scholarly journals From Knothe's Rearrangement to Brenier's Optimal Transport Map

2013 ◽  
Vol 45 (1) ◽  
pp. 64-87 ◽  
Author(s):  
Nicolas Bonnotte
Keyword(s):  
Optimal Transport ◽  
Optimal Transport Map

scholarly journals Pollution Transfer as Optimal Mass Transport Problem

2016 ◽  
Vol 8 (6) ◽  
pp. 58
Author(s):  
L. Ndiaye ◽  
Mb. Ndiaye ◽  
A. Sy ◽  
D. Seck
Keyword(s):  
Porous Media ◽  
Optimal Transport ◽  
Existence Of Solution ◽  
Nonlinear Pde ◽  
Mathematical Framework ◽  
Sufficient Condition ◽  
Mass Transportation ◽  
Optimal Mass Transport ◽  
Transportation Theory ◽  
Optimal Transport Map

In this paper, we use mass transportation theory to study pollution  transfer in  porous media.  We show   the existence of a $L^2-$regular vector field defined by a $W^{1, 1}-$ optimal transport map. A sufficient condition for solvability of our model, is given by   a (non homogeneous) transport equation with  a  source defined by a measure. The mathematical framework used, allows us to  show in some specifical cases, existence of solution for  a nonlinear PDE deriving from the modelling. And we end by numerical simulations.

Freeform illumination optics construction following an optimal transport map

2016 ◽  
Vol 55 (16) ◽  
pp. 4301 ◽  
Author(s):  
Zexin Feng ◽  
Brittany D. Froese ◽  
Rongguang Liang
Keyword(s):  
Optimal Transport ◽  
Optimal Transport Map

scholarly journals Optimal transport with discrete long-range mean-field interactions

2020 ◽  
Vol 10 (02) ◽  
pp. 2050011
Author(s):  
Jiakun Liu ◽  
Grégoire Loeper
Keyword(s):  
Force Field ◽  
Long Range ◽  
Interaction Potential ◽  
Velocity Potential ◽  
Optimal Transport ◽  
Mean Field ◽  
External Force Field ◽  
Intermediate Time ◽  
Optimal Transport Map ◽  
Intermediate Density

We study an optimal transport problem where, at some intermediate time, the mass is either accelerated by an external force field or self-interacting. We obtain the regularity of the velocity potential, intermediate density, and optimal transport map, under the conditions on the interaction potential that are related to the so-called Ma–Trudinger–Wang condition from optimal transport [X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problems, Arch. Ration. Mech. Anal. 177 (2005) 151–183.].

On Hölder continuity-in-time of the optimal transport map towards measures along a curve

2011 ◽  
Vol 54 (2) ◽  
pp. 401-409 ◽  
Author(s):  
Nicola Gigli
Keyword(s):  
Optimal Transport ◽  
Hölder Continuity ◽  
Holder Continuity ◽  
Continuous Curve ◽  
Hölder Continuous ◽  
Absolutely Continuous ◽  
Reference Measure ◽  
Optimal Transport Map

AbstractWe discuss the problem of the regularity-in-time of the map t ↦ Tt ∊ Lp(ℝd, ℝd; σ), where Tt is a transport map (optimal or not) from a reference measure σ to a measure μt which lies along an absolutely continuous curve t ↦ μt in the space ($(\mathscr{P}_p(\mathbb{R}^d),W_p)$)). We prove that in most cases such a map is no more than 1/p-Hölder continuous.

scholarly journals Freeform optical beam shaping following an optimal transport map

2015 ◽  
Author(s):  
Zexin Feng ◽  
Brittany D. Froese ◽  
Lei Huang ◽  
Mali Gong ◽  
Guofan Jin ◽  
...  
Keyword(s):  
Optimal Transport ◽  
Beam Shaping ◽  
Optical Beam ◽  
Optimal Transport Map

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 Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability

2020 ◽  
Vol 24 ◽  
pp. 703-717
Author(s):  
Aurélien Alfonsi ◽  
Benjamin Jourdain
Keyword(s):  
Optimal Transport ◽  
Direct Proof ◽  
Invariant Function ◽  
Wasserstein Distance ◽  
Collège De France ◽  
Fréchet Differential ◽  
Probabilistic Proof ◽  
Optimal Coupling ◽  
Optimal Transport Map ◽  
Appl Math

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance W22(μ,ν) between two probability measures μ and ν with finite second order moments on ℝd is the composition of a martingale coupling with an optimal transport map 𝛵. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between μ and 𝛵#μ. Next, we give a direct proof that σ ↦ W22(σ,ν) is differentiable at μ in the Lions (Cours au Collège de France. 2008) sense iff there is a unique optimal coupling between μ and ν and this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savaré (Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005) and Ambrosio and Gangbo (Comm. Pure Appl. Math., 61:18–53, 2008) that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu (J. Math. Pures Appl. 125:119–174, 2019). Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant function F on L2(Ω, ℙ; ℝd) is enough for the Fréchet differential at X to be a measurable function of X.

Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ping Chen
Keyword(s):  
Cost Function ◽  
Optimal Transport ◽  
Geodesic Distance ◽  
Variational Problems ◽  
Strictly Convex ◽  
Monotonicity Properties ◽  
Monge Problem ◽  
Transport Maps ◽  
Optimal Transport Map

AbstractIn this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function \lVert y-x\rVert_{\mathrm{sc}} in R^{n} or 𝑇 is an optimal transport map for the Monge problem with cost function d(x,y), the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function \lVert y-x\rVert_{\mathrm{sc}}. Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function \lVert y-x\rVert_{\mathrm{sc}} further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others.

scholarly journals Optimal transport and unitary orbits in C⁎-algebras

2021 ◽  
Vol 281 (5) ◽  
pp. 109068
Author(s):  
Bhishan Jacelon ◽  
Karen R. Strung ◽  
Alessandro Vignati
Keyword(s):  
Optimal Transport ◽  
C Algebras
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