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SUMMARY:Variational and numerical resolution of det(D²u) = f(u)
DTSTART:20161207T125000Z
DTEND:20161207T134000Z
DTSTAMP:20241015T005900Z
UID:indico-event-1748@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: Filippo SANTAMBROGIO (Université Paris-Sud)\n\nFor
a given measure \\mu on the Euclidean space\, we can look for a convex fun
ction u such that the image of the density f(u) through Du is \\mu (here f
is a given function from R to R_+\, typically decreasing enough). In the
case where f(t)=e^{-t} we have the moment measures problem\, but for negat
ive powers we have interesting problems\, linked to other questions in con
vex and affine geometry. In the case where \\mu is uniform on a convex set
K in the plane and f(t)=t^{-4}\, the convex function u can be used to con
struct an affine hemi-sphere based on the polar convex set K*\, for instan
ce. Also\, in the case where \\mu is uniform on a set\, the problem is equ
ivalent to finding a suitable solution of Det(D^2u)=f(u). In the talk\, I
will briefly explain how to cast these problems as JKO-like optimization p
roblems involving optimal transport\, and explain a first idea of how to u
se the semidiscrete numerical methods which have been used for steps of th
e JKO scheme to get an approximation of the solutions of these problems. T
hen\, I will explain how to improve this approach in a way which better fi
ts the problem\, thus obtaining a true discretization of this moment measu
re problem\, where the measure \\mu has simply been replaced by a finitely
supported approximation of it\, which is no more specifically linked to o
ptimal transport. For this discretization\, I will present numerical resul
ts and proofs of convergence\, coming from an ongoing work in collaboratio
n with B. Klartag and Q. Mérigot\n\nhttps://indico.math.cnrs.fr/event/174
8/
LOCATION:Amphithéâtre Léon Motchane (IHES)
URL:https://indico.math.cnrs.fr/event/1748/
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