Abstract
AbstractWe 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.
Funder
Directorate for Mathematical and Physical Sciences
Publisher
Springer Science and Business Media LLC
Reference29 articles.
1. Villani, C.: Topics in Optimal Transportation, vol. 58. American Mathematical Society, Providence (2021)
2. Chiappori, P., McCann, R.J., Pass, B.: Multi-to one-dimensional optimal transport. Commun. Pure Appl. Math. 70(12), 2405–2444 (2017)
3. Galichon, A.: Optimal Transport Methods in Economics. Princeton University Press, Princeton (2016)
4. Nenna, L., Pass, B.: Variational problems involving unequal dimensional optimal transport. J. Math. Pures Appl. 139, 83–108 (2020)
5. Cullen, M.J.P.: A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow. World Scientific, London (2006)