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BEGIN:VEVENT
SUMMARY:Discrete approximation of the Griffith functional by adaptative fi
nite elements
DTSTART;VALUE=DATE-TIME:20220706T080000Z
DTEND;VALUE=DATE-TIME:20220706T083000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6984@indico.math.cnrs.fr
DESCRIPTION:Speakers: Élise Bonhomme (Université Paris-Saclay)\, Jean-Fr
ançois Babadjian\n\nThis joint work with Jean-François Babadjian is devo
ted to showing a discrete adaptative finite element approximation result f
or the isotropic two-dimensional Griffith energy arising in fracture mecha
nics. The problem is addressed in the geometric measure theoretic framewor
k of generalized special functions of bounded deformation which correspond
s to the natural energy space for this functional. It is proved to be appr
oximated in the sense of $\\Gamma$-convergence by a sequence of integral f
unctionals defined on continuous piecewise affine functions. The main feat
ure of this result is that the mesh is part of the unknown of the problem\
, and it gives enough flexibility to recover isotropic surface energies.\n
\nhttps://indico.math.cnrs.fr/event/7044/contributions/6984/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6984/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rigidity results for measurable sets
DTSTART;VALUE=DATE-TIME:20220704T073000Z
DTEND;VALUE=DATE-TIME:20220704T083000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6976@indico.math.cnrs.fr
DESCRIPTION:Speakers: Dorin Bucur (Université Savoie Mont Blanc)\, Ilaria
Fragalà\n\nLet $\\Omega \\subset \\mathbb{R}^d$ be a set with finite Leb
esgue measure such that\, for a fixed radius $r>0$\, the Lebesgue measure
of $\\Omega \\cap B _ r (x)$ is equal to a positive constant when $x$ va
ries in the essential boundary of $\\Omega$. We prove that $\\Omega$ is a
ball (or a finite union of equal balls) provided it satisfies a nondegene
racy condition\, which holds in particular for any set of diameter larger
than $r$ which is either open and connected\, or of finite perimeter and i
ndecomposable. This is a joint work with Ilaria Fragalà.\n\nhttps://indic
o.math.cnrs.fr/event/7044/contributions/6976/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6976/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Curvature functionals for surfaces with fractional regularity
DTSTART;VALUE=DATE-TIME:20220704T090000Z
DTEND;VALUE=DATE-TIME:20220704T093000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6977@indico.math.cnrs.fr
DESCRIPTION:Speakers: Reza Pakzad (University of Pittsburgh )\n\nThe curva
ture functionals (such as the Willmore functional) are usually defined und
er $W^{2\,2}$ regularity assumptions on the given surface. We will explain
how this assumption could be relaxed to the fractional Sobolev setting $W
^{1+s\, 2/s}$ for $s>1/2$\, and will discuss the related problematics of a
fractional variational plate model in nonlinear elasticity.\n\nhttps://in
dico.math.cnrs.fr/event/7044/contributions/6977/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6977/
END:VEVENT
BEGIN:VEVENT
SUMMARY:From local energy bounds to dimensional estimates in a reduced mod
el for type-I superconductors
DTSTART;VALUE=DATE-TIME:20220704T120000Z
DTEND;VALUE=DATE-TIME:20220704T130000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6978@indico.math.cnrs.fr
DESCRIPTION:Speakers: Berardo Ruffini\, Guido De Philippis\, Michael Goldm
an (Université Paris Cité)\n\nIn the limit of vanishing but moderate ext
ernal magnetic field\, we derived a few years ago together with S. Conti\,
F. Otto and S. Serfaty a branched transport problem from the full Ginzbur
g–Landau model. In this regime\, the irrigated measure is the Lebesgue m
easure and\, at least in a simplified 2d setting\, it is possible to prove
that the minimizer is a self-similar branching tree. In the regime of eve
n smaller magnetic fields\, a similar limit problem is expected but this t
ime the irrigation of the Lebesgue measure is not imposed as a hard constr
aint but rather as a penalization. While an explicit computation of the mi
nimizers seems here out of reach\, I will present some ongoing project wit
h G. De Philippis and B. Ruffini relating local energy bounds to dimension
al estimates for the irrigated measure.\n\nhttps://indico.math.cnrs.fr/eve
nt/7044/contributions/6978/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6978/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Derivation of surface tension of grain boundaries in polycrystals
DTSTART;VALUE=DATE-TIME:20220706T070000Z
DTEND;VALUE=DATE-TIME:20220706T080000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6983@indico.math.cnrs.fr
DESCRIPTION:Speakers: Emanuele Spadaro\, Adriana Garroni (Sapienza Univers
ity of Rome)\n\nInspired by a recent result of Lauteri and Luckhaus\, we d
erive\, via Gamma convergence\, a surface tension model for polycrystals i
n dimension two. The starting point is a semi-discrete model accounting fo
r the possibility of having crystal defects. The presence of defects is mo
delled by incompatible strain fields with quantised curl. In the limit as
the lattice spacing tends to zero we obtain an energy for grain boundaries
that depends on the relative angle of the orientations of the two neighbo
uring grains. The energy density is defined through an asymptotic cell pro
blem formula. By means of the bounds obtained by Lauteri and Luckhaus we a
lso show that the energy density exhibits a logarithmic behaviour for smal
l angle grain boundaries in agreement with the classical Read and Shockley
formula.\nThe talk is based on a paper in preparation in collaboration wi
th Emanuele Spadaro.\n\nhttps://indico.math.cnrs.fr/event/7044/contributio
ns/6983/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6983/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A free discontinuity problem in fluid mechanics
DTSTART;VALUE=DATE-TIME:20220704T140000Z
DTEND;VALUE=DATE-TIME:20220704T143000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6985@indico.math.cnrs.fr
DESCRIPTION:Speakers: Mickaël Nahon (Université Savoie Mont Blanc)\n\nWe
consider an incompressible Stokes fluid contained in a box $B$ that flows
around an obstacle $K\\subset B$ with a Navier boundary condition on $\\p
artial K$. I will present existence and partial regularity results for the
minimization of the drag of $K$ among all obstacles of given volume.\n\nh
ttps://indico.math.cnrs.fr/event/7044/contributions/6985/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6985/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tamped functions: A rearrangement in dimension 1
DTSTART;VALUE=DATE-TIME:20220704T130000Z
DTEND;VALUE=DATE-TIME:20220704T133000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6980@indico.math.cnrs.fr
DESCRIPTION:Speakers: Ludovic Godard-Cadillac (Nantes Université)\n\nWe d
efine a new rearrangement\, called rearrangement by tamping\, for non-nega
tive measurable functions defined on $\\mathbb{R}_+$. This rearrangement h
as many properties in common with the well-known Schwarz non-increasing re
arrangement such as the Pólya–Szegő inequality. \nContrary to the Schw
arz rearrangement\, the tamping also preserves the homogeneous Dirichlet b
oundary condition of a function. This presentation aims at presenting the
construction of the rearrangement by tamping (with an algorithmic approach
) and some recent developments around this idea.\n\nhttps://indico.math.cn
rs.fr/event/7044/contributions/6980/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6980/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Controlling nonconvexity and nonlinearity in gradient flows: two m
ethods and two model problems
DTSTART;VALUE=DATE-TIME:20220704T143000Z
DTEND;VALUE=DATE-TIME:20220704T153000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6982@indico.math.cnrs.fr
DESCRIPTION:Speakers: Felix Otto (Max Planck Institute for Mathematics in
the Sciences)\, Richard Schubert\, Maria G. Westdickenberg (RWTH Aachen Un
iversity)\n\nTogether with Felix Otto\, Richard Schubert\, and other colla
borators\, we have developed two different energy-based methods to capture
convergence rates and metastability of gradient flows. We will present th
e methods and their application to the two model problems that drove their
development: the 1-d Cahn–Hilliard equation and the Mullins–Sekerka e
volution. Both methods can be viewed as quantifying “how nonconvex“ or
“how nonlinear“ a problem can be while still retaining the optimal co
nvergence rates\, i.e.\, the rates for the convex or linear problem. Our f
ocus is on fairly large (ill-prepared) initial data.\n\nhttps://indico.mat
h.cnrs.fr/event/7044/contributions/6982/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6982/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dividing a set in half
DTSTART;VALUE=DATE-TIME:20220704T093000Z
DTEND;VALUE=DATE-TIME:20220704T103000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6979@indico.math.cnrs.fr
DESCRIPTION:Speakers: Alan Chang\, Giovanni Alberti (Università di Pisa)\
n\nIn this talk I will consider the following problem of isoperimetric typ
e:\n\nGiven a set E in $\\mathbb{R}^d$ with finite volume\, is it possible
to find an hyperplane $P$ that splits $E$ in two parts with equal volume\
, and such that the area of the cut (that is\, the intersection of $P$ an
d $E$) is of the expected order\, namely $(vol(E))^{1-1/d}$?\n\nWe can sho
w that the answer is positive if the dimension $d$ is $3$ or higher\, but\
, somewhat surprisingly\, our proof breaks down completely in dimension $d
=2$\, and we do not know what happens in this case.\n(However we know that
the answer is positive even for $d=2$ if we allow cuts that are not exact
ly planar\, but close to planar.)\n\nThis is a work in progress with Alan
Chang (Princeton University).\n\nhttps://indico.math.cnrs.fr/event/7044/co
ntributions/6979/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6979/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Exploiting convex duality in the calculus of variations
DTSTART;VALUE=DATE-TIME:20220706T130000Z
DTEND;VALUE=DATE-TIME:20220706T133000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6981@indico.math.cnrs.fr
DESCRIPTION:Speakers: Lukas Koch (MPI for MiS\, Leipzig)\, Cristiana De Fi
lippis\, Jan Kristensen\n\nI will recall the classical theory of convex du
ality and explain how this can be used to obtain regularity statements in
the study of minimisers of the problem \n$$\\mathrm{min}_{u\\in W^{1\,
p}(\\Omega)}\\int_\\Omega F(x\,\\mathrm{D} u)\\mathrm{d} x.$$\nIn particul
ar\, I will comment on recent results obtained in collaboration with Crist
iana de Filippis (Parma) and Jan Kristensen (Oxford) concerning the validi
ty of the Euler–Lagrange equations for extended real-valued integrands $
F$ satisfying no upper growth condition as well as concerning integrands $
F$ satisfying controlled duality $(p\,q)$-growth. The main example of inte
grands $F$ satisfying controlled duality $(p\,q)$-growth are convex polyno
mials.\n\nhttps://indico.math.cnrs.fr/event/7044/contributions/6981/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6981/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vortex sheet solutions for the Ginzburg–Landau system in cylinde
rs
DTSTART;VALUE=DATE-TIME:20220705T120000Z
DTEND;VALUE=DATE-TIME:20220705T130000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6987@indico.math.cnrs.fr
DESCRIPTION:Speakers: Radu Ignat (Université Toulouse III - Paul Sabatier
)\n\nWe consider the Ginzburg-Landau energy $E_\\epsilon$ for $\\mathbb{R
}^M$-valued maps defined in a cylinder $B^N\\times (0\,1)^n$ satisfying th
e degree-one vortex boundary condition on $\\partial B^N\\times (0\,1)^n$
in dimensions $M\\geq N\\geq 2$ and $n\\geq 1$. The aim is to study the ra
dial symmetry of global minimizers of this variational problem. We prove t
he following: if $N\\geq 7$\, then for every $\\epsilon>0$\, there exists
a unique global minimizer which is given by the non-escaping radially symm
etric vortex sheet solution $u_\\epsilon(x\,z)=(f_\\epsilon(|x|) \\frac{x}
{|x|}\, 0_{\\mathbb{R}^{M-N}})$\, $\\forall x\\in B^N$ that is invariant i
n $z\\in (0\,1)^n$. If $2\\leq N \\leq 6$ and $M\\geq N+1$\, the following
dichotomy occurs between escaping and non-escaping solutions: there exist
s $\\epsilon_N>0$ such that\n\n$\\bullet$ if $\\epsilon\\in (0\, \\epsilon
_N)$\, then every global minimizer is an escaping radially symmetric vorte
x sheet solution of the form $R \\tilde u_\\epsilon$ where $\\tilde u_\\ep
silon(x\,z)=(\\tilde f_{\\epsilon}(|x|) \\frac{x}{|x|}\, 0_{\\mathbb{R}^{M
-N-1}}\, g_{\\epsilon}(|x|))$ is invariant in $z$-direction with $g_\\epsi
lon>0$ in $(0\,1)$ and $R\\in O(M)$ is an orthogonal transformation keepin
g invariant the space $\\mathbb{R}^N\\times \\{0_{\\mathbb{R}^{M-N}}\\}$\;
\n\n$\\bullet$ if $\\epsilon\\geq \\epsilon_N$\, then the non-escaping rad
ially symmetric vortex sheet solution $u_\\epsilon(x\,z)=(f_\\epsilon(|x|)
\\frac{x}{|x|}\, 0_{\\mathbb{R}^{M-N}})$\, $\\forall x\\in B^N\, z\\in (0
\,1)^n$ is the unique global minimizer\; moreover\, there are no bounded e
scaping solutions in this case.\n\nWe also discuss the problem of vortex
sheet $\\mathbb{S}^{M-1}$-valued harmonic maps.\n\nhttps://indico.math.cnr
s.fr/event/7044/contributions/6987/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6987/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fractional Allen–Cahn systems with multi-well potential and nonl
ocal minimal partitions
DTSTART;VALUE=DATE-TIME:20220705T130000Z
DTEND;VALUE=DATE-TIME:20220705T133000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6988@indico.math.cnrs.fr
DESCRIPTION:Speakers: Thomas Gabard (Université Paris-Est Créteil )\n\nT
he aim of this talk is to present results on the asymptotic analysis of a
fractional version of the vectorial Allen–Cahn equation with multiple-we
ll in arbitrary dimension. In contrast to usual Allen–Cahn equations\, t
he Laplace operator is replaced by the fractional Laplacian as defined in
Fourier space. Our results concern the singular limit $\\varepsilon\\to 0$
and show that arbitrary solutions with uniformly bounded energy converge
both in the energetic and geometric sense to nonlocal minimal partitions i
n $\\Omega$. The notion of nonlocal minimal partition corresponds to the s
tationary version of the nonlocal minimizing clusters introduced by M. Col
ombo & F. Maggi (2017) and A. Cesaroni & M. Novaga (2020)\, and generalizi
ng the nonlocal minimal surfaces of L. Caffarelli\, J.M. Roquejoffre\, & O
. Savin (2010).\n\nhttps://indico.math.cnrs.fr/event/7044/contributions/69
88/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6988/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A Lagrangian description of entropy solutions of the eikonal equat
ion
DTSTART;VALUE=DATE-TIME:20220705T143000Z
DTEND;VALUE=DATE-TIME:20220705T153000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6990@indico.math.cnrs.fr
DESCRIPTION:Speakers: Elio Marconi (EPFL)\, Xavier Lamy ( Université Toul
ouse III - Paul Sabatier )\n\nWe consider the behaviour as $\\varepsilon \
\to 0^+$ of the following family of functionals introduced by P. Aviles an
d Y. Giga:\n$$\nF_\\varepsilon(u\,\\Omega):= \\int_{\\Omega} \\left( \\var
epsilon |\\nabla^2 u|^2 + \\frac{1}{\\varepsilon}\\left|1-|\\nabla u|^2\\r
ight|^2\\right)dx\, \\qquad \\mbox{where }\\Omega\\subset \\mathbb{R}^2.\n
$$\nFunctions with equi-bounded energy as $\\varepsilon \\to 0$ are pre-co
mpact in $L^1(\\Omega)$ and all the limits belong to the class of the so
called 'entropy solutions' of the eikonal equation $|\\nabla u| =1$ in $\\
Omega$.\n\nWe introduce a Lagrangian description of these solutions and we
investigate their fine properties.\nAs a corollary we obtain that if $\\O
mega$ is an ellipse\, then minimizers of $F_\\varepsilon(\\cdot\, \\Omega)
$ in the space \n$$\n \\left\\{ u\\in W^{2\,2}(\\Omega) : u=0 \\mbox{ and
} \\frac{\\partial u}{\\partial n} = -1 \\mbox{ at }\\partial \\Omega\\rig
ht\\}\n$$\nconverge to $u_\\ast := \\mathrm{dist}(\\cdot\, \\partial \\Ome
ga)$.\n\nMoreover we get a sharp quantitative version of the result in Jab
in–Otto–Perthame (2002)\, stating that the only bounded simply connect
ed domain $\\Omega$ admitting zero energy states with Dirichlet boundary c
onditions is the disk.\n\nPart of the work is done in collaboration with X
avier Lamy.\n\nhttps://indico.math.cnrs.fr/event/7044/contributions/6990/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6990/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Strong convergence results for total variation regularized inverse
problems in a low noise regime
DTSTART;VALUE=DATE-TIME:20220705T090000Z
DTEND;VALUE=DATE-TIME:20220705T093000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6992@indico.math.cnrs.fr
DESCRIPTION:Speakers: Romain Petit (Université Paris Dauphine)\, Yohann D
e Castro\, Vincent Duval\n\nWe consider an imaging inverse problem which c
onsists in recovering a “simple” function from a set of noisy linear m
easurements. Our approach is variationnal: we produce an approximation of
the unknown function by solving a least squares problem with a total varia
tion regularization term. Our aim is to prove this approximation converges
to the unknown function in a low noise regime. Specifically\, we are inte
rested in a convergence of “geometric” type: convergence of the level
sets\, of the number of non-trivial level sets\, etc. This result is close
ly related to stability questions for solutions of the prescribed curvatur
e problem. This is a joint work with Vincent Duval and Yohann De Castro.\n
\nhttps://indico.math.cnrs.fr/event/7044/contributions/6992/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6992/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Variational models with data driven regularisation for inverse pro
blems
DTSTART;VALUE=DATE-TIME:20220705T070000Z
DTEND;VALUE=DATE-TIME:20220705T080000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6991@indico.math.cnrs.fr
DESCRIPTION:Speakers: Carola-Bibiane Schönlieb (University of Cambridge)\
n\nInverse problems are about the reconstruction of an unknown physical qu
antity from indirect measurements. Most inverse problems of interest are i
ll-posed and require appropriate mathematical treatment for recovering mea
ningful solutions. Variational regularization is one of the main mechanism
s to turn inverse problems into well-posed ones by adding prior informatio
n about the unknown quantity to the problem\, often in the form of assumed
regularity of solutions. Classically\, such regularization approaches are
handcrafted. Examples include Tikhonov regularization\, the total variati
on and several sparsity-promoting regularizers such as the L1 norm of Wave
let coefficients of the solution. While such handcrafted approaches delive
r mathematically and computationally robust solutions to inverse problems\
, providing a universal approach to their solution\, they are also limited
by our ability to model solution properties and to realise these regulari
zation approaches computationally. Recently\, a new paradigm has been intr
oduced to the regularization of inverse problems\, which derives regulariz
ation approaches for inverse problems in a data driven way. Here\, regular
ization is not mathematically modelled in the classical sense\, but modell
ed by highly over-parametrised models\, typically deep neural networks\, t
hat are adapted to the inverse problems at hand by appropriately selected
(and usually plenty of) training data. In this talk\, I will review some m
achine learning based regularization techniques\, present some work on uns
upervised and deeply learned convex regularisers and their application to
image reconstruction from tomographic and blurred measurements\, and finis
h by discussing some open mathematical problems.\n\nhttps://indico.math.cn
rs.fr/event/7044/contributions/6991/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6991/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Isoperimetric problems on periodic lattices
DTSTART;VALUE=DATE-TIME:20220706T090000Z
DTEND;VALUE=DATE-TIME:20220706T100000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6993@indico.math.cnrs.fr
DESCRIPTION:Speakers: Gian Paolo Leonardi (University of Trento )\, Leonar
d Kreutz\, Marco Cicalese\n\nMotivated by the crystallization issue\, we f
ocus on the minimization of Heitman–Radin potential energies for configu
rations of $N$ particles in a periodic lattice\, and in particular on the
connection with anisotropic isoperimetric problems in the suitably rescale
d limit as $N\\to\\infty$. Besides identifying the asymptotic Wulff shapes
through Gamma-convergence\, we obtain fluctuation estimates for quasimini
mizers that include the well-known $N^{3/4}$ conjecture for minimizers in
planar lattices. Our technique combines the sharp quantitative Wulff inequ
ality with a notion of quantitative closeness between discrete and continu
um problems. These results have been obtained in collaborations with Marco
Cicalese and Leonard Kreutz.\n\nhttps://indico.math.cnrs.fr/event/7044/co
ntributions/6993/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6993/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Uniform Convergence Rates for Lipschitz Learning Down to Graph Con
nectivity
DTSTART;VALUE=DATE-TIME:20220705T080000Z
DTEND;VALUE=DATE-TIME:20220705T083000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6995@indico.math.cnrs.fr
DESCRIPTION:Speakers: Tim Roith (Friedrich-Alexander-Universität Erlangen
-Nürnberg)\n\nDiscrete to continuum convergence results for graph-based l
earning have seen an increased interest in the last years. In particular\,
the connections between discrete machine learning and continuum partial d
ifferential equations or variational problems\, lead to new insights and b
etter algorithms.\n\nThis talk considers Lipschitz learning — which is t
he limit of $p$-Laplacian learning for $p$ to infinity — and introduces
new proof strategies for the discrete to continuum limit. Our framework pr
ovides a convergence result in a sparse graph regime and additionally yiel
ds convergence rates. Employing a homogenized non-local operator with a mu
ch larger bandwidth allows us to extend uniform convergence rates to any g
raph length scale strictly above graph connectivity. We will sketch the i
deas of the proof and indicate how the approach may be used in other probl
ems\, like spectral convergence of the graph Laplacian.\n\nhttps://indico.
math.cnrs.fr/event/7044/contributions/6995/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6995/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Smooth Bilevel Programming for Sparse Regularization
DTSTART;VALUE=DATE-TIME:20220706T113000Z
DTEND;VALUE=DATE-TIME:20220706T123000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6994@indico.math.cnrs.fr
DESCRIPTION:Speakers: Clarice Poon\, Gabriel Peyré (École Normale Supér
ieure )\n\nIteratively reweighted least square (IRLS) is a popular approac
h to solve sparsity-enforcing regression problems in machine learning. Sta
te of the art approaches are more efficient but typically rely on specific
coordinate pruning schemes. In this work\, we show how a surprisingly sim
ple reparametrization of IRLS\, coupled with a bilevel resolution (instead
of an alternating scheme) is able to achieve top performances on a wide r
ange of sparsity (such as Lasso\, group Lasso and trace norm regularizatio
ns)\, regularization strength (including hard constraints)\, and design ma
trices (ranging from correlated designs to differential operators). Simila
rly to IRLS\, our method only involves linear systems resolutions\, but in
sharp contrast\, corresponds to the minimization of a smooth function. De
spite being non-convex\, we show that there is no spurious minima and that
saddle points are “ridable”\, so that there always exists a descent d
irection. We thus advocate for the use of a BFGS quasi-Newton solver\, whi
ch makes our approach simple\, robust and efficient. At the end of the tal
k\, I will discuss the associated gradient flows as well as the connection
with Hessian geometry and mirror descent. This is a joint work with Clari
ce Poon (Bath Univ.). The corresponding article is available: https://arxi
v.org/abs/2106.01429. A python notebook introducing the method is availabl
e at this address: https://nbviewer.org/github/gpeyre/numerical-tours/blob
/master/python/optim_7_noncvx_pro.ipynb\n\nhttps://indico.math.cnrs.fr/eve
nt/7044/contributions/6994/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6994/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Convergence rate of general entropic optimal transport costs
DTSTART;VALUE=DATE-TIME:20220706T140000Z
DTEND;VALUE=DATE-TIME:20220706T150000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6997@indico.math.cnrs.fr
DESCRIPTION:Speakers: Guillaume Carlier (Université Paris Dauphine)\, Pau
l Pegon (Université Paris Dauphine)\, Luca Tamanini\n\nEntropic optimal t
ransport (EOT) has received a lot of attention in recent years because it
is related to efficient solvers. In this talk\, I will address the rate of
convergence of the value to the optimal transport cost as the noise param
eter vanishes. This is a joint work with Paul Pegon and Luca Tamanini.\n\n
https://indico.math.cnrs.fr/event/7044/contributions/6997/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6997/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Frobenius theorem for weak submanifolds
DTSTART;VALUE=DATE-TIME:20220705T093000Z
DTEND;VALUE=DATE-TIME:20220705T103000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6986@indico.math.cnrs.fr
DESCRIPTION:Speakers: Giovanni Alberti (Università di Pisa)\, Andrea Merl
o\, Annalisa Massaccesi (University of Padova )\n\nThe question of produci
ng a foliation of the $n$-dimensional Euclidean space with $k$-dimensional
submanifolds which are tangent to a prescribed $k$-dimensional simple vec
torfield is part of the celebrated Frobenius theorem: a decomposition in s
mooth submanifolds tangent to a given vectorfield is feasible (and then th
e vectorfield itself is said to be integrable) if and only if the vectorfi
eld is involutive. In this seminar I will summarize the results obtained i
n collaboration with G. Alberti\, A. Merlo and E. Stepanov when the smooth
submanifolds are replaced by weaker objects\, such as integral or normal
currents or even contact sets with “some“ boundary regularity. I will
also provide Lusin-type counterexamples to the Frobenius property for rect
ifiable currents. Finally\, I will try to highlight the connection between
involutivity/integrability à la Frobenius and Carnot–Carathéodory spa
ces and how to apply our techniques in this framework.\n\nhttps://indico.m
ath.cnrs.fr/event/7044/contributions/6986/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6986/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A variational approach to data-driven problems in fluid mechanics
DTSTART;VALUE=DATE-TIME:20220706T123000Z
DTEND;VALUE=DATE-TIME:20220706T130000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6996@indico.math.cnrs.fr
DESCRIPTION:Speakers: Stefan Schiffer (Bonn University)\, Richard Schubert
(University of Bonn )\, Christina Lienstromberg (Universität Stuttgart)\
n\nIn this talk\, we discuss a data-driven approach to viscous fluid mecha
nics. Typically\, in order to describe the behaviour of fluids\, two diffe
rent kinds of modelling assumptions are used. On the one hand\, there are
first principles like the balance of forces or the incompressibility condi
tion. On the other hand there are material specific constitutive laws that
describe the relation between the strain and the viscous stress of the fl
uid. Combining both\, one obtains the partial differential equations of fl
uid mechanics like the Stokes or Navier–Stokes equations. The constituti
ve laws are obtained by fitting a law from a certain class (for example li
near\, power law\, etc.) to experimental data. This leads to modelling err
ors.\n\nInstead of using a constitutive relation\, we introduce a data-dri
ven formulation that has previously been examined in the context of solid
mechanics and directly draws on material data. This leads to a variational
solution concept\, that incorporates differential constraints coming from
first principles and produces fields that are optimal in terms of closene
ss to the data. In order to derive this formulation we recast the differen
tial constraints of fluid mechanics in the language of constant-rank diffe
rential operators. We show a $\\Gamma$-convergence result for the functio
nals arising in the data-driven fluid mechanical problem which implies tha
t the method is well-adapted to the convergence of experimental data throu
gh increasing experimental accuracy. \nFurthermore\, we will see that the
data-driven solutions are consistent with PDE solutions if the data are gi
ven by a constitutive law and discuss advantages of this new solution conc
ept.\n\nhttps://indico.math.cnrs.fr/event/7044/contributions/6996/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6996/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A Γ-convergence result for non-self dual U(1)-Yang–Mills–Higg
s energies of Ginzburg–Landau type
DTSTART;VALUE=DATE-TIME:20220705T140000Z
DTEND;VALUE=DATE-TIME:20220705T143000Z
DTSTAMP;VALUE=DATE-TIME:20221007T091900Z
UID:indico-contribution-6989@indico.math.cnrs.fr
DESCRIPTION:Speakers: Federico Luigi Dipasquale (University of Verona )\n\
nLet $E \\to M$ be a Hermitian complex line bundle with structure group ${
\\rm U}(1)$ over a closed smooth orientable connected Riemannian manifold
$M$. Fix a smooth metric connection ${\\rm D}_0$ on $E$ and consider\, for
$\\varepsilon > 0$\, the non-self dual ${\\rm U}(1)$-Yang–Mills–Higgs
energies\n\n$\\displaystyle G_\\varepsilon(u_\\varepsilon\, A_\\varepsilo
n) := \\int_M \\frac{1}{2}| {\\rm D}_{A_\\varepsilon} u_\\varepsilon|^2 +
\\frac{1}{4\\varepsilon^2}\\left(1-|u_\\varepsilon|^2\\right)^2 + \\frac{1
}{2}|F_{A_\\varepsilon}|^2 \\\, {\\rm vol}_g\,\n$\n\nwhere $(u\, A) \\in W
^{1\,2}(M\,E) \\times W^{1\,2}(M\,{\\rm T}^*M)$\, ${\\rm D}_A := {\\rm D}_
0 - i A$\, and $F_A$ denotes the curvature form of ${\\rm D}_A$. The funct
ionals $G_\\varepsilon$ arise as natural generalisation of the usual Ginzb
urg–Landau energy on domains of $\\mathbb{R}^n$. \n\nThe aim of the talk
is to illustrate the following $\\Gamma$-convergence result\, obtained in
collaboration with G. Canevari and G. Orlandi (Università di Verona): as
$\\varepsilon \\to 0$\, the rescaled functionals $\\frac{G_\\varepsilon}{
|{\\log\\varepsilon}|}$ $\\Gamma$-converge\, in the flat topology of Jacob
ians\, to ($\\pi$ times) the codimension two area functional.\n\nhttps://i
ndico.math.cnrs.fr/event/7044/contributions/6989/
LOCATION:M2 building\, Cité Scientifique - Meeting room\, 1st floor (Labo
ratoire Paul Painlevé)
URL:https://indico.math.cnrs.fr/event/7044/contributions/6989/
END:VEVENT
END:VCALENDAR