Numerical Optimal Transport from 1D to 2D Using a Non-local Monge-Ampère Equation

Matthew A. Cassini, Brittany Froese Hamfeldt

Research output: Contribution to journalArticlepeer-review


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-Ampère type equation. We provide a new level-set framework for interpreting this nonlinear 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 they can easily be generalised to higher dimensions. Several challenging computational tests validate the new numerical method.

Original languageEnglish (US)
Pages (from-to)509-535
Number of pages27
Issue number2
StatePublished - Jun 2024

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Monge-Ampère type equation
  • Non-local equations
  • Numerical analysis
  • Optimal transport


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