In my talk, I will explain the approach of finding solutions to nonlinear PDEs via tug-of-war games. I will focus on the context of p-Laplacian and the non-local geometric p-Laplacian. Zoom meeting: Link
In a Hilbert space, for convex optimization, we report on recent advances regarding the acceleration of first-order algorithms. We rely on inertial dynamics with damping driven by the Hessian, and the link between continuous dynamic systems and algorithms obtained by temporal discretization. We first review the classical results, from Polyak's heavy ball with friction method…
We survey the known results concerning the minimal Riesz energy on sufficiently smooth sets, and present some new results on fractal sets, which are the key examples of non-rectifiable sets. In particular, we will talk about the connection of these problems to ergodic theory and probability theory: the key step in our proofs is applying…
A popular method to approximate a fixed point of a non-expansive map is C is the Krasnoselskii-Mann iteration. This covers a wide range of iterative methods in convex minimization, equilibria, and beyond. In the Euclidean setting, a flexible method to obtain convergence rates for this iteration is the PEP methodology introduced by Drori and Teboulle…
Mesh-free methods for boundary value problems (BVPs) can be convenient on manifolds where generating a mesh may be difficult or when the manifold is not known explicitly but is determined by data. Moreover, BVPs are important in machine learning since they provide a rigorous method of regularization for many regression problems. In this talk we…
Mean field game PDE systems were introduced by J-M. Lasry and P.-L. Lions to describe Nash equilibria in multi-agents dynamic optimization. In the simplest model, those are forward-backward systems coupling Hamilton-Jacobi with Fokker-Planck equations. In this talk I will discuss the long time behavior of second order systems in the periodic case under suitable stability…
High intensity (focused) ultrasound HIFU is used in numerous medical and industrial applications ranging from lithotripsy and thermotherapy via ultrasound cleaning and welding to sonochemistry. The relatively high amplitudes arising in these applications necessitate modeling of sound propagation via nonlinear wave equations and in this talk we will first of all dwell on this modeling aspect. Then…
Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. Block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy’s gradient curves, convergence of Newton’s flow, finite length of Tikhonov path, convergence of central paths,…
In this talk we present well posedness of Measure Differential Equations, i.e. evolution equations in the Wasserstein space of probability measures driven by dissipative probability vector fields. We take inspiration from the theory of \emph{dissipative operators} in Hilbert spaces and of Wasserstein gradient flows of geodesically convex functionals. Our approach is based on a measure-theoretic…
Iterative Bregman projections is a classical method to compute Bregman projections onto an intersection of affine sets. In statistics it was applied to the adjustment of distributions to a priori known marginals, and is best known as the Iterative proportional fitting procedures. In this talk I will present novel results concerning such classical method as well…
In this joint ongoing work with Nicolas Forcadel (INSA Rouen) we study traffic flows models with a bifurcation. The model consists in a single incoming road divided after a junction into several outgoing ones. There are basically two classes of models to describe this situation: microscopic models, which explain how each vehicle behaves in function…
In this talk we propose a model for the simulation of retinal prostheses based on the use of organic polymer nanoparticles (NP). The model consists of a nonlinearly coupled system of elliptic partial differential equations accounting for: (1) light photoconversion into free charged carriers in the NP bulk; (2) charge transport in the NP bulk…
We improve Cheeger's lower bound for the first nonzero eigenvalue of the Laplacian on compact Riemannian manifolds with Ricci curvature bounded from below. Zoom link: https://ncsu.zoom.us/j/8027642791?pwd=d1lNaWZyUW4zeUFvaTA5VmlsTWtjdz09
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…