Results 1 -
7 of
7
From Knothe’s transport to Brenier’s map and a continuation method for optimal transport
- SIAM J. Math. An
"... A simple procedure to map two probability measures in Rd is the so-called Knothe-Rosenblatt rearrangement, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of sol ..."
Abstract
-
Cited by 10 (2 self)
- Add to MetaCart
A simple procedure to map two probability measures in Rd is the so-called Knothe-Rosenblatt rearrangement, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.
Rectifiability of Optimal Transportation Plans
, 2010
"... The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is C² with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge’s optimal transportati ..."
Abstract
-
Cited by 8 (5 self)
- Add to MetaCart
The purpose of this note is to show that the solution to the Kantorovich optimal transportation problem is supported on a Lipschitz manifold, provided the cost is C² with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge’s optimal transportation problem satisfy a change of variables equation almost everywhere.
Rearrangements of vector valued functions, with application to atmospheric and oceanic ows
"... This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniquene ..."
Abstract
-
Cited by 5 (2 self)
- Add to MetaCart
(Show Context)
This paper establishes the equivalence of four de nitions of two vector valued functions being rearrangements, and gives a characterisation of the set of rearrangements of a prescribed function. The theory of monotone rearrangement ofavector valued function is used to show the existence and uniqueness of the minimiser of an energy functional arising from a model for atmospheric and oceanic ow. At each xed time solutions are shown to be equal to the gradient of a convex function, verifying the conjecture of Cullen, Norbury and Purser. Key words Rearrangement of functions, semigeostrophic, variational problems, generalised solution.
Convexity of the support of the displacement interpolation: counterexamples
, 2015
"... Given two smooth and positive densities ρ0, ρ1 on two compact convex sets K0,K1, respec-tively, we consider the question whether the support of the measure ρt obtained as the geodesic interpolant of ρ0 and ρ1 in the Wasserstein space W2(Rd) is necessarily convex or not. We prove that this is not the ..."
Abstract
-
Cited by 1 (0 self)
- Add to MetaCart
(Show Context)
Given two smooth and positive densities ρ0, ρ1 on two compact convex sets K0,K1, respec-tively, we consider the question whether the support of the measure ρt obtained as the geodesic interpolant of ρ0 and ρ1 in the Wasserstein space W2(Rd) is necessarily convex or not. We prove that this is not the case, even when ρ0 and ρ1 are uniform measures. 1
Optimal Transport for Applied Mathematicians Calculus of Variations, PDEs and Modelling
"... 1.1 Kantorovich and Monge problems................ 9 1.2 Duality.............................. 17 1.3 Strictly convex costs....................... 21 ..."
Abstract
- Add to MetaCart
1.1 Kantorovich and Monge problems................ 9 1.2 Duality.............................. 17 1.3 Strictly convex costs....................... 21
