## Seminars and Colloquia by Series

### 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.

### Dynamical Path Planning Methods For Control Problems in Unknown Environment

Series
SIAM Student Seminar
Time
Monday, November 27, 2017 - 15:10 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Haoyan ZhaiSchool of Mathematics, Georgia Institute of Technology
In this talk, we provide a deterministic algorithm for robotic path finding in unknown environment and an associated graph generator use only potential information. Also we will generalize the algorithm into a path planning algorithm for certain type of optimal control problems under some assumptions and will state some approximation methods if certain assumption no longer holds in some cases. And we hope to prove more theoretical results for those algorithms to guarantee the success.

### Local Hausdorff dimension and measure

Series
SIAM Student Seminar
Time
Friday, February 12, 2016 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
John DeverGeorgia Institute of Technology
A local Hausdorff dimension is defined on a metric space. We study its properties and use it to define a local measure. We show that in many circumstances we can recover the global Hausdorff dimension from the local one. We give an example of a compact metric space with a continuum of local dimension values. We define the dimension of a measure and connect the definition to that of local Hausdorff dimension and measure for a class of spaces called (variable) Ahlfors Q-regular. Very little background knowledge, aside from basic familiarity with metric spaces, will be assumed.

### THE EIGENVALUES OF THE LAPLACIAN ON DOMAINS WITH SMALL SLITS

Series
SIAM Student Seminar
Time
Tuesday, November 24, 2015 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hagop TossounianGeorgia Institute of Technology
This is a summary of result of LUC HILLAIRET AND CHRIS JUDGE.

### On the inverse of some sign matrices and on the Moments sliding vector field on the intersection of several manifolds: nodally attractive case

Series
SIAM Student Seminar
Time
Friday, October 23, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Fabio DifonzoGeorgia Institute of Technology
In this paper, we consider selection of a sliding vector fieldof Filippov type on a discontinuity manifold $\Sigma$ of co-dimension 3(intersection of three co-dimension 1 manifolds). We propose an extension of the “moments vector field”to this case, and - under the assumption that $\Sigma$ is nodally attractive -we prove that our extension delivers a uniquely definedFilippov vector field. As it turns out, the justification of our proposed extension requiresestablishing invertibility of certain sign matrices. Finally,we also propose the extension of the moments vector field todiscontinuity manifolds of co-dimension 4 and higher.

### Periodic Eigendecomposition and its application in nonlinear dynamics

Series
SIAM Student Seminar
Time
Friday, October 17, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Xiong DingSchool of Physics, Georgia Tech
Periodic eigendecomposition algorithm for calculating eigenvectors of a periodic product of a sequence of matrices, an extension of the periodic Schur decomposition, is formulated and compared with the recently proposed covariant vectors algorithms. In contrast to those, periodic eigendecomposition requires no power iteration and is capable of determining not only the real eigenvectors, but also the complex eigenvector pairs. Its effectiveness, and in particular its ability to resolve eigenvalues whose magnitude differs by hundreds of orders, is demonstrated by applying the algorithm to computation of the full linear stability spectrum of periodic solutions of Kuramoto-Sivashinsky system.