Optimal Transport and the Monge Ampère equation

The Monge Ampère equation is an nonlinear second-order elliptic partial differential equation. The Monge Problem optimizes a cost function over the set of transport maps that push forward a fixed probability measure, $\mu$, to another fixed probability measure, $\nu$. Given sufficient regularity conditions on the two measures, one can show that solving the corresponding Monge Ampère equation is equivalent to being a transport map for $\mu$ and $\nu$.

Deriving the Monge Ampère equation from the Monge Problem

The appropriate variant of the Monge Problem for this situation is

$\inf _{T}\left\{F(T):=\int _{X}|x-T(x)|^{2}d\mu \right\}$

Where the infimum is taken over all transport maps, $T$, that push forward $\mu$ to $\nu$. Consider the case when $\mu ,\nu$ are absolutely continuous, where $\mu$ is the starting measure, and $\nu =T_{\#}\mu$. Say $f$ and $g$ are the densities of $\mu$ and $\nu$ respectively. Moreover, let $T:X\to X$ be $C^{1}$ and injective. The change of variables formula tells us that $T$ pushes forward $\mu$ to $\nu$ if and only if`

$g(y)={\frac {f(T^{-1}(y))}{\det(DT(T^{-1}(y)))}}$

For this particular variant of the Monge problem, the cost function is quadratic. It can be shown that this implies when $T$ is optimal, $T=\nabla u$ where $u$ is convex. If we require $u$ to be strictly convex, $T$ is guaranteed to be injective. Once $\nabla u$ is substituted for $T$ in the change of variables formula, one obtains

$\det(D^{2}u(x))={\frac {f(x)}{g(\nabla u(x))}}$

The above equation is a type of Monge Ampère equation. Note that since $u$ from above is assumed to be convex, the Jacobian term is positive. Moreover, $\det D^{2}(u(x))$ has a monotonicity property when $D^{2}(u(x))\leq D^{2}(u(y))$. The specific boundary problem that is relevant to our Monge Problem is $\nabla u(\Omega )=\Omega '$. In this case, $\Omega$ is the domain for the measure $\mu$, and $\Omega '$ is the domain for the measure $\nu$. When $u$ is a homeomorphism, this implies that $u(\partial \Omega )=\partial \Omega '$. Solutions to this boundary value problem and the Monge Ampère equation gives us information about the optimal transport map. For example, regularity of the solutions to our boundary value problem implies regularity for the optimal transport map.

Notable Properties of the Monge Ampère equation

Properties of solutions to the Monge Ampère equation give us information about the optimal transport map. This relationship stems from the following regularity result by Caffarelli.

Theorem. If $f$ and $g$ are $C^{0,\alpha }(\Omega )$ and are both bounded from above and from below on the whole $\Omega$ by positive constants and $\Omega$ is a convex open set, then the unique Brenier solution of $u$ of the Monge Ampère equation belongs to $C^{2,\alpha }(\Omega )\cap C^{1,\alpha }{\overline {\Omega }}$, and $u$ satisfies the equation in the classical sense.

$u$ satisfies the Monge Ampère equation in the Brenier sense if $(\nabla u)_{\#}(f\cdot {\mathcal {L}}^{d})=(g\cdot {\mathcal {L}}^{d})$. $u$ is a classical solution if it satisfies the Monge Ampère equation at every point in $\Omega$ and the corresponding boundary value conditions. Note that being a Brenier solution implies that $\nabla u$ is a transport map for appropriate $f$ and $g$. In particular, we would like $f$ and $g$ to define probability measures absolutely continuous to the Lebesgue measure. This implies that given the appropriate regularity conditions on $\mu$ and $\nu$ for our Monge Problem, we can study classical solutions to the corresponding Monge Ampère equation instead.