Optimal Mass Transport on Metric Graphs

Mazón, José M.; Rossi, Julio D.; Toledo, Julián

doi:10.1137/140995611

Cited by 8 publications

(7 citation statements)

References 15 publications

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“…(since the mass transfer only happens where there is some mass to transfer from/into) and, in turn, ♭ , ♯ have compact support (see (34)). Finally, the dominated convergence theorem allows us to write…”

Section: Numerical Resultsmentioning

confidence: 99%

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Comparing comparisons between vehicular traffic states in microscopic and macroscopic first‐order models

Cristiani

Saladino

2018

Math Methods in App Sciences

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In this paper, we deal with the analysis of the solutions of traffic flow models at multiple scales in both cases of a single road and road networks. We are especially interested in measuring the distance between traffic states (as they result from the mathematical modeling) and investigating whether these distances are somehow preserved passing from the microscopic to the macroscopic scale. By means of both theoretical and numerical investigations, we show that, on a single road, the notion of Wasserstein distance fully catches the human perception of distance independently of the scale, while in the case of networks it partially loses its nice properties. KEYWORDS earth mover's distance, follow-the-leader model, LWR model, many-particle limit, multi-path model, networks, traffic flow, Wasserstein distance INTRODUCTIONIn this paper, we deal with the analysis of the solutions of traffic flow models at multiple scales. More precisely, we are interested in measuring the distance between traffic states (as they result from the mathematical modeling) and investigating whether these distances are somehow preserved passing from the microscopic to the macroscopic scale. We will consider both cases of a single road and road networks.Connections between microscopic (agent-based) and macroscopic (fluid dynamics) traffic flow models are already well established. Aw et al, 1 Greenberg, 2 and Di Francesco et al 3 investigated the many-particle limit in the framework of second-order traffic models, deriving the macroscopic Aw-Rascle-Zhang (ARZ) 4,5 model from a particular second-order microscopic Follow-the-Leader (FtL) 6,7 model. On the other hand, Colombo and Rossi, 8 Rossi, 9 Di Francesco and Rosini, 10 and Di Francesco et al 11 investigated the many-particle limit in the framework of first-order traffic models, deriving the Ligthill-Whitham-Richards (LWR) 12,13 model as the limit of a specific first-order FtL model. Let us also mention the papers by Forcadel et al 14 and Forcadel and Salazar,15 which investigate the many-particle limit exploiting the link between conservation laws and Hamilton-Jacobi equations.Moving to road networks, analogous connections are rarer. This is probably because of the fact that macroscopic traffic models on networks are in general ill-posed, since the conservation of mass is not sufficient alone to characterize a unique solution at junctions. This ambiguity makes more difficult to find the right limit of the microscopic model, which, in turn, can be defined in different ways near the junctions. In this context let us mention the paper by Cristiani and Sahu, 16 which investigates the many-particle limit of a first-order FtL model suitably extended to a road network. The corresponding macroscopic model appears to be the extension of the LWR model on networks introduced by Hilliges and Weidlich 17 and then extensively studied by Briani and Cristiani 18 and Bretti et al. 19 This paper focuses in particular on the comparison of solutions to the equations associated to first-order traffic models....

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Section: Numerical Resultsmentioning

confidence: 99%

“…19 the complete procedure, recently used also in the context of traffic flow . More sophisticated characterizations based on a fluid‐dynamic approach, p ‐Laplacian, or a variational approach are also available.…”

Section: Ingredientsmentioning

confidence: 99%

Comparing comparisons between vehicular traffic states in microscopic and macroscopic first‐order models

Cristiani

Saladino

2018

Math Methods in App Sciences

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“…Recent characterizations also seem to be unfit for this goal. Let us mention in this regard the variational approach proposed by Mazón et al [25], which generalizes to networks the results by Evans and Gangbo [12]. It can be also shown [25] that the Wasserstein distance has a nice link with the p-Laplacian operator.…”

Section: Introductionmentioning

confidence: 84%

Sensitivity analysis of the LWR model for traffic forecast on large networks using Wasserstein distance

Briani

Cristiani

Iacomini

2018

Communications in Mathematical Sciences

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In this paper we investigate the sensitivity of the LWR model on network to its parameters and to the network itself. The quantification of sensitivity is obtained by measuring the Wasserstein distance between two LWR solutions corresponding to different inputs. To this end, we propose a numerical method to approximate the Wasserstein distance between two density distributions defined on a network. We found a large sensitivity to the traffic distribution at junctions, the network size, and the network topology.

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“…≤ 0, (19) where in the last inequality we have used the convexity of L(f, •); taking into account that L(f, •) is in addition strictly convex, we see that a strict inequality prevails in (19), leading to a contradiction, unless q = q 1 = −q 2 ⇐⇒ q 1 = q 2 = 0.…”

Section: Let Us Definementioning

confidence: 96%

“…We remark that a central role in our construction is played by probability measures, defined on a sort of tangent bundle of the graph: they constitute the relaxed framework for the variational problems under consideration. There is a broad interest in the recent literature on probability measures supported on graphs/networks, see for instance [7,19]. One of the goal being, for instance, to extend mean field games models to graphs (see [1,6,14,15]).…”

Section: Introductionmentioning

confidence: 99%

Aubry-Mather theory on graphs

Siconolfi,

Sorrentino

2021

Preprint

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Optimal Mass Transport on Metric Graphs

Cited by 8 publications

References 15 publications

Comparing comparisons between vehicular traffic states in microscopic and macroscopic first‐order models

Comparing comparisons between vehicular traffic states in microscopic and macroscopic first‐order models

Sensitivity analysis of the LWR model for traffic forecast on large networks using Wasserstein distance

Aubry-Mather theory on graphs

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