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-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 language | English (US) |
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Pages (from-to) | 509-535 |
Number of pages | 27 |
Journal | Matematica |
Volume | 3 |
Issue number | 2 |
DOIs | |
State | Published - Jun 2024 |
All Science Journal Classification (ASJC) codes
- General Mathematics
Keywords
- Monge-Ampère type equation
- Non-local equations
- Numerical analysis
- Optimal transport