Seminars and Colloquia by Series

Toward algorithms for linear response and sampling

Series
CDSNS Colloquium
Time
Friday, April 14, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Nisha ChandramoorthyGeorgia Tech

Zoom Link: Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: Linear response refers to the smooth change in the statistics of an observable in a dynamical system in response to a smooth parameter change in the dynamics. The computation of linear response has been a challenge, despite work pioneered by Ruelle giving a rigorous formula in Anosov systems. This is because typical linear perturbation-based methods are not applicable due to their instability in chaotic systems. Here, we give a new differentiable splitting of the parameter perturbation vector field, which leaves the resulting split Ruelle's formula amenable to efficient computation. A key ingredient of the overall algorithm, called space-split sensitivity, is a new recursive method to differentiate quantities along the unstable manifold.

In the second part, we discuss a new KAM method-inspired construction of transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.

Self-similar blow up profiles for fluids via physics-informed neural networks

Series
CDSNS Colloquium
Time
Friday, April 7, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and online
Speaker
Javier Gomez SerranoBrown University

Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Abstract: In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for different equations in fluid dynamics. The new numerical framework is shown to be both robust and readily adaptable to several situations.

Joint work with Yongji Wang, Ching-Yao Lai and Tristan Buckmaster.

Hill Four-Body Problem with oblate bodies

Series
CDSNS Colloquium
Time
Friday, March 17, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Wai Ting LamFAU

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

G. W. Hill made major contributions to Celestial Mechanics. One of them is to develop his lunar theory as an alternative approach for the study of the motion of the Moon around the Earth, which is the classical Lunar Hill problem. The mathematical model we study is one of the extensions of the classical Hill approximation of the restricted three-body problem. Considering a restricted four body problem, with a hierarchy between the bodies: two larger bodies, a smaller one and a fourth infinitesimal body, we encounter the shapes of the three heavy bodies via oblateness. We first find that the triangular central configurations of the three heavy bodies is a scalene triangle. Through the application of the Hill approximation, we obtain the limiting Hamiltonian that describes the dynamics of the infinitesimal body in a neighborhood of the smaller body. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter and the Jupiter’s Trojan asteroid Hektor. 

A Dynamical Systems Approach for Most Probable Escape Paths over Periodic Boundaries

Series
CDSNS Colloquium
Time
Friday, March 10, 2023 - 15:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Emmanuel FleurantinUNC, GMU

https://gatech.zoom.us/j/98358157136 

Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler-Lagrange system, which we call the River.  Using the Maslov index we identify a subset of the River which is comprised of local minimizers.  The Onsager-Machlup (OM) functional, which is treated as a perturbation of the Friedlin-Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the talk is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.

Exploring global dynamics and blowup in some nonlinear PDEs

Series
CDSNS Colloquium
Time
Friday, February 24, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Jonathan JaquetteBrown University

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.

In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.

Some results on a simple model of kinetic theory

Series
CDSNS Colloquium
Time
Friday, February 17, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006; Zoom streaming available
Speaker
Federico BonettoGeorgia Tech

Please Note: Zoom link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

In 1955, Mark Kac introduced a simple model to study the evolution of a gas of particles undergoing pairwise collisions. Although extremely simplified in such a way to be rigorously treatable, the model maintains interesting aspects of gas dynamics. In recent years, we worked with M. Loss and others to extend the analysis to more "realistic" versions of the original model.

I will introduce the Kac model and present some standard and more recent results. These results refer to a system with a fixed number of particles and at fixed kinetic energy (micro canonical ensemble) or temperature (canonical ensemble). I will introduce a "Grand Canonical" version of the Kac system and discuss new results on it.

The controllability function method and the feedback synthesis problem for a robust linear system

Series
CDSNS Colloquium
Time
Friday, February 10, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Tetiana RevinaV. N. KARAZIN KHARKIV NATIONAL UNIVERSITY

https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

The talk is about controllability for uncertain linear systems. Our approach is 
based on the Controllability Function (CF) method proposed by V.I. Korobov in 
1979. The CF method is a development of the Lyapunov function method and the 
dynamic programming method. The CF includes both approaches at a certain values 
of parameters. The main advance of the CF method is finiteness of the time of motion 
(settling-time function). 
In the talk the feedback synthesis problem for a chain of integrators system 
with continuous bounded unknown perturbations is considered. This problem consist 
in constructing a control in explicit form which depends on phase coordinates and 
steers an arbitrary initial point from a neighborhood of the origin to the origin in a 
finite time (settling-time function). Besides the control is satisfies some preassigned 
constrains. The range of the unknown perturbations such that the control solving the 
synthesis problem for the system without the perturbations also solves the synthesis 
problem for the perturbed system are found. This study shows the relations between 
the range of perturbations and the bounds of the settling-time function.
In particular the feedback synthesis problem for the motion of a material 
point with allowance for friction is solved. 
Keywords: chain of integrators, finite-time stability, robust control, settling 
time estimation, uncertain systems, unknown bounded perturbation

Nonsingular Poisson suspensions

Series
CDSNS Colloquium
Time
Friday, February 3, 2023 - 11:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Oleksandr DanilenkoInstitute for Low Temperature Physics and Engineering

 https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09

Let T be an invertible measure preserving transformation of a standard infinite measure space (X,m). Then a Poisson suspension (X*,m*,T*) of the dynamical system (X,m,T) is a well studied object in ergodic theory (especially for the last 20 years). It has physical applications as a model for the ideal gas consisting of countably many non-interacting particles. A natural problem is to develop a nonsingular counterpart of the theory of Poisson suspensions. The following will be enlightened in the talk:

--- description of the m-nonsingular (i.e. preserving the equivalence class of m) transformations T such that T* is m*-nonsingular
---algebraic and topological properties of the group of all m*-nonsingular Poisson suspensions
--- an interplay between dynamical properties of T and T*
--- an example of a "phase transition" in the ergodic properties of T* depending on the scaling of m
--- applications to Kazhdan property (T), stationary (nonsingular) group actions and the Furstenberg entropy.

(joint work with Z. Kosloff and E. Roy)

 

Differential encoding of sensory information across cortical microcircuitry

Series
CDSNS Colloquium
Time
Friday, January 27, 2023 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006 and Online
Speaker
Hannah ChoiGeorgia Tech

Please Note: https://gatech.zoom.us/j/98358157136

Mammalian cortical networks are known to process sensory information utilizing feedforward and feedback connections along the cortical hierarchy as well as intra-areal connections between different cortical layers. While feedback and feedforward signals have distinct layer-specific connectivity motifs preserved across species, the computational relevance of these connections is not known. Motivated by predictive coding theory, we study how expected and unexpected visual information is encoded along the cortical hierarchy, and how layer-specific feedforward and feedback connectivity contributes to differential, context-dependent representations of information across cortical layers, by analyzing experimental recordings of neural populations and also by building a recurrent neural network (RNN) model of the cortical microcircuitry. Experimental evidence shows that information about identity of the visual inputs and expectations are encoded in different areas of the mouse visual cortex, and simulations with our RNNs which incorporate biologically plausible connectivity motifs suggest that layer-specific feedforward and feedback connections may be the key contributor to this differential representation of information.
 

Statistical and non-statistical dynamics in doubly intermittent maps

Series
CDSNS Colloquium
Time
Friday, November 11, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
In-person in Skiles 005
Speaker
Stefano LuzzattoAbdus Salam International Centre for Theoretical Physics (ICTP)

Please Note: In-person. Streaming available via zoom: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

 

We introduce a large family of one-dimensional full branch maps which generalize the classical “intermittency maps” by admitting two neutral fixed points and possibly also critical points and/or singularities. We study the statistical properties of these maps for various parameter values, including the existence (and non-existence) of physical measures, and their properties such as decay of correlations and limit theorems. In particular we describe a new mechanism for relatively persistent non-statistical chaotic dynamics. This is joint work with Douglas Coates and Muhammad Mubarak.

Pages