A Volumetric Approach to Monge's Optimal Transport on Surfaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917666110832640 |
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| author | Tsai, Richard Turnquist, Axel G. R. |
| author_facet | Tsai, Richard Turnquist, Axel G. R. |
| contents | We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in $\mathbb{R}^3$, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the original problem on the surface, we define a new Optimal Transport problem on a thin tubular region, $T_ε$, adjacent to the surface. This extension offers enhanced flexibility and simplicity for numerical discretization on Cartesian grids. The Optimal Transport mapping and potential function computed on $T_ε$ are consistent with the original problem on surfaces. We demonstrate that, with the proposed volumetric approach, it is possible to use simple and straightforward numerical methods to solve Optimal Transport for $Γ= \mathbb{S}^2$ and the $2$-torus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01745 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Volumetric Approach to Monge's Optimal Transport on Surfaces Tsai, Richard Turnquist, Axel G. R. Numerical Analysis 65N06 (Primary), 49Q22, 35R01, 35J15, 35J96, 35J60, 58J05 (Secondary) We propose a volumetric formulation for computing the Optimal Transport problem defined on surfaces in $\mathbb{R}^3$, found in disciplines like optics, computer graphics, and computational methodologies. Instead of directly tackling the original problem on the surface, we define a new Optimal Transport problem on a thin tubular region, $T_ε$, adjacent to the surface. This extension offers enhanced flexibility and simplicity for numerical discretization on Cartesian grids. The Optimal Transport mapping and potential function computed on $T_ε$ are consistent with the original problem on surfaces. We demonstrate that, with the proposed volumetric approach, it is possible to use simple and straightforward numerical methods to solve Optimal Transport for $Γ= \mathbb{S}^2$ and the $2$-torus. |
| title | A Volumetric Approach to Monge's Optimal Transport on Surfaces |
| topic | Numerical Analysis 65N06 (Primary), 49Q22, 35R01, 35J15, 35J96, 35J60, 58J05 (Secondary) |
| url | https://arxiv.org/abs/2310.01745 |