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CCF Seminar Nicolas Garcia
Carnegie Mellon University
Title: On the rate of convergence of the uniform Monte Carlo integration error via optimal transportation.
Abstract: In this talk I will discuss the problem of finding the convergence rate of the Monte Carlo error when considering the set of Lipschitz continuous functions with Lipschitz constant less than one. The Kantorovich duality result from Optimal transportation theory implies that this error is equal to the Monge distance between the Lebesgue measure in the unit cube and the empirical measure associated to a sequence of i.i.d. random variables uniformly distributed in the cube. Because of this, it is then natural to consider the transportation distance between those two measures in order to obtain the Monte Carlo error. In the talk I will present an optimal transportation approach to obtain sharp estimates on this distance. It turns out that this approach creates the same transformations in the cube that were proposed by Ajtai, Komlós and Tusnády in 1984. They were the first ones who obtained the estimates I will present. If time permits I will also talk about how the estimates on the uniform Monte Carlo error allow us to approach some variational problems when functionals are determined by random data.
Date: Monday, April 15, 2013
Time: 5:00 pm
Location: Wean Hall 6423
Submitted by: Dmitry Kramkov