One of the major open problems in homogenization of Hamilton-Jacobi equations is to under deep properties of the effective Hamiltonian. In this talk, I will present some recent progress. In particular, consider the effective Hamiltonian associated with the mechanical Hamiltonian H(p,x)=(|p|^2)/2+V(x). We can show that for generic V, the effective Hamiltonian is piecewise 1d in…
On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace leads to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify the rate of convergence to this profile uniformly in relative error, showing the rate is either exponentially fast (with a rate constant predicted by…
In this talk, we first propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. To this end we introduce a dynamical system for which we prove that its trajectories asymptotically converge…
In this talk we consider the pressureless Euler system in dimension greater than or equal to two. Several works have been devoted to the search for solutions which satisfy the following adhesion or sticky particle principle: if two particles of the fluid do not interact, then they move freely keeping constant velocity, otherwise they join…
Zoom meeting: Link
Recently there has been considerable research into the stability of shocks in systems of conservation laws, with stability understood in some square-integrable sense. In this talk I will give some background on systems of nonlinear hyperbolic partial differential equations (known as conservation laws), and on the issues concerning well-posedness. There are reasons that the still-unsolved…
Augmented Lagrangians were first employed in an algorithm for solving nonlinear programming problems with equality constraints. However, the approach was soon extended to inequality constraints and shown in the case of convex programming to correspond to applying the proximal point algorithm to solve a dual problem. Recent developments make it possible now to articulate that…
We study problems of optimal transport, by embedding them in a general functional analytic framework of convex optimization. This provides a unified treatment of alarge class of related problems in probability theory and allows for generalizations of the classical problem formulations. General results on convex duality yield dual problems and optimality conditions for these problems.…
A nonlinear model is developed for fluid-poroelastic structure interaction with quasi-Newtonian fluids that exhibit a shear-thinning property. The flow in the fluid region is described by the Stokes equations and in the poroelastic medium by the quasi-static Biot model. Equilibrium and kinematic conditions are imposed on the interface. A mixed Darcy formulation is employed, resulting…
In recent years, novel optimization ideas have been applied to several inverse problems in combination with machine learning approaches, to improve the inversion by optimally choosing different quantities/functions of interest. A fruitful approach in this sense is bilevel optimization, where the inverse problems are considered as lower-level constraints, while on the upper-level a loss function based…
Tropical convex sets arise as ``log-limits'' of parametric families of classical convex sets. The tropicalizations of polyhedra and spectrahedra are of special interest, since they can be described in terms of deterministic and stochastic games with mean payoff. In that way, one gets a correspondence between classes of zero-sum games, with an unsettled complexity, and classes…
We consider the density properties of divergence-free vector fields b in L^1(,BV(^2)) which are ergodic/weakly mixing/strongly mixing: this means that their Regular Lagrangian Flow X_t is an ergodic/weakly mixing/strongly mixing measure preserving map when evaluated at t=1. Our main result is that there exists a G-set U made of divergence free vector fields such that – The map T associating b with…
Among all drum heads of a fixed area, a circular drum head produces the vibration of lowest frequency. The general dimensional analogue of this fact is the Faber-Krahn inequality: balls have the smallest principal Dirichlet eigenvalue among subsets of Euclidean space with a fixed volume. I will discuss new quantitative stability results for the Faber-Krahn…
The well-posedness of Riccati equations plays a central role in the study of the optimal control problem with quadratic functionals for linear partial differential equations (PDEs). Indeed, it allows the synthesis of the optimal control by solving the Riccati equation corresponding to the minimization problem, and then of the closed-loop equation. In this lecture I…
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 L² 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. Zoom Meeting: Link
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…
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…
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.…
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…
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…