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 </math> is the starting measure, and <math> \nu = T_\# \mu </math>. | Consider the case when <math> \mu, \nu </math> are absolutely continuous, where <math> \mu </math> is the starting measure, and <math> \nu = T_\# \mu </math>. Say <math> f </math> and <math> g </math> are the densities of <math> \mu </math> and <math> \nu </math> respectively. Moreover, let <math> T </math> be <math> C^1 </math> and injective. | ||
==References== | ==References== | ||
Revision as of 00:00, 9 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 . Say and are the densities of and respectively. Moreover, let be and injective.
