# Applied Mathematics Seminar

## Xavier Blanc (LJLL et Matherials)

Thursday, October 17th, 10h — Salle de séminaire du CERMICS

Schémas préservant l’asymptotique sur maillages coniques

On considère un système hyperbolique linéaire appelé équation
de la chaleur hyperbolique. Ce système est une approxlmation très
utilisée de l’équation du transfert radiatif, en particulier dans les
expériences de fusion par confinement inertiel (laser méga-joule). Dans
la limite d’une forte interaction du rayonnement avec la matière, cette
équation dégénère vers une équation de diffusion. Nous présenterons des
schémas volumes finis qui respectent cette limite, d’abord sur des
maillages polygonaux, puis sur des maillages dont les arêtes sont
définies par des coniques. Il s’agit d’un travail en commun avec V.
Delmas et P. Hoch (CEA).

## Pierre Bellec (Rugers University)

Thursday, October 17th, 15h — Salle de séminaire du CERMICS

First order expansion of convex regularized estimators

We consider first order expansions of convex penalized estimators in
high-dimensional regression problems with random designs. Our setting includes
linear regression and logistic regression as special cases. For a given
penalty function $h$ and the corresponding penalized estimator $\hat\beta$, we
construct a quantity $\eta$, the first order expansion of $\hat\beta$, such that
the distance between $\hbeta$ and $\eta$ is an order of magnitude smaller than
the estimation error $\|\hat{\beta} – \beta^*\|$. In this sense, the first
order expansion $\eta$ can be thought of as a generalization of influence
functions from the mathematical statistics literature to regularized estimators
in high-dimensions. Such first order expansion implies that the risk of
$\hat{\beta}$ is asymptotically the same as the risk of $\eta$ which leads to a
precise characterization of the MSE of $\hat\beta$; this characterization takes a
particularly simple form for isotropic design. Such first order expansion also
leads to confidence intervals based on $\hat{\beta}$. We provide sufficient
conditions for the existence of such first order expansion for three
regularizers: the Lasso in its constrained form, the lasso in its penalized
form, and the Group-Lasso. The results apply to general loss functions under
some conditions and those conditions are satisfied for the squared loss in
linear regression and for the logistic loss in the logistic model.

Joint work with Arun K Kuchibhotla (UPenn)

## Horia Cornean (Aalborg)

Tuesday, October 29, 10h — Salle de séminaire du CERMICS

Parseval frames of exponentially localized magnetic Wannier functions

Abstract: Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension d \le 3, we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model m occupied energy bands by a real-analytic and {{\mathbb {Z}}}^{d}-periodic family \left\{ P(\mathbf{k}) \right\} _{\mathbf{k}\in {{\mathbb {R}}}^{d}} of orthogonal projections of rank m. A moving orthonormal basis of {{\,\mathrm{Ran}\,}}P(\mathbf{k}) consisting of real-analytic and {{\mathbb {Z}}}^d-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of P vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of m-1orthonormal, real-analytic, and {{\mathbb {Z}}}^d-periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of m+1 real-analytic and {{\mathbb {Z}}}^d-periodic Bloch vectors which generate {{\,\mathrm{Ran}\,}}P(\mathbf{k}). Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, {{\mathbb {Z}}}^d-periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by 2d discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schrödinger operators as well.
This is joint work with Domenico Monaco and Massimo Moscolari.

## Sonia Fliss (ENSTA)

Thursday, November 7th, 10h — Salle de séminaire du CERMICS

TBA

## David Herzog (Iowa State University)

Thursday, November 21st, 10h — Salle de séminaire du CERMICS

TBA

## Denis Talay (Inria Sophia-Antipolis)

Wednesday, November 27th, 14h30 — Salle F106

TBA