scholarly journals Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

2020 ◽  
Author(s):  
Mohit Bansil ◽  
Jun Kitagawa
Keyword(s):  
Optimal Transport ◽  
Convex Geometry ◽  
Regularity Theory ◽  
Uniform Norm ◽  
Quantitative Stability ◽  
Wirtinger Inequality ◽  
Dual Variables ◽  
Transport Problems ◽  
Monge Ampere ◽  
Stability Results

Abstract We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory.

scholarly journals Nonlinear systems coupled through multi-marginal transport problems

2019 ◽  
Vol 31 (3) ◽  
pp. 450-469
Author(s):  
M. LABORDE
Keyword(s):  
Nonlinear Systems ◽  
Optimal Transport ◽  
Gradient Flow ◽  
Uniqueness Result ◽  
System Of Nonlinear Equations ◽  
Planning Model ◽  
Highly Nonlinear ◽  
Transport Problems ◽  
Dimension One ◽  
Monge Ampere

In this paper, we introduce a dynamical urban planning model. This leads to the study of a system of nonlinear equations coupled through multi-marginal optimal transport problems. A first example consists in solving two equations coupled through the solution to the Monge–Ampère equation. We show that theWasserstein gradient flow theory provides a very good framework to solve these highly nonlinear systems. At the end, a uniqueness result is presented in dimension one based on convexity arguments.

scholarly journals On discontinuity of planar optimal transport maps

2015 ◽  
Vol 07 (02) ◽  
pp. 239-260 ◽  
Author(s):  
Otis Chodosh ◽  
Vishesh Jain ◽  
Michael Lindsey ◽  
Lyuboslav Panchev ◽  
Yanir A. Rubinstein
Keyword(s):  
Optimal Transport ◽  
Main Idea ◽  
Regularity Theory ◽  
Probability Measures ◽  
Bounded Domains ◽  
Quadratic Cost ◽  
Extrinsic Geometry ◽  
Discontinuous Map ◽  
Transport Maps ◽  
Monge Ampere

Consider two bounded domains Ω and Λ in ℝ2, and two sufficiently regular probability measures μ and ν supported on them. By Brenier's theorem, there exists a unique transportation map T satisfying T#μ = ν and minimizing the quadratic cost ∫ℝn ∣T(x) - x∣2 dμ(x). Furthermore, by Caffarelli's regularity theory for the real Monge–Ampère equation, if Λ is convex, T is continuous. We study the reverse problem, namely, when is T discontinuous if Λ fails to be convex? We prove a result guaranteeing the discontinuity of T in terms of the geometries of Λ and Ω in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of ∂Λ to distinguish between Brenier and Alexandrov weak solutions of the Monge–Ampère equation. We also use this approach to give a new proof of a result due to Wolfson and Urbas. We conclude by revisiting an example of Caffarelli, giving a detailed study of a discontinuous map between two explicit domains, and determining precisely where the discontinuities occur.

Quantitative stability and error estimates for optimal transport plans

2020 ◽  
Author(s):  
Wenbo Li ◽  
Ricardo H Nochetto
Keyword(s):  
Error Estimates ◽  
Convergence Rates ◽  
Optimal Transport ◽  
Quantitative Stability ◽  
Fully Discrete ◽  
Discrete Algorithms ◽  
Transport Maps ◽  
Transport Problems ◽  
Optimal Transport Map ◽  
Continuous Measures

Abstract Optimal transport maps and plans between two absolutely continuous measures $\mu$ and $\nu$ can be approximated by solving semidiscrete or fully discrete optimal transport problems. These two problems ensue from approximating $\mu$ or both $\mu$ and $\nu$ by Dirac measures. Extending an idea from Gigli (2011, On Hölder continuity-in-time of the optimal transport map towards measures along a curve. Proc. Edinb. Math. Soc. (2), 54, 401–409), we characterize how transport plans change under the perturbation of both $\mu$ and $\nu$. We apply this insight to prove error estimates for semidiscrete and fully discrete algorithms in terms of errors solely arising from approximating measures. We obtain weighted $L^2$ error estimates for both types of algorithms with a convergence rate $O(h^{1/2})$. This coincides with the rate in Theorem 5.4 in Berman (2018, Convergence rates for discretized Monge–Ampère equations and quantitative stability of optimal transport. Preprint available at arXiv:1803.00785) for semidiscrete methods, but the error notion is different.

scholarly journals Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Siran Li
Keyword(s):  
Banach Space ◽  
Discrete Geometry ◽  
Thin Shell ◽  
Euclidean Geometry ◽  
Convex Geometry ◽  
Regularity Theory ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Space Theory ◽  
Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

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