# From Knothe's Transport to Brenier's Map and a Continuation Method for Optimal Transport

@article{Carlier2009FromKT, title={From Knothe's Transport to Brenier's Map and a Continuation Method for Optimal Transport}, author={Guillaume Carlier and Alfred Galichon and Filippo Santambrogio}, journal={SIAM J. Math. Anal.}, year={2009}, volume={41}, pages={2554-2576} }

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…

## 79 Citations

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

- MathematicsSIAM J. Math. Anal.
- 2013

It is proved that on the torus (to avoid boundary issues), when all the data are smooth, the evolution is also smooth and is entirely determined by a PDE for the Kantorovich potential (which determines the map) with a subtle initial condition.

Multi-physics optimal transportation and image interpolation

- Computer Science
- 2015

How some physics can be added to the optimal transportation theory, how to construct algorithms to compute solutions to the corresponding optimization problems and how to apply the proposed methods to image interpolation are studied.

Benamou-Brenier and other continuous numerical methods

- Mathematics
- 2015

In this chapter we present some numerical methods to solve optimal transport problems. The most famous method is for sure the one due to J.-D. Benamou and Y. Brenier, which transforms the problem…

Stochastic optimal transportation problem and related topics

- Mathematics, Computer Science
- 2014

The Knothe-R Rosenblatt process is introduced as a stochastic optimal control analogue of the KnotheRosenblatt rearrangement and it is shown that it can be approximated by a minimizer of a class of Stochastic optimal transportation problems with a small parameter.

A Newton algorithm for semi-discrete optimal transport

- Computer Science, MathematicsJournal of the European Mathematical Society
- 2019

The purpose of this article is to bridge the gap between theory and practice by introducing a damped Newton's algorithm which is experimentally efficient and by proving the global convergence of this algorithm with optimal rates.

Causal Transport in Discrete Time and Applications

- MathematicsSIAM J. Optim.
- 2017

A dynamic programming principle is established that links the causal transport problem to the transport problem for general costs recently considered by Gozlan et al. and gives conditions under which the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal analogue to the Brenier's map.

An algorithm for optimal transport between a simplex soup and a point cloud

- Computer Science, MathematicsSIAM J. Imaging Sci.
- 2018

The convergence with linear speed of a damped Newton's algorithm is proved to solve the optimal transport problem as the resolution of a non-linear system where one wants to prescribe the quantity of mass in each cell of the so-called Laguerre diagram.

A Hierarchical Approach to Optimal Transport

- Computer ScienceSSVM
- 2013

An extension of the auction algorithm is presented that exploits the regularity of the otherwise arbitrary cost function and takes into account a sparse subset of possible assignment pairs while still guaranteeing global optimality of the solution.

Optimal transport and barycenters for dendritic measures

- Mathematics, Computer Science
- 2019

A variant of the Wasserstein distance on the space of probability measures, specially designed to deal with measures whose support has a dendritic, or treelike structure with a particular direction of orientation, is introduced.

Particle flow inspired by Knothe-Rosenblatt transport for nonlinear filters

- MathematicsDefense, Security, and Sensing
- 2013

We derive a new algorithm for particle flow corresponding to Bayes’ rule that was inspired by Knothe- Rosenblatt transport, which is well known in transport theory. We emphasize that our flow is not…

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