**O**ptimal transport :
**T**heory and **A**pplications

to cosmological **R**econstruction and **I**mage
processing

**L'optimisation et ses environs : une journée
franco-russe au laboratoire Poncelet**

Российско-французский семинар / Russian-French workshop

Laboratoire J.-V. Poncelet, 7 mai 2010

10:00–10:30 *Coffee*

10:30–11:15 M. Zelikin

11:20–12:05 J. Salomon

12:10–12:55 L. Lokutsievski

12:55–14:00 *Lunch*

14:00–14:45 I. Ekeland

14:50–15:35 A. Gasnikov

15:35–15:55 *Coffee*

15:55–16:40 G. Magaril-Ilyaev

16:45–17:30 A. Kolesnikov

**Pricing quality: equilibrium in hedonic markets**
(I. Ekeland, Canada Research Chair in Mathematical Economics,
University of British Columbia, Vancouver, Canada)

A market is in equilibrium if demand equals supply.
Traditionally, this is understood to mean that, for every available
good, the quantity produced is equal to the quantity consumed. But
many goods come in indivisible units and you cannot have more than
one: the typical example is a job, but houses or cars are very close
to that. Such goods are called hedonic, and their main
characteristic then is quality, described by a multidimensional
parameter. The talk will aim at describing equilibrium in such
markets. We will show that it leads to an extension of the optimal
transportation problem: given two measured spaces
(*X*, *m*) and (*Y*, *n*) and a third
space *Z*, among all maps *f* and *g* from
*X* and *Y* to *Z* such that
*f*(*m*) = *g*(*n*), find the ones
which minimize a certain integral.

**Mathematical modeling of traffic flow**
(A. Gasnikov, with A. Buslaev)

**Geometric applications of optimal transport**
(A. Kolesnikov)

**Optimal probabilistic search**
(L. Lokutsievski)

I will tell about vortex singularities which appear in optimal trajectories of standard search problems. These singularities in the 1-dimensional case have a connection with chattering control (Fuller's problem for example). In the case of search on strongly convex sets exact differential equations of optimal trajectories will be constructed. Also some questions about existence and uniqueness will be discussed.

**Optimal recovery of linear operators from inaccurate data** (G. Magaril-Ilyaev, with
K. Ossipenko)

**Local matching indicators for concave transport costs**
(__J. Salomon__, with __J. Delon__ et
__A. Sobolevski__)

In this talk, we introduce a class of indicators that enable to
compute efficiently optimal transport plans associated to arbitrary
distributions of *N* demands and *N* supplies
in **R** in the case where the cost function is concave.
The computational cost of these indicators is small and independent
of *N*. Using them recursively according to a particular
algorithm allows to find an optimal transport plan in less than
*N*^{2} evaluations of the cost function.

**Field theories in multiple integral
minimization** (M. Zelikin)

Last modified: Fri Apr 30 12:47:37 MSD 2010