Numerical Optimal Transport from 1D to 2D using a Non-local Monge-Amp\`ere Equation

TL;DR

This paper introduces a novel numerical method for solving optimal transport problems between different-dimensional densities, specifically from 1D to 2D, using a non-local Monge-Ampère equation and a new level set framework.

Contribution

It presents a new discretisation combining monotone finite difference schemes with a variable-support Dirac delta, enabling consistent and monotone solutions for 1D to 2D optimal transport.

Findings

Method successfully solves challenging 1D to 2D transport problems.

Numerical tests validate the accuracy and robustness of the proposed approach.

Framework can be extended to higher dimensions.

Abstract

We consider the numerical solution of the optimal transport problem between densities that are supported on sets of unequal dimension. Recent work by McCann and Pass reformulates this problem into a non-local Monge-Amp\`ere type equation. We provide a new level set framework for interpreting this non-linear PDE. We also propose a novel discretisation that combines carefully constructed monotone finite difference schemes with a variable-support discrete version of the Dirac delta function. The resulting method is consistent and monotone. These new techniques are described and implemented in the setting of 1D to 2D transport, but can easily be generalised to higher dimensions. Several challenging computational tests validate the new numerical method.