Posted January 27, 2024

Last modified March 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (click here to join)
Sergey Dashkovskiy , Julius-Maximilians-Universität Würzburg

Stability Properties of Dynamical Systems Subjected to Impulsive Actions

We consider several approaches to study stability and instability properties of infinite dimensional impulsive systems. The approaches are of Lyapunov type and provide conditions under which an impulsive system is stable. In particular we will cover the case, when discrete and continuous dynamics are not stable simultaneously. Also we will handle the case when both the flow and jumps are stable, but the overall system is not. We will illustrate these approaches by means of several examples.

Posted March 19, 2024

Computational Mathematics Seminar

until 3:30 pm Digital Media Center 1034
Quoc Tran-Dinh, UNC Chapel Hill

Boosting Convergence Rates for Fixed-Point and Root-Finding Algorithms

Approximating a fixed-point of a nonexpansive operator or a root of a nonlinear equation is a fundamental problem in computational mathematics, which has various applications in different fields. Most classical methods for fixed-point and root-finding problems such as fixed-point or gradient iteration, Halpern's iteration, and extragradient methods have a convergence rate of at most O(1/square root k) on the norm of the residual, where k is the iteration counter. This convergence rate is often obtained via appropriate constant stepsizes. In this talk, we aim at presenting some recent development to boost the theoretical convergence rates of many root-finding algorithms up to O(1/k). We first discuss a connection between the Halpern fixed-point iteration in fixed-point theory and Nesterov's accelerated schemes in convex optimization for solving monotone equations involving a co-coercive operator (or equivalently, fixed-point problems of a nonexpansive operator). We also study such a connection for different recent schemes, including extra anchored gradient method to obtain new algorithms. We show how a faster convergence rate result from one scheme can be transferred to another and vice versa. Next, we discuss various variants of the proposed methods, including randomized block-coordinate algorithms for root-finding problems,which are different from existing randomized coordinate methods in optimization. Finally, we consider the applications of these randomized coordinate schemes to monotone inclusions and finite-sum monotone inclusions. The algorithms for the latter problem can be applied to many applications in federated learning.

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Megan Fairchild, Louisiana State University

TBA

Posted April 1, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Kevin Schreve, Louisiana State University

TBA

Posted January 6, 2024

Last modified March 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (click here to join)
Madalena Chaves, Centre Inria d'Université Côte d'Azur

Coupling, Synchronization Dynamics, and Emergent Behavior in a Network of Biological Oscillators

Biological oscillators often involve a complex network of interactions, such as in the case of circadian rhythms or cell cycle. Mathematical modeling and especially model reduction help to understand the main mechanisms behind oscillatory behavior. In this context, we first study a two-gene oscillator using piecewise linear approximations to improve the performance and robustness of the oscillatory dynamics. Next, motivated by the synchronization of biological rhythms in a group of cells in an organ such as the liver, we then study a network of identical oscillators under diffusive coupling, interconnected according to different topologies. The piecewise linear formalism enables us to characterize the emergent dynamics of the network and show that a number of new steady states is generated in the network of oscillators. Finally, given two distinct oscillators mimicking the circadian clock and cell cycle, we analyze their interconnection to study the capacity for mutual period regulation and control between the two reduced oscillators. We are interested in characterizing the coupling parameter range for which the two systems play the roles "controller-follower".

Posted January 28, 2024

Last modified April 1, 2024

Mathematical Physics and Representation Theory Seminar

2:30 pm – 3:20 pm Lockett 233
Greg Parker, Stanford University

$\mathbb Z_2$-harmonic spinors as limiting objects in geometry and topology

$\mathbb Z_2$-harmonic spinors are singular solutions of Dirac-type equations that allow topological twisting around a submanifold of codimension 2. These objects arise as limits at the boundary of various moduli spaces in several distinct areas of low-dimensional topology, gauge/Floer theory, and enumerative geometry. The first part of this talk will introduce these objects, and discuss the various contexts in which they arise and the relationship between them. The second part of the talk will focus on the deformations of $\mathbb Z_2$-harmonic spinors when varying background parameters as a model for the novel analytic problems presented by these objects. In particular, the deformations of the singular submanifold play a role, giving the problem some characteristics similar to a free-boundary-value problem and leading to a hidden elliptic pseudo-differential operator that governs the geometry of the moduli spaces.

Posted February 21, 2024

Last modified April 12, 2024

Applied Analysis Seminar Questions or comments?

3:30 pm Lockett 232
Ben Seeger, The University of Texas at Austin

Equations on Wasserstein space and applications

The purpose of this talk is to give an overview of recent work involving differential equations posed on spaces of probability measures and their use in analyzing mean field limits of controlled multi-agent systems, which arise in applications coming from macroeconomics, social behavior, and telecommunications. Justifying this continuum description is often nontrivial and is sensitive to the type of stochastic noise influencing the population. We will describe settings for which the convergence to mean field stochastic control problems can be resolved through the analysis of the well-posedness for a certain Hamilton-Jacobi-Bellman equation posed on Wasserstein spaces, and how this well-posedness allows for new convergence results for more general problems, for example, zero-sum stochastic differential games of mean-field type.

Posted January 18, 2024

Informal Geometry and Topology Seminar Questions or comments?

1:30 pm Lockett 233
Krishnendu Kar, Louisiana State University

TBA

Posted January 31, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Morgan Weiler, Cornell University

TBA

Posted January 17, 2024

Last modified March 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (click here to join)
Tobias Breiten, Technical University of Berlin

On the Approximability of Koopman-Based Operator Lyapunov Equations

Computing the Lyapunov function of a system plays a crucial role in optimal feedback control, for example when the policy iteration is used. This talk will focus on the Lyapunov function of a nonlinear autonomous finite-dimensional dynamical system which will be rewritten as an infinite-dimensional linear system using the Koopman operator. Since this infinite-dimensional system has the structure of a weak-* continuous semigroup in a specially weighted Lp-space one can establish a connection between the solution of an operator Lyapunov equation and the desired Lyapunov function. It will be shown that the solution to this operator equation attains a rapid eigenvalue decay, which justifies finite rank approximations with numerical methods. The usefulness for numerical computations will also be demonstrated with two short examples. This is joint work with Bernhard Höveler (TU Berlin).

Posted September 24, 2023

Last modified March 3, 2024

Algebra and Number Theory Seminar Questions or comments?

3:20 pm – 4:10 pm Lockett 233 or click here to attend on Zoom
Ana Bălibanu, Louisiana State University

TBA

Posted February 1, 2024

Last modified February 11, 2024

Geometry and Topology Seminar Seminar website

3:30 pm Lockett 233
Jean-François Lafont, The Ohio State University

TBA

Posted January 16, 2024

Last modified March 4, 2024

Control and Optimization Seminar Questions or comments?

11:30 am – 12:20 pm Zoom (click here to join)
Jorge Poveda, University of California, San Diego
Donald P. Eckman, NSF CAREER, and AFOSR Young Investigator Program Awardee

Multi-Time Scale Hybrid Dynamical Systems for Model-Free Control and Optimization

Hybrid dynamical systems, which combine continuous-time and discrete-time dynamics, are prevalent in various engineering applications such as robotics, manufacturing systems, power grids, and transportation networks. Effectively analyzing and controlling these systems is crucial for developing autonomous and efficient engineering systems capable of real-time adaptation and self-optimization. This talk will delve into recent advancements in controlling and optimizing hybrid dynamical systems using multi-time scale techniques. These methods facilitate the systematic incorporation and analysis of both "exploration and exploitation" behaviors within complex control systems through singular perturbation and averaging theory, resulting in a range of provably stable and robust algorithms suitable for model-free control and optimization. Practical engineering system examples will be used to illustrate these theoretical tools.