Minimal convex extensions and finite difference discretization of the quadratic Monge-Kantorovich problem
This work provides a provably convergent numerical method for a specific class of Optimal Transport problems, offering a fast adaptive approach for practitioners.
The authors adapt the MA-LBR scheme to solve the Optimal Transport problem between two absolutely continuous measures where the target has convex support, proving convergence and demonstrating the method's ability to capture a minimal Brenier solution.
We present an adaptation of the MA-LBR scheme to the Monge-Amp{è}re equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the Optimal Transport problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid stepsize vanishes and we show with numerical experiments that it is able to reproduce subtle properties of the Optimal Transport problem.