NCSU Differential Equations/Nonlinear Analysis Seminar Schedule Fall 2022

Wednesday, August 24, 15:00-16:00, Zoom meeting: Link

Speaker: Jacopo Schino
Title: Orbital stability of ground states to Schrödinger equations with mass constraint
Abstract: With a variational approach, I will discuss the existence and orbital stability of standing-wave solutions (i.e., with a specific time-dependence) with minimal energy (so-called ground states) to a non-linear Schrödinger equation where the  norm is prescribed. I will focus on the simpler case where the energy is bounded below and show a novel approach that simplifies the proof.

Wednesday, August 31, 15:00-16:00, Zoom meeting: Link

Speaker: Paul Manns, TU Dortmund, Germany
Title: On total variation regularization for PDE-constrained optimization with integer controls
Abstract: We study the effect of total variation regularization on PDE-constrained optimization problems, where the control input functions may only attain finitely many integer values. The regularization helps to avoid undesirable effects such as chattering behavior. In particular, the weak-* compactness of the feasible set in the space of functions of bounded variation allows to derive a meaningful stationarity concept and a trust-region algorithm that produces iterates such that all limits are stationary under mild assumptions on the governing PDE. We sketch a possible approach for discretizing the trust-region subproblems and point out the arising difficulties, in particular for the case that the dimension of the domain is larger than one.

Wednesday, September 07, 15:00-16:00, SAS Hall 4201, Zoom meeting: Link

Speaker: Michael Malisoff, LSU, USA
Title: Event-Triggered Control Using a Positive Systems Approach
Abstract: Control systems are a class of dynamical systems that contain forcing terms. When control systems are used in engineering applications, the forcing terms can represent forces that can be applied to the systems. Then the feedback control problem consists of finding formulas for the forcing terms, which are functions that can depend on the state of the systems, and which ensure a prescribed qualitative behavior of the dynamical systems, such as global asymptotic convergence towards an equilibrium point. Then the forcing terms are called feedback controls. Traditional feedback control methods call for continuously changing the feedback control values, or changing their values at a sequence of times that are independent of the state of the control systems. This can lead to unnecessarily frequent changes in control values, which can be undesirable in engineering applications. This motivated the development of event-triggered control, whose objective is to find formulas for feedback controls whose values are only changed when it is essential to change them in order to achieve a prescribed system behavior. This talk summarizes the speaker’s recent research on event-triggered control theory and applications in marine robotics, which is collaborative with Corina Barbalata, Zhong-Ping Jiang, and Frederic Mazenc. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.

Wednesday, September 14, 15:00-16:00, Zoom meeting: Link

Speaker: Maria Teresa Chiri, Queen’s University
Title: Controlling the spread of invasive biological species
Abstract: We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population. Our goal is to derive a simpler model, describing the controlled evolution of a contaminated set. We first analyze the optimal control of 1-dimensional traveling wave profiles. Using Stokes’ formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. Then we introduce a family of optimization problems for a moving set and show how these can be derived from the original parabolic problems, by taking a sharp interface limit. In connection with moving sets, we show some results on controllability, existence of optimal strategies, and necessary conditions.
 This is a joint work with Prof. Alberto Bressan (Penn State University) and Dr. Najmeh Salehi (Saint Mary’s College).

Wednesday, September 21, 15:00-16:00, Zoom meeting: Link

Speaker: Ryan Murray, NCSU
Title: Adversarially robust classification, non-local perimeters, and geometric flows
Abstract: Classification is a fundamental task in data science and machine learning, and in the past ten years there have been significant improvements on classification tasks (e.g. via deep learning). However, recently there have been a number of works demonstrating that these improved algorithms can be “fooled” using specially constructed adversarial examples. In turn, there has been increased attention given to creating machine learning algorithms which are more robust against adversarial attacks. 

In this talk I will describe a recently proposed variational framework for understanding adversarial robustness. This variational problem has deep connections with optimal transportation, isoperimetric problems, and mean curvature flow. I’ll discuss some of our recent results which make these connections and establish a delicate regularity theory for optimal solutions to this problem. I’ll also discuss recent algorithmic advances which can detect and track topological changes that are induced by the presence of an adversary in the variational problem.

Wednesday, Oct 05, 15:00-16:00, Zoom meeting: Link

Speaker: Alexei Novikov, Penn State University, USA
Title: Long-time behavior of a randomly perturbed oscillator
Abstract: We consider a long-time behavior of a stochastically forced nonlinear oscillator. In a long-time limit the force converges to fractional Brownian motion, a process that has memory. In contrast, we show that the  limit of the nonlinear oscillator driven by this force converges to diffusion driven by standard (not fractional) Brownian motion, and thus retains no memory in the scaling limit. We were motivated by
conjectured universality of large time diffusive limits of particle transport in incompressible flows with small noise. I will explain that the main role of the noise at large times is to enable switching between different regions of the flow, the precise nature of noise is unimportant and this leads to universality.


Wednesday, Oct 12, 15:00-16:00, Zoom meeting: Link

Speaker: Theodore D. Drivas, Stony Brook University, USA
Title: Remarks on the long-time dynamics of 2D Euler
Abstract: We will discuss some old and new results concerning the long-time behavior of solutions to the two-dimensional incompressible Euler equations. Specifically, we discuss whether steady states can be isolated, wandering for solutions starting nearby certain steady states, singularity formation at infinite time, and finally some results/conjectures on the infinite-time limit near and far from equilibrium.

Wednesday, Oct 19, 15:00-16:00, Zoom meeting: Link

Speaker: Michel De Lara, Cermics, École des Ponts ParisTech, France
Title:  Hidden Convexity in the $l_0$ Pseudonorm
Abstract: The so-called $l_0$ pseudonorm counts the number of nonzero components of a vector. It is standard in sparse optimization problems. However, as it is a discontinuous and nonconvex function, the l0 pseudonorm cannot be satisfactorily handled with the Fenchel conjugacy. In this talk, we review a series of recent results on a class of Capra
(Constant Along Primal Rays) conjugacies that reveal hidden convexity in the $l_0$ pseudonorm.

First, we present the Euclidean Capra-conjugacy. We show that it is suitable to analyze the $l_0$ pseudonorm, as this latter is “convex” in the sense of generalized convexity
(equal to its biconjugate). We present mathematical expressions of the Capra-subdifferential of the $l_0$ pseudonorm, and graphical representations.

In a second part, we provide different extensions. We introduce the class of Capra-conjugacies defined by means of norms. We show that such Capra-conjugacies are suitable to analyze, not only the $l_0$ pseudonorm, but provide convex lower bounds for 0-homogeneous functions. We will also point out how to tackle the rank matrix function.

Finally, we discuss how the theory opens the way for possible algorithms in sparse optimization problems.

Wednesday, Oct 26, 15:00-16:00, Zoom meeting: Link

Speaker: Peter W. Michor, University of Vienna, Austria
Title: Whitney manifold germs as source for manifolds of mappings
Abstract: During the preparation of a foundational chapter on manifolds
of mappings for a book on geometric continuum mechanics I found out
that the following object behaves surprisingly well as source of a manifold
of mappings:

— A Whitney manifold germ M˜ ⊃ M consists of an open manifold M˜
together with a closed subset M ⊂ M˜ which is the closure of its open
interior, such that there exists a continuous linear extension operator
from the space of Whitney jets on M to the space of smooth functions
C∞(M˜ ), with their natural locally convex topologies. This concept is
local in M˜ , due to recent advances for the existence of continuous
Whitney extension operators by D. Vogt, M. Tidten, L. Frerick, and
J. Wengenroth. This notion is more general than all existing notions:
domains with Lipschitz boundary or H¨older boundary, the manifolds with
rough boundary of Roberts and Schmeding

— The following concepts are very well behaved: Smooth mappings into
manifolds. Vector bundles. Fiber bundles. The space of vector fields on
M tangent to the boundary is a convenient Lie algebra, with ”Lie group”
(in a weakened sense) the group of diffeomorphisms of M.

Wednesday, Nov 02, 15:00-16:00, Zoom meeting: Link

Speaker: Guillaume Carlier, CEREMADE, Université Paris-Dauphine
Title: A refined Fenchel-Young inequality and applications to optimal transport
and convex duality
Abstract: In this talk, I will first present a very simple quantitative form of the Young-Fenchel inequality.  I will then discuss some applications: a short proof of the Brøndsted-Rockafellar in Hilbert spaces and a primal-dual attainment for perturbed convex minimization problems. I will finally explain how this inequality (or some generalizations) can be used for quantitative stability of optimal transport and entropic optimal transport (this part is based on a joint work with Paul Pegon and Luca Tamanini).

Wednesday, Nov 09, 15:00-16:00, Zoom meeting: Link

Speaker: Marco Antonio López Cerdá, Universidad de Alicante
Title: A survey of the subdifferential of the supremum function.
Featured applications
Abstract: This talk presents various characterizations of the
subdifferential of the pointwise supremum of an arbitrary family of
convex functions, as well as some featured applications. Starting by the
maximum generality framework, we move after to particular contexts in
which some continuity and compacity assumptions are either imposed or
inforced via processes of compactification of the index set and
regularization of the data functions. Some relevant applications of the
general results are presented, in particular to derive rules for the
subdifferential of the sum, and for convexifying a general
(unconstrained) optimization problem. The last part deals with some
specific constraint qualifications for the convex optimization problem
with an arbitrary set of constraints, and also contains different sets
of KKT-type optimality conditions appealing to the subdifferential of
the supremum function.