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Optimal transport :
Theory and Applications
to cosmological Reconstruction and Image
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 N2 evaluations of the cost function.
Field theories in multiple integral minimization (M. Zelikin)