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Numerical Solution of Monge–Kantorovich Equations via a Dynamic Formulation

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Abstract

We extend our previous work on a biologically inspired dynamic Monge–Kantorovich model (Facca et al. in SIAM J Appl Math 78:651–676, 2018) and propose it as an effective tool for the numerical solution of the \(L^{1}\)-PDE based optimal transportation model. We first introduce a new Lyapunov-candidate functional and show that its derivative along the solution trajectory is strictly negative. Moreover, we are able to show that this functional admits the optimal transport density as a unique minimizer, providing further support to the conjecture that our dynamic model is time-asymptotically equivalent to the Monge–Kantorovich equations governing \(L^{1}\) optimal transport. Remarkably, this newly proposed Lyapunov-candidate functional can be effectively used to calculate the Wasserstein-1 (or earth mover’s) distance between two measures. We numerically solve these equations via a simple approach based on standard forward Euler time stepping and linear Galerkin finite element. The accuracy and robustness of the proposed solver is verified on a number of test problems of mixed complexity also in comparison with other approaches proposed in the literature. Numerical results show that the proposed scheme is very efficient and accurate for the calculation the Wasserstein-1 distances.

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References

  1. Ambrosio, L.: Lecture notes on optimal transport problems. In: Mathematical Aspects of Evolving Interfaces. Springer, pp. 1–52 (2003)

  2. Barrett, J.W., Prigozhin, L.: A mixed formulation of the Monge–Kantorovich equations. Math. Model. Num. Anal. 41, 1041–1060 (2007)

    Article  MathSciNet  Google Scholar 

  3. Bartels, S., Schön, P.: Adaptive approximation of the Monge–Kantorovich problem via primal-dual gap estimates. ESAIM-Math. Model. Num. 51, 2237–2261 (2017)

    Article  MathSciNet  Google Scholar 

  4. Beirão da Veiga, L., Manzini, G., Putti, M.: Post processing of solution and flux for the nodal mimetic finite difference method. Num. Methods PDE 31, 336–363 (2014)

    Article  MathSciNet  Google Scholar 

  5. Benamou, J.-D., Brenier, Y.: A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84, 375–393 (2000)

    Article  MathSciNet  Google Scholar 

  6. Benamou, J.-D., Brenier, Y., Guittet, K.: The Monge–Kantorovitch mass transfer and its computational fluid mechanics formulation. Int. J. Numer. Methods Fluids, 40:21–30. ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 2001) (2002)

  7. Benamou, J.-D., Carlier, G.: Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations. J. Opt. Theory Appl. 167, 1–26 (2015)

    Article  MathSciNet  Google Scholar 

  8. Benamou, J.-D., Carlier, G., Cuturi, M., Nenna, L., Peyré, G.: Iterative Bregman projections for regularized transportation problems. SIAM J. Sci. Comput. 37, A1111–A1138 (2015)

    Article  MathSciNet  Google Scholar 

  9. Bergamaschi, L., Facca, E., Martínez, A., Putti, M.: Spectral preconditioners for the efficient numerical solution of a continuous branched transport model. J. Comput. Appl. Math. 354, 259–270 (2018)

    Article  MathSciNet  Google Scholar 

  10. Bochev, P., Lehoucq, R.B.: On the finite element solution of the pure Neumann problem. SIAM Rev. 47, 50–66 (2005)

    Article  MathSciNet  Google Scholar 

  11. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. In: Springer Series in Computational Mathematics. Springer, Berlin (2013)

  12. Bonifaci, V., Mehlhorn, K., Varma, G.: Physarum can compute shortest paths. J. Theor. Biol. 309, 121–133 (2012)

    Article  MathSciNet  Google Scholar 

  13. Bouchitté, G., Buttazzo, G., Seppecher, P.: Shape optimization solutions via Monge–Kantorovich equation. C. R. Acad. Sci. Paris Sér. I Math 324, 1185–1191 (1997)

    Article  MathSciNet  Google Scholar 

  14. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

    Book  Google Scholar 

  15. Buttazzo, G., Stepanov, E.: On regularity of transport density in the Monge–Kantorovich problem. SIAM J. Control Optim 42, 1044–1055 (2003)

    Article  MathSciNet  Google Scholar 

  16. Cuturi, M.: Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances, ArXiv e-prints (2013)

  17. Delzanno, G.L., Finn, J.M.: Generalized Monge–Kantorovich optimization for grid generation and adaptation in \(l_{p}\). SIAM J. Sci. Comput. 32, 3524–3547 (2010)

    Article  MathSciNet  Google Scholar 

  18. Delzanno, G.L., Finn, J.M.: The fluid dynamic approach to equidistribution methods for grid adaptation. Comput. Phys. Commun. 182, 330–346 (2011)

    Article  MathSciNet  Google Scholar 

  19. Evans, L.C., Gangbo, W.: Differential equations methods for the Monge–Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137, 1–66 (1999)

    MathSciNet  MATH  Google Scholar 

  20. Facca, E., Cardin, F., Putti, M.: Towards a stationary Monge–Kantorovich dynamics: The Physarum Polycephalum experience. SIAM J. Appl. Math. 78, 651–676 (2018)

    Article  MathSciNet  Google Scholar 

  21. Feldman, M., McCann, R.J.: Uniqueness and transport density in Monge’s mass transportation problem. Calc. Var. Partial Differ. 15, 81–113 (2002)

    Article  MathSciNet  Google Scholar 

  22. Flamary, R., Courty, N.: Pot python optimal transport library, (2017)

  23. Fragalà, I., Gelli, M.S., Pratelli, A.: Continuity of an optimal transport in Monge problem. J. Math. Pure Appl. 84, 1261–1294 (2005)

    Article  MathSciNet  Google Scholar 

  24. Jacobs, M., Léger, F., Li, W., Osher, S.: Solving large-scale optimization problems with a convergence rate independent of grid size, arXiv, (2018)

  25. Larson, M.G., Niklasson, A.J.: A conservative flux for the continuous Galerkin method based on discontinuous enrichment. Calcolo 41, 1–12 (2004)

    Article  MathSciNet  Google Scholar 

  26. Li, W., Ryu, E.K., Osher, S., Yin, W., Gangbo, W.: A parallel method for Earth Mover’s distance. J. Scient. Comput. 75, 182–197 (2018)

    Article  MathSciNet  Google Scholar 

  27. Liu, J., Yin, W., Li, W., Chow, Y. T.: Multilevel optimal transport: a fast approximation of Wasserstein-1 distances, arXiv preprint arXiv:1810.00118, (2018)

  28. Nakagaki, T., Yamada, H., Toth, A.: Maze-solving by an amoeboid organism. Nature 407, 470–470 (2000)

    Article  Google Scholar 

  29. Perrot, M., Courty, N., Flamary, R., Habrard, A.: Mapping estimation for discrete optimal transport, In: Advances in Neural Information Processing Systems, pp. 4197–4205 (2016)

  30. Putti, M., Cordes, C.: Finite element approximation of the diffusion operator on tetrahedra. SIAM J. Sci. Comput. 19, 1154–1168 (1998)

    Article  MathSciNet  Google Scholar 

  31. Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations. Springer Series in Computational Mathematics, vol. 23. Springer-Verlag, Berlin (1994)

  32. Ryu, E.K., Chen, Y., Li, W., Osher, S.: Vector and matrix optimal mass transport: Theory, algorithm, and applications. SIAM J. Sci. Comput. 40, A3675–A3698 (2018)

    Article  MathSciNet  Google Scholar 

  33. Santambrogio, F.: Optimal transport for applied mathematicians, vol. 87 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling

  34. Solomon, J., Rustamov, R., Guibas, L., Butscher, A.: Earth mover’s distances on discrete surfaces. ACM Transactions on Graphics (TOG) 33, 67 (2014)

    Article  Google Scholar 

  35. Tero, A., Kobayashi, R., Nakagaki, T.: A mathematical model for adaptive transport network in path finding by true slime mold. J. Theor. Biol. 244, 553–564 (2007)

    Article  MathSciNet  Google Scholar 

  36. Vardi, Y., Zhang, C.-H.: A modified Weiszfeld algorithm for the Fermat-Weber location problem. Math. Prog. 90, 559–566 (2001)

    Article  MathSciNet  Google Scholar 

  37. Villani, C.: Optimal transport, old and new. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer-Verlag, Berlin (2009)

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Acknowledgements

This work was partially funded by the the UniPD-SID-2016 project “Approximation and discretization of PDEs on Manifolds for Environmental Modeling” and by the EU-H2020 project “GEOEssential-Essential Variables workflows for resource efficiency and environmental management”, project of “The European Network for Observing our Changing Planet (ERA-PLANET)”, GA 689443.

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Authors and Affiliations

  1. Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore, Piazza dei Cavalieri, 3, Pisa, Italy

    Enrico Facca

  2. Gran Sasso Science Institute, Viale F. Crispi, 7, 67100, L’Aquila, Italy

    Sara Daneri

  3. Department of Mathematics “Tullio Levi-Civita”, University of Padua, via Trieste 63, Padua, Italy

    Franco Cardin & Mario Putti

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  1. Enrico Facca
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  2. Sara Daneri
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  3. Franco Cardin
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  4. Mario Putti
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Correspondence to Mario Putti.

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Facca, E., Daneri, S., Cardin, F. et al. Numerical Solution of Monge–Kantorovich Equations via a Dynamic Formulation. J Sci Comput 82, 68 (2020). https://doi.org/10.1007/s10915-020-01170-8

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  • DOI: https://doi.org/10.1007/s10915-020-01170-8

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