Optimal Transport and the Monge Ampère equation: Difference between revisions
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:<math> \inf_{T} \left \{ F(T) := \int_{X} |x - T(x)|^2 d \mu \right \} </math> | :<math> \inf_{T} \left \{ F(T) := \int_{X} |x - T(x)|^2 d \mu \right \} </math> | ||
Consider the case when <math> \mu, \nu </math> are absolutely continuous, where <math> \mu < | Consider the case when <math> \mu, \nu </math> are absolutely continuous, where <math> \mu </math> is the starting measure, and <math>\nu(B) = \mu(T^{-1}(B))</math> for every Borel set <math> B </math>. | ||
==References== | ==References== | ||
Revision as of 23:51, 8 May 2020
The Monge Ampère equation[1] is an nonlinear second-order elliptic partial differential equation. Given sufficient conditions on an optimal transport problem, the Monge Ampère equation defines a change of variables formula for the pushforward of a transport map. In particular, the Monge Ampère equation applies to a variant of the Monge Problem where the cost function is quadratic.
Deriving the Monge Ampère equation from the Monge Problem
The appropriate variant of the Monge Problem for this situation is
Consider the case when are absolutely continuous, where is the starting measure, and for every Borel set .
