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

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