Seminars and Colloquia by Series

Quantitative acceleration of convergence to invariant distribution by irreversibility in diffusion processes

Series
PDE Seminar
Time
Tuesday, December 5, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yuqing WangGeorgia Tech

Sampling from the Gibbs distribution is a long-standing problem studied across various fields. Among many sampling algorithms, Langevin dynamics plays a crucial role, particularly for high-dimensional target distributions. In practical applications, accelerating sampling dynamics is always desirable. It has long been studied that adding an irreversible component to reversible dynamics, such as Langevin, can accelerate convergence. Concrete constructions of irreversible components have also been explored in specific scenarios. However, a general strategy for such construction is still elusive. In this talk, I will introduce the concept of leveraging irreversibility to accelerate general dynamics, along with the quantification of irreversible dynamics. Our theory is mainly based on designing a modified entropy functional originally developed for linear kinetic equations (Dolbeault et al., 2015).

The most likely evolution of diffusing and vanishing particles: Schrodinger Bridges with unbalanced marginals

Series
PDE Seminar
Time
Tuesday, November 21, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yongxin ChenGeorgia Tech

Stochastic flows of an advective-diffusive nature are ubiquitous in biology and the physical sciences. Of particular interest is the problem to reconcile observed marginal distributions with a given prior posed by E. Schroedinger in 1932/32 and known as the Schroedinger Bridge Problem (SBP). It turns out that Schroedinger’s problem can be viewed as a problem in large deviations, a modeling problem, as well as a control problem. Due to the fundamental significance of this problem, interest in SBP and in its deterministic (zero-noise limit) counterpart of Optimal Transport (OT) has in recent years enticed a broad spectrum of disciplines, including physics, stochastic control, computer science, and geometry. Yet, while the mathematics and applications of SBP/OT have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention; the problem to interpolate between “unbalanced” marginals has been approached by introducing source/sink terms into the transport equations, in an adhoc manner, chiefly driven by applications in image registration. Nevertheless, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schroedinger’s dictum; that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated law represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this talk is to present recent results on stochastic evolutions with losses, whereupon particles are “killed” (jump into a coffin/extinction state) according to a probabilistic law, and thereby mass is gradually lost along their stochastically driven flow. Through a suitable embedding we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism in the style of Schroedinger, given a prior law that allows for losses, we explore the most probable evolution of particles along with the most likely killing rate as the particles transition between the specified marginals. Our approach differs sharply from previous work involving a Feynman-Kac multiplicative reweighing of the reference measure which, as we argue, is far from Schroedinger’s quest. We develop a suitable Schroedinger system of coupled PDEs' for this problem, an iterative Fortet-IPF-Sinkhorn algorithm for computations, and finally formulate and solve a related fluid-dynamic control problem for the flow of one-time marginals where both the drift and the new killing rate play the role of control variables. Joint work with Tryphon Georgiou and Michele Pavon.

Onsager's conjecture in 2D

Series
PDE Seminar
Time
Tuesday, November 14, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Razvan-Octavian RaduPrinceton University

I will begin by describing the ideas involved in the Nash iterative constructions of solutions to the Euler equations. These were introduced by De Lellis and Szekelyhidi (and developed by many authors) in order to tackle the flexible side of the Onsager conjecture. Then, I will describe Isett’s proof of the conjecture in the 3D case, and highlight the simple reason for which the strategy will not work in 2D. Finally, I will describe a construction of non-conservative solutions that works also in 2D (this is joint work with Vikram Giri).

Dynamics of kink clusters for scalar fields in dimension 1+1

Series
PDE Seminar
Time
Tuesday, November 7, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jacek JendrejCNRS and LAGA, Universite Sorbonne Paris Nord

We consider classical scalar fields in dimension 1+1 with a
self-interaction potential being a symmetric double-well. Such a model
admits non-trivial static solutions called kinks and antikinks. A kink
cluster is a solution approaching, for large positive times, a
superposition of alternating kinks and antikinks whose velocities
converge to 0 and mutual distances grow to infinity. Our main result is
a determination of the asymptotic behaviour of any kink cluster at the
leading order.
Our results are partially inspired by the notion of "parabolic motions"
in the Newtonian n-body problem. I will present this analogy and mention
its limitations. I will also explain the role of kink clusters as
universal profiles for formation of multi-kink configurations.
This is a joint work with Andrew Lawrie.

Long time behavior in cosmological Einstein-Belinski-Zakharov spacetimes

Series
PDE Seminar
Time
Tuesday, October 31, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Claudio MuñozUniversidad de Chile

In this talk I will present some recent results in collaboration with Jessica Trespalacios where we consider Einstein-Belinski-Zakharov spacetimes and prove local and global existence, long time behavior of possibly large solutions and some applications to gravisolitons of Kasner type.

The convergence problem in mean field control

Series
PDE Seminar
Time
Tuesday, October 17, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joe JacksonUniversity of Chicago

This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" N-particle control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be C^1, even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the N-particle value functions towards the value function of the corresponding MFC problem.

Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density

Series
PDE Seminar
Time
Tuesday, September 26, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Wenren ZhaoNYU Abu Dhabi

In this talk, I will discuss my recent research on the asymptotic stability and inviscid damping of 2D monotone shear flows with non-constant density in inhomogeneous ideal fluids within a finite channel. More precisely, I proved that if the initial perturbations belong to the Gevrey-2- class, then linearly stable monotone shear flows in inhomogeneous ideal fluids are also nonlinear asymptotically stable. Furthermore, inviscid damping is proved to hold, meaning that the perturbed velocity converges to a shear flow as time approaches infinity.

Exploiting low-dimensional structures in machine learning and PDE simulations

Series
PDE Seminar
Time
Tuesday, September 19, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wenjing LiaoGeorgia Tech

Many data in real-world applications are in a high-dimensional space but exhibit low-dimensional structures. In mathematics, these data can be modeled as random samples on a low-dimensional manifold. I will talk about machine learning tasks like regression and classification, as well as PDE simulations. We consider deep learning as a tool to solve these problems. When data are sampled on a low-dimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. Our results demonstrate that deep neural networks can utilize low-dimensional geometric structures of data in machine learning and PDE simulations.

Spectral stability for periodic waves in some Hamiltonian systems

Series
PDE Seminar
Time
Tuesday, September 12, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Atanas StefanovUniversity of Alabama at Birmingham

A lot of recent work in the theory of partial differential equations has focused on the existence and stability properties of special solutions for Hamiltonian PDE’s.  

We review some recent works (joint with Hakkaev and Stanislavova), for spatially periodic traveling waves and their stability properties. We concentrate on three examples, namely the Benney system, the Zakharov system and the KdV-NLS model. We consider several standard explicit solutions, given in terms of Jacobi elliptic functions. We provide explicit and complete description of their stability properties. Our analysis is based on the careful examination of the spectral properties of the linearized operators, combined with recent advances in the Hamiltonian instability index formalism.

Selection of standing waves at small energy for NLS with a trapping potential in 1 D

Series
PDE Seminar
Time
Tuesday, September 5, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Scipio CuccagnaUniversita` di Trieste

Due to linear superposition, solutions of a Linear Schrodinger Equation with a trapping potential,  produce a discrete  quasiperiodic part. When  a nonlinear perturbation is turned on,  it is known in principle, and proved in various situations,  that at small energies there is a phenomenon of standing wave selection where, up to radiation,  quasiperiodicity breaks down and there is convergence to a periodic wave.  We will discuss  this phenomenon in 1 D, where cubic nonlinearities are long range perturbations of the linear equations. Our aim is to show that a very effective framework to see these phenomena is provided by   a combination of the dispersion theory of  Kowalczyk, Martel and Munoz  along with  Maeda's  notion of Refined Profile.

Pages