We are concerned with the rigorous justification of the  so-called hard congestion limit from a compressible system with singular pressure towards a mixed  compressible-incompressible system modeling partially congested dynamics, in the framework of BV solutions. We will consider small BV perturbations of reference solutions constituted by (possibly interacting) large interfaces, and we will  analyze the dynamics of…

In this talk we present some recent result about the long time behavior of the solutions for some diffusion processes on a metric graph.  We study  evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat…

The talk is about oscillatory integral operators on manifolds.  Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.

In this talk, I will talk about some recent well-posedness and stability results for several fluid models in 2D. More precisely, I will discuss the global well-posedness for the 2D Boussinesq equations with fractional dissipation. For the Oldroyd-B model, we show that small smooth data lead to global and stable solutions. When Navier-Stokes is coupled…

In this talk, we discuss second-order optimality conditions for optimal control problems. This analysis is very important when we study the stability of the solution to the control problem with respect to small perturbations of the data. It is also crucial for proving superlinear or quadratic convergence of numerical algorithms for solving the problem, as…

In several shape optimization problems one has to deal with cost functionals of the form ${\cal F}(\Omega)=F(\Omega)+kG(\Omega)$, where $F$ and $G$ are two shape functionals with a different monotonicity behavior and $\Omega$ varies in the class of domains with prescribed measure. In particular, the cost functional ${\cal F}(\Omega)$ is not monotone with respect to $\Omega$…

I will describe how an invariant measure with Cauchy-distributed marginals arises from a supercritical stochastic heat equation with an additional, independent additive noise. Joint work with Chiranjib Mukherjee. Zoom meeting: Link

In this talk I will first review recent results which characterize adversarial training (AT) of binary classifiers as nonlocal perimeter regularization. Then I will speak about a probabilistic generalization of AT which also admits such a geometric interpretation, albeit with a different nonlocal perimeter. Using suitable relaxations one can prove the existence of solutions for…

We examine the asymptotic behaviors of solutions to Hamilton-Jacobi equations while varying the underlying domains. We establish a connection between the convergence of these solutions and the regularity of the additive eigenvalues in relation to the domains. To accomplish this, we introduce a framework based on Mather measures that enables us to compute the one-sided derivative…

We consider inverse problems concerning the one-dimensional viscous Burgers equation and some related nonlinear systems (involving heat effects, variable density, and fluid-solid interaction). We are dealing with inverse problems in which the goal is to find the size of the spatial interval from some appropriate boundary observations. Depending on the properties of the initial and…

I will discuss certain systems of transport type whose coefficients depend nonlinearly on the solution. Applications of such systems range from the modeling of pressure-less gases to the study of mean field games in a discrete state space. I will identify a notion of weak solution within the class of coordinate-wise decreasing functions, a condition…

Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust to adversarial perturbations of data. In response to this, a popular paradigm that has been developed to enforce robustness of learning models is adversarial training (AT), but this paradigm introduces many computational and theoretical difficulties. Recent…

 In this talk, I shall present a relaxation formula and duality theory for the multi-marginal Coulomb cost that appears in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which…

Feasibility models are a powerful approach to many real-world problems where simply finding a point that comes close enough to meeting many, sometimes contradictory demands is enough. In this talk I will outline the theoretical foundations for the convergence analysis of fixed point iterations of expansive mappings, and show how this specializes to fundamental algorithms…

A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; we can think of it as a spinning bubble of air in water. In this talk, we present a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. The…

We will consider various models in hydrodynamic, including the 2d Navier-Stokes, Boussinesq equations, and a Brinkman-Forchheimer-Navier-Stokes equations in 3d. These models are driven by an external stochastic Brownian perturbation. We will implement space-time numerical schemes and prove their convergence. We will show some rates of convergence as well. Furthermore, we will show the difference between…

Recent developments in the practice of numerical programming require optimization problems not only to be solved numerically, but also to be differentiated. This allows to integrate the computational operation of evaluating a solution in larger models, which are themselves trained or optimized using gradient methods. Most well known applications include bilevel optimization and implicit input-output…