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DTSTAMP:20260424T052101Z
UID:1759735800@ist.ac.at
DTSTART:20251006T093000
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DESCRIPTION:Speaker: Filippo Quattrocchi\nhosted by Ylva Götberg\nAbstract
 : The theory of optimal transport provides an elegant and powerful descrip
 tion of many evolution equations as gradient flows. The primary objective 
 of this thesis is to adapt and extend the theory to deal with important eq
 uations that are not covered by the classical framework\, specifically bou
 ndary value problems and kinetic equations. Additionally\, we establish ne
 w results in periodic homogenization for discrete dynamical optimal transp
 ort and in quantization of measures.    Section 1.1 serves as an invita
 tion to the classical theory of optimal transport\, including the main def
 initions and a selection of well-established theorems. Sections 1.1-1.5 in
 troduce the main results of this thesis\, outline the motivations\, and re
 view the current state of the art.    In Chapter 2\, we consider the Fo
 kker--Planck equation on a bounded set with positive Dirichlet boundary co
 nditions. We construct a time-discrete scheme involving a modification of 
 the Wasserstein distance and\, under weak assumptions\, prove its converge
 nce to a solution of this boundary value problem. In dimension 1\, we show
  that this solution is a gradient flow in a suitable space of measures.  
   Chapter 3 presents joint work with Giovanni Brigati and Jan Maas. We i
 ntroduce a new theory of optimal transport to describe and study particle 
 systems at the mesoscopic scale. We prove adapted versions of some fundame
 ntal theorems\, including the Benamou--Brenier formula and the identificat
 ion of absolutely continuous curves of measures.    Chapter 4 presents 
 joint work with Lorenzo Portinale. We prove convergence of dynamical trans
 portation functionals on periodic graphs in the large-scale limit\, when t
 he cost functional is asymptotically linear. Additionally\, we show that d
 iscrete 1-Wasserstein distances converge to 1-Wasserstein distances constr
 ucted from crystalline norms on R^d.    Chapter 5 concerns optimal empi
 rical quantization: the problem of approximating a measure by the sum of n
  equally weighted Dirac deltas\, so as to minimize the error in the p-Wass
 erstein distance. Our main result is an analog of Zador's theorem\, provid
 ing asymptotic bounds for the minimal error as n tends to infinity.
LOCATION:Central Bldg / O1 / Mondi 2a (I01.O1.008) and Zoom\, ISTA
ORGANIZER:
SUMMARY:Filippo Quattrocchi: Thesis Defense: Optimal Transport Methods for 
 Kinetic Equations\, Boundary Value Problems\, and Discretization of Measur
 es
URL:https://talks-calendar.ista.ac.at/events/6016
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