A convex numerical scheme for the Evans-Gangbo transport-density system, in one, two, and three dimensions
Abstract
We give a constructive numerical treatment of Monge's optimal transport problem in 1D, 2D, 3D, following Evans-Gangbo: the map is built from a Kantorovich potential u and density a solving −div(a∇ u) = f+−f-, via Ṫ = −a∇ u(T)/[(1−t)f+(T)+tf-(T)], T(0,x)=x. In 1D this gives a closed form. In 2D--left unfinished in an earlier report--we obtain (u,a) via a convex Beckmann flow problem (second-order-cone programming), cross-checked against a dual LP, then build the map by direct integration or a ray-based 1D "clock" ODE with explicit density per ray. Validation uses strong duality, mass conservation, and vanishing-regularization limits, on a translation-invariant case and two 2D examples (a radial cone, a checkerboard) checked by symmetry; both show boundary-concentrated mass and non-unique flux where supports touch. We extend to 3D (theory unchanged), matching the 2D answer to six digits and reproducing the same phenomena on a radial ball, exposing a limit of the 2D visualization style. Movies illustrating the transport maps are available as ancillary files on the arXiv page of this submission.
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BibTeX
@misc{ndiaye2026convex,
title = {{A convex numerical scheme for the Evans-Gangbo transport-density system, in one, two, and three dimensions}},
author = {Babacar Mbaye Ndiaye and Diaraf Seck},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29987},
eprint = {2609.29987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29987},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A convex numerical scheme for the Evans-Gangbo transport-density system, in one, two, and three dimensions
AU - Babacar Mbaye Ndiaye
AU - Diaraf Seck
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29987
UR - https://arxiv.org/abs/2609.29987
AB - We give a constructive numerical treatment of Monge's optimal transport problem in 1D, 2D, 3D, following Evans-Gangbo: the map is built from a Kantorovich potential u and density a solving −div(a∇ u) = f^+−f^-, via Ṫ = −a∇ u(T)/[(1−t)f^+(T)+tf^-(T)], T(0,x)=x. In 1D this gives a closed form. In 2D--left unfinished in an earlier report--we obtain (u,a) via a convex Beckmann flow problem (second-order-cone programming), cross-checked against a dual LP, then build the map by direct integration or a ray-based 1D "clock" ODE with explicit density per ray. Validation uses strong duality, mass conservation, and vanishing-regularization limits, on a translation-invariant case and two 2D examples (a radial cone, a checkerboard) checked by symmetry; both show boundary-concentrated mass and non-unique flux where supports touch. We extend to 3D (theory unchanged), matching the 2D answer to six digits and reproducing the same phenomena on a radial ball, exposing a limit of the 2D visualization style. Movies illustrating the transport maps are available as ancillary files on the arXiv page of this submission.
ER -