Tous les événements à venir

Consider a control system 𝛛t f + Af = Bu. Assume that 𝛱 is

a projection and that you can control both the systems

 𝛛t f + 𝛱Af = 𝛱Bu,

 𝛛t f + (1-𝛱)Af = (1-𝛱)Bu.

Can you conclude that the first system itself is controllable ? We

cannot expect it in general. But in a joint work with Andreas Hartmann,

we managed to do it for the half-heat equation. It turns out that the

property we need for our case is:

 If 𝛺 satisfies some cone condition, the set {f+g, f∈L²(𝛺), g∈L²(𝛺),

f is holomorphic, g is anti-holomorphic} is closed in L²(𝛺).

The first proof by Friedrichs consists of long computations, and is

very "complex analysis". But a later proof by Shapiro uses quite

general coercivity estimates proved by Smith, whose proof uses some

tools from algebra : Hilbert's nullstellensatz and/or primary ideal

decomposition.

In this first talk, we will introduce the algebraic tools needed and

present Smith's coercivity inequalities. In a second talk, we will

explain how useful these inequalities are to study the control

properties of the half-heat equation.

We discuss a new swarm-based gradient descent (SBGD) method for non-convex optimization. The swarm consists of agents, each is identified with position $x$ and mass $m$. There are three key aspects to the SBGD dynamics: (i) persistent transition of mass from agents at high to lower ground; (ii) a random marching direction, aligned with the steepest gradient descent; and (iii) a time stepping protocol which decreases with $m$.

The interplay between positions and masses leads to dynamic distinction between `heavier leaders’ near local minima, and `lighter explorers’ which explore for improved position with large(r) time steps. Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer. 

We give a light talk on optimality of shapes in geometry and physics. First, we recollect classical geometric results that the disk has the largest area (respectively, the smallest perimeter) among all domains of a given perimeter (respectively, area). Second, we recall that the circular drum has the lowest fundamental tone among all drums of a given area or perimeter and reinterpret the result in a quantum-mechanical language of nanostructures. In parallel, we discuss the analogous optimality of square among all rectangles in geometry and physics. As the main body of the talk, we present our recent attempts to prove the same spectral-geometric properties in relativistic quantum mechanics, where the mathematical model is a matrix-differential (Dirac) operator with complex (infinite-mass) boundary conditions. It is frustrating that such an illusively simple and expected result remains unproved and apparently out of the reach of current mathematical tools.

Neural style transfer (NST) is a deep learning technique that produces an unprecedentedly rich style transfer from a style image to a content image. It is particularly impressive when it comes to transferring style from a painting to an image. NST was originally achieved by solving an optimization problem to match the global statistics of the style image while preserving the local geometric features of the content image. The two main drawbacks of this original approach is that it is computationally expensive and that the resolution of the output images is limited by high GPU memory requirements. Many solutions have been proposed to both accelerate NST and produce images with larger size. However, our investigation shows that these accelerated methods all compromise the quality of the produced images in the context of painting style transfer. Indeed, transferring the style of a painting is a complex task involving features at different scales, from the color palette and compositional style to the fine brushstrokes and texture of the canvas. This paper provides a solution to solve the original global optimization for ultra-high resolution (UHR) images, enabling multiscale NST at unprecedented image sizes. This is achieved by spatially localizing the computation of each forward and backward passes through the VGG network. Extensive qualitative and quantitative comparisons, as well as a user study, show that our method produces style transfer of unmatched quality for such high-resolution painting styles. By a careful comparison, we show that state of the art fast methods are still prone to artifacts, thus suggesting that fast painting style transfer remains an open problem.

Joint work with Lara Raad, José Lezama and Jean-Michel Morel.

The Beurling--Selberg extremal approximation problems aim to find optimal unisided bandlimited approximations of a target function of bounded variation. We present an extension of the Beurling--Selberg problems, which we call “of higher-order,” where the approximation residual is constrained to faster decay rates in the asymptotic, ensuring the smoothness of their Fourier transforms. Furthermore, we harness the solution’s properties to bound the extremal singular values of confluent Vandermonde matrices with nodes on the unit circle. As an application to sparse super-resolution, this enables the derivation of a simple minimal resolvable distance, which depends only on the properties of the point-spread function, above which stability of super-resolution can be guaranteed.

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Une singularité de dimension $d$ est quasi-ordinaire par rapport à une projection finie $X$ -----> ${\mathbb C}^d$ si le discriminant de la projection est un diviseur à croisements normaux. Les singularités quasi-ordinaires sont au cœur de l'approche de Jung de la résolution des singularités en caractéristique zéro. En caractéristiques positives, elles ne sont pas très utiles du point de vue de la résolution des singularités, le problème de leurs résolutions étant presque aussi compliqué que le problème de résolution des singularités en général. En utilisant une version pondérée du polyèdre caractéristique de Hironaka (ou tout simplement la géométrie des équations) et des plongements successifs dans des espaces affines de "grandes" dimensions, nous introduisons la notion de singularités Teissier qui coïncide avec les singularités quasi-ordinaires en caractéristiques zéro, mais qui en est différente en caractéristiques positives. Nous démontrons qu'une singularité Teissier définie sur un corps de caractéristique positive est la fibre spéciale d'une famille équisingulière sur une courbe de caractéristique mixte dont la fibre générique (en caractéristique zéro donc) a des singularités quasi-ordinaires. Ici, L'équisingularité de la famille correspond à l'existence d'une résolution plongée simultanée.

Travail en collaboration avec Bernd Schober.

The regular model of a curve is a key object in the study of the arithmetic of the curve, as information about the special fiber of a regular model provides information about its generic fiber (such as rational points through the Chabauty-Coleman method, index, Tamagawa number of the Jacobian, etc). Every curve has a somewhat canonical regular model obtained from the quotient of a regular semistable model by resolving only singularities of a special type called quotient singularities. We will discuss in this talk what is known about the resolution graphs of $Z/pZ$-quotient singularities in the wild case, when $p$ is also the residue characteristic. The possible singularities that can arise in this process are not yet completely understood, even in the case of elliptic curves in residue characteristic 2.

TBA

Annulé

Simuler numériquement de manière précise l'évolution des interfaces séparant différents milieux est un enjeu crucial dans de nombreuses applications (multi-fluides, fluide-structure, etc). La méthode MOF (moment-of-fluid), extension de la méthode VOF (volume-of-fluid), utilise une reconstruction affine des interfaces par cellule basée sur les fractions volumiques et les centroïdes de chaque phase. Cette reconstruction d'interface est solution d'un problème de minimisation sous contrainte de volume. Ce problème est résolu dans la littérature par des calculs géométriques sur des polyèdres qui ont un coût important en 3D. On propose dans cet exposé une nouvelle approche du calcul de la fonction objectif et de ses dérivées de manière complètement analytique dans le cas de cellules hexaédriques rectangulaires et tétraédriques en 3D. Les résultats numériques montrent un gain important en temps de calcul.

We study the growth of the resolvent of a Toeplitz operator $T_b$, defined on the Hardy space, in terms of the distance to its spectrum $\s(T_b)$. We are primarily interested in the case when the symbol $b$ is a Laurent polynomial (\emph{i.e., } the matrix $T_b$ is banded). We show that for an arbitrary such symbol the growth of the resolvent is quadratic, and under certain additional assumption it is linear. We also prove the quadratic growth of the resolvent for a certain class of non-rational symbols.

This is a joint work with S. Kupin and A. Vishnyakova.