Optimal Transport and the Monge Ampère equation: Difference between revisions
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== Deriving the Monge Ampère equation from the Monge Problem == | == Deriving the Monge Ampère equation from the Monge Problem == | ||
The appropriate variant of the Monge Problem for this situation is | |||
:<math> \inf_{T} \left \{ F(T) := \int_{X} |x - T(x)|^2 d \mu \right \} </math> | |||
==References== | ==References== | ||
Revision as of 23:44, 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
