Numerical Solution of the $L^1$-Optimal Transport Problem on Surfaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866929372113403904 |
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| author | Berti, Luca Facca, Enrico Putti, Mario |
| author_facet | Berti, Luca Facca, Enrico Putti, Mario |
| contents | In this article we study the numerical solution of the $L^1$-Optimal Transport Problem on 2D surfaces embedded in $R^3$, via the DMK formulation introduced in [FaccaCardinPutti:2018]. We extend from the Euclidean into the Riemannian setting the DMK model and conjecture the equivalence with the solution Monge-Kantorovich equations, a PDE-based formulation of the $L^1$-Optimal Transport Problem.
We generalize the numerical method proposed in [FaccaCardinPutti:2018,FaccaDaneriCardinPutti:2020] to 2D surfaces embedded in $\REAL^3$ using the Surface Finite Element Model approach to approximate the Laplace-Beltrami equation arising from the model. We test the accuracy and efficiency of the proposed numerical scheme, comparing our approximate solution with respect to an exact solution on a 2D sphere. The results show that the numerical scheme is efficient, robust, and more accurate with respect to other numerical schemes presented in the literature for the solution of ls$L^1$-Optimal Transport Problem on 2D surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_06479 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Numerical Solution of the $L^1$-Optimal Transport Problem on Surfaces Berti, Luca Facca, Enrico Putti, Mario Numerical Analysis Optimization and Control In this article we study the numerical solution of the $L^1$-Optimal Transport Problem on 2D surfaces embedded in $R^3$, via the DMK formulation introduced in [FaccaCardinPutti:2018]. We extend from the Euclidean into the Riemannian setting the DMK model and conjecture the equivalence with the solution Monge-Kantorovich equations, a PDE-based formulation of the $L^1$-Optimal Transport Problem. We generalize the numerical method proposed in [FaccaCardinPutti:2018,FaccaDaneriCardinPutti:2020] to 2D surfaces embedded in $\REAL^3$ using the Surface Finite Element Model approach to approximate the Laplace-Beltrami equation arising from the model. We test the accuracy and efficiency of the proposed numerical scheme, comparing our approximate solution with respect to an exact solution on a 2D sphere. The results show that the numerical scheme is efficient, robust, and more accurate with respect to other numerical schemes presented in the literature for the solution of ls$L^1$-Optimal Transport Problem on 2D surfaces. |
| title | Numerical Solution of the $L^1$-Optimal Transport Problem on Surfaces |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2106.06479 |