Numerical computation of generalized Wasserstein distances with applications to traffic model analysis

Maya Briani; Emiliano Cristiani; Giovanni Franzina; Francesca L. Ignoto; Maya Briani; Emiliano Cristiani; Giovanni Franzina; Francesca L. Ignoto

doi:10.3934/nhm.2025048

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1108-1144. doi: 10.3934/nhm.2025048

Previous Article Next Article

Research article

Numerical computation of generalized Wasserstein distances with applications to traffic model analysis

1.
Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Rome, Italy
2.
Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Rome, Italy

Received: 05 April 2025 Revised: 20 September 2025 Accepted: 09 October 2025 Published: 24 October 2025

Generalized Wasserstein distances allow us to quantitatively compare two continuous or atomic mass distributions with equal or different total masses. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the past few years, giving some insights into their physical meaning. After that, we explore their usage in the context of a sensitivity analysis of differential models for traffic flow. The quantification of the models' sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.
- Generalized Wasserstein distance,
- Wasserstein distance,
- traffic modeling,
- sensitivity analysis,
- linear programming,
- nonlinear programming,
- computational methods
Citation: Maya Briani, Emiliano Cristiani, Giovanni Franzina, Francesca L. Ignoto. Numerical computation of generalized Wasserstein distances with applications to traffic model analysis[J]. Networks and Heterogeneous Media, 2025, 20(4): 1108-1144. doi: 10.3934/nhm.2025048

Related Papers:

Abstract

Generalized Wasserstein distances allow us to quantitatively compare two continuous or atomic mass distributions with equal or different total masses. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the past few years, giving some insights into their physical meaning. After that, we explore their usage in the context of a sensitivity analysis of differential models for traffic flow. The quantification of the models' sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.

References

[1]	E. Cristiani, B. Piccoli, A. Tosin, Multiscale Modeling of Pedestrian Dynamics, Springer Cham, 978-3-319-06620-2, 2014. https://doi.org/10.1007/978-3-319-06620-2
[2]	M. Briani, E. Cristiani, E. Iacomini, Sensitivity analysis of the LWR model for traffic forecast on large networks using Wasserstein distance, Commun. Math. Sci., 16 (2018), 123–144. https://doi.org/10.4310/CMS.2018.v16.n1.a6 doi: 10.4310/CMS.2018.v16.n1.a6
[3]	E. Cristiani, M. C. Saladino, Comparing comparisons between vehicular traffic states in microscopic and macroscopic first-order models, Math. Methods Appl. Sci., 42 (2019), 918–934. https://doi.org/10.1002/mma.5395 doi: 10.1002/mma.5395
[4]	A. Figalli, N. Gigli, A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions, J. Math. Pures Appl., 94 (2010), 107–130. https://doi.org/10.1016/j.matpur.2009.11.005 doi: 10.1016/j.matpur.2009.11.005
[5]	B. Piccoli, F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335–358. https://doi.org/10.1007/s00205-013-0669-x doi: 10.1007/s00205-013-0669-x
[6]	S. Kondratyev, L. Monsaingeon, D. Vortnikov, A new optimal transport distance on the space of finite Radon measures, Adv. Differ. Equ., 21 (2016), 1117–1164.
[7]	L. Chizat, G. Peyré, B. Schmitzer, F. X. Vialard, An interpolating distance between optimal transport and Fisher–Rao metrics, Found. Comput. Math., 18 (2018), 1–44. https://doi.org/10.1007/s10208-016-9331-y doi: 10.1007/s10208-016-9331-y
[8]	L. Chizat, G. Peyré, B. Schmitzer, F. X. Vialard, Unbalanced optimal transport: Dynamic and Kantorovich formulations, J. Funct. Anal., 274 (2018), 3090–3123. https://doi.org/10.1016/j.jfa.2018.03.008 doi: 10.1016/j.jfa.2018.03.008
[9]	M. Liero, A. Mielke, G. Savaré, Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures, Invent. Math., 211 (2018), 969–1117. https://doi.org/10.1007/s00222-017-0759-8 doi: 10.1007/s00222-017-0759-8
[10]	Z. Ma, X. Wei, X. Hong, H. Lin, Y. Qiu, Y. Gong, Learning to count via unbalanced optimal transport, Proc. AAAI Conf. Artif. Intell., 35 (2021), 2319–2327. https://doi.org/10.1609/aaai.v35i3.16332 doi: 10.1609/aaai.v35i3.16332
[11]	G. Savaré, G. E. Sodini, A relaxation viewpoint to Unbalanced Optimal Transport: Duality, optimality and Monge formulation, J. Math. Pures Appl., 188 (2024), 114–178. https://doi.org/10.1016/j.matpur.2024.05.009 doi: 10.1016/j.matpur.2024.05.009
[12]	L. Lombardini, F. Rossi, Obstructions to extension of Wasserstein distances for variable masses, Proc. Am. Math. Soc., 150 (2022), 4879–4890. https://doi.org/10.1090/proc/16030 doi: 10.1090/proc/16030
[13]	S. Serfaty, L. Ambrosio, E. Mainini, Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 217–246. https://doi.org/10.1016/j.anihpc.2010.11.006 doi: 10.1016/j.anihpc.2010.11.006
[14]	E. Mainini, A description of transport cost for signed measures, J. Math. Sci., 181 (2012), 837–855. https://doi.org/10.1007/s10958-012-0718-2 doi: 10.1007/s10958-012-0718-2
[15]	C. Villani, Optimal Transport, Springer Berlin, Heidelberg, 2009. https://doi.org/10.1007/978-3-540-71050-9
[16]	C. Villani, Topics in Optimal Transportation, American Mathematical Society, Lyon, France, 2003. https://doi.org/10.1090/gsm/058
[17]	F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser Cham, 2015. https://doi.org/10.1007/978-3-319-20828-2
[18]	S. M. Sinha, Mathematical Programming, Elsevier Cham, 2006. https://doi.org/10.1016/B978-81-312-0376-7.X5000-3
[19]	L. G. Hanin, Kantorovich–Rubinstein norm and its application in the theory of Lipschitz spaces, Proc. Am. Math. Soc., 115 (1992), 345–352.
[20]	L. Kantorovich, G. S. Rubinstein, On a space of totally additive functions, Vestn. Leningr. Univ., 13 (1958), 52–59.
[21]	B. Piccoli, F. Rossi, On properties of the generalized Wasserstein distance, Arch. Ration. Mech. Anal., 222 (2016), 1339–1365. https://doi.org/10.1007/s00205-016-1026-7 doi: 10.1007/s00205-016-1026-7
[22]	L. Caffarelli, R. J. McCann, Free boundaries in optimal transport and Monge–Ampère obstacle problems, Ann. Math., 171 (2010), 673–730. https://doi.org/10.4007/annals.2010.171.673 doi: 10.4007/annals.2010.171.673
[23]	A. Figalli, The optimal partial transport problem, Arch. Ration. Mech. Anal., 195 (2010), 533–560. https://doi.org/10.1007/s00205-008-0212-7 doi: 10.1007/s00205-008-0212-7
[24]	J. D. Benamou, Y. Brenier, A computational fluid mechanics solutions to the Monge–Kantoriovich mass transfer problem, Numer. Math., 84 (2000), 375–393. https://doi.org/10.1007/s002110050002 doi: 10.1007/s002110050002
[25]	M. J. Lighthill, G. B. Whitham, On kinematic waves Ⅱ. A theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. Ser. A, 229 (1955), 317–345. https://doi.org/10.1098/rspa.1955.0089 doi: 10.1098/rspa.1955.0089
[26]	P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42–51. https://doi.org/10.1287/opre.4.1.42
[27]	A. Aw, M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916–938. https://doi.org/10.1137/S0036139997332099 doi: 10.1137/S0036139997332099
[28]	H. M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Res. Part B, 36 (2002), 275–290. https://doi.org/10.1016/S0191-2615(00)00050-3 doi: 10.1016/S0191-2615(00)00050-3
[29]	M. Briani, E. Cristiani, E. Onofri, Inverting the fundamental diagram and forecasting boundary conditions: how machine learning can improve macroscopic models for traffic flow, Adv. Comput. Math., 50 (2024), 115. https://doi.org/10.1007/s10444-024-10206-8 doi: 10.1007/s10444-024-10206-8

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)