has been done on estimating the parameters of a Hawkes process given a data set and creating variants of the process for different applications.

 

In this talk, I will be proposing a new variant of a Hawkes process that takes into account the effect of police activity on the underlying crime rate and an algorithm for estimating its parameters given a crime data set.

Post-grazing dynamics of a vibro-impacting energy generator

Series
SIAM Student Seminar
Time
Tuesday, November 3, 2020 - 16:00 for 1 hour (actually 50 minutes)
Location
Online at https://bluejeans.com/893955256
Speaker
Larissa SerdukovaMathematics & Statistics Department, University of Reading

 

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric. 

Bluejeans link: https://bluejeans.com/893955256

Post-grazing dynamics of a vibro-impacting energy generator--Postponed

Series
SIAM Student Seminar
Time
Friday, March 27, 2020 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Larissa SerdukovaGT Math

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.

Learning functions varying along an active subspace

Series
SIAM Student Seminar
Time
Friday, February 7, 2020 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hao LiuGT Math

Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This work studies regression of a $s$-Hölder function $f$ in $\mathbb{R}^D$ which varies along an active subspace of dimension $d$ while $d\ll D$. A direct approximation of $f$ in $\mathbb{R}^D$ with an $\varepsilon$ accuracy requires the number of samples $n$ in the order of $\varepsilon^{-(2s+D)/s}$. In this work, we modify the Generalized Contour Regression (GCR) algorithm to estimate the active subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the active subspace, but its sample complexity is an open question. Our modified GCR improves the efficiency over the original GCR and leads to a mean squared estimation error of $O(n^{-1})$ for the active subspace, when $n$ is sufficiently large. The mean squared regression error of $f$ is proved to be in the order of $\left(n/\log n\right)^{-\frac{2s}{2s+d}}$, where the exponent depends on the dimension of the active subspace $d$ instead of the ambient space $D$. This result demonstrates that GCR is effective in learning low-dimensional active subspaces. The convergence rate is validated through several numerical experiments.

This is a joint work with Wenjing Liao.

Spin Dynamics: Algorithms and Spin of Planets

Series
SIAM Student Seminar
Time
Friday, October 25, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Renyi ChenGT Math

In this talk, we will focus on the spin dynamics of rigid bodies.
Algorithm part: There are many algorithms designed for N body simulations. 
But, to study the climates of a planet, we need to extend the simulation from point mass bodies to rigid bodies.
In the N-rigid-body simulations, we will consider the orientation and angular momentum of the rigid body to understand the spin.
In terms of the algorithm, symplectic integrators are designed by splitting methods. 
Physical part: We studied the spin dynamics of an Earth-like planet in circumbinary systems.
Canonical Delaunay variables and Andoyer variables are applied to split the variables to be slow part and fast part.
Applying averaging method, we approximated the spin dynamics.
From the approximated dynamics, we may draw some interesting physical conclusions.
 

The Kac Model and (Non-)Equilibrium Statistical Mechanics

Series
SIAM Student Seminar
Time
Friday, October 4, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Prof. Federico Bonetto (Distinguished Speaker) GT Math

In 1959 Mark Kac introduced a simple model for the evolution 
of a gas of hard spheres undergoing elastic collisions. The main 
simplification consisted in replacing deterministic collisions with 
random Poisson distributed collisions.

It is possible to obtain many interesting results for this simplified 
dynamics, like estimates on the rate of convergence to equilibrium and 
validity of the Boltzmann equation. The price paid is that this system 
has no space structure.

I will review some classical results on the Kac model and report on an 
attempt to reintroduce some form of space structure and non-equilibrium 
evolution in a way that preserve the mathematical tractability of the 
system.
 

Joint SIAM Student Conference

Series
SIAM Student Seminar
Time
Saturday, April 7, 2018 - 10:30 for 8 hours (full day)
Location
Skiles 005
Speaker
Graduate StudentsGeorgia Institute of Technology, Clemson University, Emory University, University of Alabama at Birmingham
This joint SIAM student conference is organized by the SIAM Student Chapter at School of Mathematics, Georgia Tech together with SIAM chapters at Clemson University, Emory University and University of Alabama at Birmingham. Detailed schedule and information can be found at jssc.math.gatech.edu.

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