Optimal Mass Transport Problem

• Introduction:

  
  
  • The optimal mass transport problem also know as Monge-Kantrovich problem (MKP) first arises in the engineering community and it concerns finding an optimal way of moving a pile of sand from one configuration to another with minimum total work.
  
  

• L² Monge-Kantorovich Problem:

  
  
  • The $L^2$ MKP states that given two positive density functions $\rho_0$ and $\rho_1$, find the mapping $\phi$ that transfers $\rho_0$ to $\rho_1$ and minimizes the energy functional
    \[ C(\phi) = \int_{\mathbb{R}^d}\left|\phi(x) - x\right|^2\rho_0(x) d x.~~~~~~\qquad\]
  • The map $\phi$ realizes the transfer of $\rho_0$ to $\rho_1$ if \[ \int_{\phi^{-1}(\Omega)} \rho_0(x)dx = \int_{\Omega}\rho_1(x)\, dx, \qquad \forall \; \Omega\subset \mathbb{R}^d~~~~~~\qquad~~~~~~\qquad \]
  • This leads to \[ \rho_1(\phi) \textrm{det} \left(\frac{\partial \phi}{\partial x}\right) = \rho_0.\]
  • Under some regularity conditions, there is a unique $ \phi$ that solves the $L^2$ MKP and it is characterized as $\phi = \nabla \Psi.$
  
  
  • The solution of the $L^2$ MKP reduces to solving the Monge Apère equation (MAE): \[ \rho_1\left(\nabla \Psi\right) \textrm{det}\left( D^2 \Psi\right) = \rho_0,\] where $D^2 \Psi$ is the Hessian matrix of $\Psi$.
  
  
  • In my recent work (with Russell and Williams), we study the numerical solution of MAE.