Seminars and Colloquia by Series

Dual representation of polynomial modules with applications to partial differential equations

Series
Dissertation Defense
Time
Friday, April 15, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 or ONLINE
Speaker
Marc HärkönenGeorgia Tech

In 1939, Wolfgang Gröbner proposed using differential operators to represent ideals in a polynomial ring. Using Macaulay inverse systems, he showed a one-to-one correspondence between primary ideals whose variety is a rational point, and finite dimensional vector spaces of differential operators with constant coefficients. The question for general ideals was left open. Significant progress was made in the 1960's by analysts, culminating in a deep result known as the Ehrenpreis-Palamodov fundamental principle, connecting polynomial ideals and modules to solution sets of linear, homogeneous partial differential equations with constant coefficients. 

This talk aims to survey classical results, and provide new constructions, applications, and insights, merging concepts from analysis and nonlinear algebra. We offer a new formulation generalizing Gröbner's duality for arbitrary polynomial ideals and modules and connect it to the analysis of PDEs. This framework is amenable to the development of symbolic and numerical algorithms. We also study some applications of algebraic methods in problems from analysis.

Link: https://gatech.zoom.us/j/95997197594?pwd=RDN2T01oR2JlaEcyQXJCN1c4dnZaUT09

Optimal Motion Planning and Computational Optimal Transport

Series
Dissertation Defense
Time
Friday, April 8, 2022 - 13:00 for
Location
Skiles 006
Speaker
Haodong SunGeorgia Institute of Technology

In this talk, we focus on designing computational methods supported by theoretical properties for optimal motion planning and optimal transport (OT). 

Over the past decades, motion planning has attracted large amount of attention in robotics applications. Given certain
configurations in the environment, the objective is to find trajectories which move the robot from one position to the other while satisfying given constraints. We introduce a new method to produce smooth and collision-free trajectories for motion planning task. The proposed model leads to short and smooth trajectories with advantages in numerical computation. We design an efficient algorithm which can be generalized to robotics applications with multiple robots.

The idea of optimal transport naturally arises from many application scenarios and provides powerful tools for comparing probability measures in various types. However, obtaining the optimal plan is generally a computationally-expensive task, sometimes even intractable. We start with the entropy transport problem as a relaxed version of original optimal transport problem with soft marginals, and propose an efficient algorithm to obtain the sample approximation for the optimal plan. We also study an inverse problem of OT and present the computational methods for learning the cost function from the given optimal transport plan. 
 

Capillary Gravity Water Waves Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping

Series
Dissertation Defense
Time
Friday, April 8, 2022 - 09:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Xiao LiuGeorgia Institute of Technology

Please Note: https://bluejeans.com/421317143/2787

We consider the 2-dim capillary gravity water wave problem -- the free boundary problem of the Euler equation with gravity and surface tension -- of finite depth x2 \in (-h,0) linearized at a uniformly monotonic shear flow U(x2). Our main results consist of two aspects, eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity wave, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes, we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain the stronger decay for the remainders.

A Self-limiting Hawkes Process: Interpretation, Estimation, and Use in Crime Modeling

Series
Dissertation Defense
Time
Friday, April 1, 2022 - 13:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 268
Speaker
Jack OlindeGeorgia Institute of Technology

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, we propose a new variant of a Hawkes process, called a self-limiting 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.  We show that the self-limiting Hawkes process fits real crime data just as well, if not better, than the standard Hawkes model.  We also show that the self-limiting Hawkes process fits real financial data at least as well as the standard Hawkes model.

 

Heat kernel pull back metrics, non-collapsed spaces and convexity

Series
Dissertation Defense
Time
Friday, April 1, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Xingyu ZhuGeorgia Tech

We present in this talk some results concerning the metric measure spaces with lower Ricci curvature bounds. 

Firstly, we extend the technique of smoothing Riemannian metric by heat kernel pull back metrics to non-compact setting, and use it to solve a conjecture of De Philippis-Gigli. This is joint work with Brena-Gigli-Honda. Secondly, we study the second term in the short time expansion of the heat kernel pull back metrics and the connection with non-collapsed spaces. This is joint work with Honda. Finally, we use the 1D localization technique to extend some convexity results on the regular set and in the interior of such metric measure spaces.

Link: https://gatech.zoom.us/j/5491403383?pwd=Um1NM05MeWJMRnNuVHViQ1NWdHFaZz09 

Application of optimal transport theory on numerical computation, analysis, and dynamical systems on graph

Series
Dissertation Defense
Time
Wednesday, March 23, 2022 - 14:00 for
Location
ONLINE
Speaker
Shu LiuGeorgia Institute of Technology

Abstract: 

In this talk, we mainly focus on the applications of optimal transport theory from the following two aspects:

(1)Based on the theory of Wasserstein gradient flows, we develop and analyze a numerical method proposed for solving high-dimensional Fokker-Planck equations (FPE). The gradient flow structure of FPE allows us to derive a finite-dimensional ODE by projecting the dynamics of FPE onto a finite-dimensional parameter space whose parameters are inherited from certain generative model such as normalizing flow. We design a bi-level minimization scheme for time discretization of the proposed ODE. Such algorithm is sampling-based, which can readily handle computations in high-dimensional space. Moreover, we establish theoretical bounds for the asymptotic convergence analysis as well as the error analysis for our proposed method.

(2)Inspired by the theory of Wasserstein Hamiltonian flow, we present a novel definition of stochastic Hamiltonian process on graphs as certain kinds of inhomogeneous Markov process. Such definition is motivated by lifting to the probability space of the graph and considering the Hamiltonian dynamics on this probability space. We demonstrate some examples of the stochastic Hamiltonian process in classical discrete problems, such as the optimal transport problems and Schrödinger bridge problems (SBP).

The Bluejeans link: https://bluejeans.com/982835213/2740

Stability and Instability of the Kelvin-Stuart Cat's Eyes Flow to the 2D Euler's Equation

Series
Dissertation Defense
Time
Friday, January 28, 2022 - 09:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Shasha LiaoGeorgia Tech

The linear stability of a family of Kelvin-Stuart Cat's eyes flows of 2D Euler equation was studied both analytically and numerically. We proved linear stability under co-periodic perturbations and linear instability under multi-periodic perturbations. These results were first obtained numerically using spectral methods and then proved analytically.

The Bluejeans link is: https://bluejeans.com/353383769/0224

Domains of Analyticity and Gevrey estimates in weakly dissipative systems.

Series
Dissertation Defense
Time
Friday, August 27, 2021 - 12:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Adrian Perez BustamanteGeorgia Tech

We consider the problem of following quasi-periodic tori in perturbations of Hamiltonian systems which involve friction and external forcing.
In the first part, we study a family of dissipative standard maps of the cylinder for which the dissipation is a function of a small complex parameter of perturbation, $\varepsilon$.  We compute perturbative expansions formally in $\varepsilon$ and use them to estimate the shape of the domains of analyticity of invariant circles as functions of $\varepsilon$. We also give evidence that the functions might belong to a Gevrey class. The numerical computations we perform support conjectures on the shape of the domains of analyticity.

In the second part, we study rigorously the(divergent) series of formal expansions of the torus obtained using Lindstedt method.   We show that, for some systems in the literature, the series is Gevrey. We hope that the method of proof can be of independent interest: We develop KAM estimates for the divergent series. In contrast with the regular KAM method, we loose control of all the domains, so that there is no convergence, but we can generate enough control to show that the series is Gevrey.

https://bluejeans.com/417759047/0103

Fast Algorithm for Invariant Circle and their Stable Manifolds: Rigorous Results and Efficient Implementations

Series
Dissertation Defense
Time
Friday, July 9, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE: Bluejeans: https://gatech.bluejeans.com/6489493135
Speaker
Yian YaoGeorgia Tech

In this dissertation, we present, analyze, and implement a quadratically convergent algorithm to compute the invariant circle and the foliation by stable manifolds for 2-dimensional maps. The 2-dimensional maps we are considering are mainly motivated by oscillators subject to periodic perturbation.

The algorithm is based on solving an invariance equation using a quasi-Newton method, and the algorithm works irrespective of whether the dynamics on the invariant circle conjugates to a rotation or is phase-locked, and thus we expect only finite regularity on the invariant circle but analytic on the stable manifolds.

More specifically, the dissertation is divided into the following two parts.

In the theoretical part, we derive our quasi-Newton algorithm and prove that starting from an initial guess that satisfies the invariance equation very approximately, the algorithm converges quadratically to a true solution which is close to the initial guess. The proof of the convergence is based on an abstract Nash-Moser Implicit Function Theorem specially tailored for this problem. 

In the numerical part, we discuss some implementation details regarding our algorithm and implemented it on the dissipative standard map. We follow different continuation paths along the perturbation and drift parameter and explore the "bundle merging" scenario when the hyperbolicity of the map losses due to the increase of the perturbation. For non-resonant eigenvalues, we also generalize the algorithm to 3-dimension and implemented it on the 3-D Fattened Arnold Family.

Physical Billiards and Open Dynamical Systems

Series
Dissertation Defense
Time
Thursday, July 8, 2021 - 10:00 for 2 hours
Location
https://bluejeans.com/675272964/8610
Speaker
Hassan AttarchiGeorgia Institute of Technology

This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a ``hole" of positive measure in the phase space which some portion of points in phase space escapes through that hole at each iterate of the dynamical system map. Here, we study the escape rate (a quantity that presents at what rate points in phase space escape through the hole) and various estimations of the escape rate of an open dynamical system. We uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Moreover, exact computations of escape rate and its estimations are present for the skewed tent map and Arnold’s cat map.

In the second part of the thesis, we study physical billiards where the moving particle has a finite nonzero size. In contrast to mathematical billiards where a trajectory is excluded when it hits a corner point of the boundary, in physical billiards reflection of the physical particle (a ball) off a visible corner point is well-defined. Initially, we study properties of such reflections in a physical billiards. Our results confirm that the reflection considered in the literature about physical billiards are indeed no-slip friction-free (elastic) collisions.

In the third part of the thesis, we study physical Ehrenfests' wind-tree models, where we show that physical wind-tree models are dynamically richer than the well-known Lorentz gas model. More precisely, when we replace the point particle by a physical one (a ball), the wind-tree models show a new superdiffusive regimes that never been observed in any other model such as Lorentz gas.

Finally, we prove that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle.

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