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Title: Optimal Transport: Some Analytic and Geometric Aspects
Authors: Chen, Eric
Advisors: Chang, Sun-Yung Alice
Contributors: Ache, Antonio
Department: Mathematics
Class Year: 2014
Abstract: Let µ and ν be probability measures on R n , and let c : R n ×R n → [0, ∞) be a cost function. In its basic form, the optimal transport problem is to find a map T : R n → R n pushing the measure µ forward to ν, minimizing the cost of transportation R Rn c(x, T(x)) dµ(x). We review the reformulation of this problem in the context of the Kantorovich duality and describe some results on optimal transport with the quadratic cost function c(x, y) = |x−y| 2/2, in particular Brenier’s polar factorization theorem and its consequences. Finally, we discuss some results on the regularity of optimal maps in the quadratic cost setting and give a new condition guaranteeing the discontinuity of optimal maps in R n .
Extent: 93 pages
Type of Material: Princeton University Senior Theses
Language: en_US
Appears in Collections:Mathematics, 1934-2020

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