Talk:Optimal Transport and the Monge Ampère equation: Difference between revisions
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* The explanation of the relationship to optimal transport is a little confusing. It might be better to explain what a transport map is (perhaps linking to the page on the Monge Problem on the OT wiki) and say something along the lines of ``under sufficient regularity assumptions on the measures mu and nu, the condition that a transport map pushes forward mu to nu can be equivalently formulated in terms of the transport map solving a type of Monge-Ampere equation`` | * The explanation of the relationship to optimal transport is a little confusing. It might be better to explain what a transport map is (perhaps linking to the page on the Monge Problem on the OT wiki) and say something along the lines of ``under sufficient regularity assumptions on the measures mu and nu, the condition that a transport map pushes forward mu to nu can be equivalently formulated in terms of the transport map solving a type of Monge-Ampere equation`` | ||
* The only time that we need the cost to be quadratic is to reduce to the case that the transport map is given by the gradient of a convex function. Maybe it's best to save explaining this technicality to the next section. | * The only time that we need the cost to be quadratic is to reduce to the case that the transport map is given by the gradient of a convex function. Maybe it's best to save explaining this technicality to the next section. | ||
==Deriving the Monge Ampere Equation== | |||
* Emphasize that the infimum is over all transport maps pushing forward mu to nu. | |||
* There is a type-o -- an extra > sign. | |||
* ``tell us that T pushes forward mu to nu if and only if`` | |||
* ``The above equation is a type of Monge-Ampere equation`` | |||
==Notable Properties of the Monge-Ampere equation== | |||
* Move the discussion of the boundary conditions to the previous section, and start with ``Properties of solutions to the Monge Ampere equation give us information about the optimal transport map`` | |||
* Cite Villani for the theorem you quote and mention Caffarelli's contribution | |||
* Cite Villani for the notion of Brenier solutions | |||
Revision as of 22:45, 13 May 2020
Beginning
- Cite specific pages where Santambrogio discusses Monge Ampere, since this is only a small part of that chapter
- The explanation of the relationship to optimal transport is a little confusing. It might be better to explain what a transport map is (perhaps linking to the page on the Monge Problem on the OT wiki) and say something along the lines of ``under sufficient regularity assumptions on the measures mu and nu, the condition that a transport map pushes forward mu to nu can be equivalently formulated in terms of the transport map solving a type of Monge-Ampere equation``
- The only time that we need the cost to be quadratic is to reduce to the case that the transport map is given by the gradient of a convex function. Maybe it's best to save explaining this technicality to the next section.
Deriving the Monge Ampere Equation
- Emphasize that the infimum is over all transport maps pushing forward mu to nu.
- There is a type-o -- an extra > sign.
- ``tell us that T pushes forward mu to nu if and only if``
- ``The above equation is a type of Monge-Ampere equation``
Notable Properties of the Monge-Ampere equation
- Move the discussion of the boundary conditions to the previous section, and start with ``Properties of solutions to the Monge Ampere equation give us information about the optimal transport map``
- Cite Villani for the theorem you quote and mention Caffarelli's contribution
- Cite Villani for the notion of Brenier solutions