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Minimal flows, divergence constraints, and transport density

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Optimal Transport for Applied Mathematicians

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 87))

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Abstract

In this chapter we connect the Monge-Kantorovich problem, which is mainly “Lagrangian” in spirit, with other models which could describe optimal transport issues, in Eulerian form. This primarily concerns the original Monge cost | x − y | , rather than other power costs. We see the equivalence with Beckmann’s continuous transportation model, which consists in \(\min \int \vert \mathbf{w}\vert \mathrm{d}x\,:\, \nabla \cdot \mathbf{w} =\mu -\nu \}\). We also analyze properties and summability of the optimal w, in relation to the notion of transport density and we develop a framework of transport problems via measures on paths. These ideas are then applied, in the discussion section, to two variants models, which provide a useful modeling for other transport-related phenomena: traffic congestion and branched structures.

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Notes

  1. 1.

    In this way the flow-minimization problem corresponding to costs of the form | x − y | p is transformed into the so-called Benamou-Brenier problem, which we will discuss in Chapters 5 and 6, but we do not push this analogy to further conclusions.

  2. 2.

    The strange notation w [γ] is chosen so as to distinguish from the object w Q that we will introduce in Section 4.2.3.

  3. 3.

    Pay attention to the use of the gradient of the Kantorovich potential u: we are using the result of Lemma 3.6 which provides differentiability of u on transport rays.

  4. 4.

    Their proof is somehow intermediate between that of [143] and the one we present here: indeed, approximation by atomic measures is also performed in [208], as here, but on both the source and the target measure, which requires a geometric analysis of the transport rays as in [143].

  5. 5.

    Note that this is just an example of Nash equilibrium with a continuum of players, as we will see in Section 7.4.3 and, more particularly, in Box 7.3.

  6. 6.

    For a survey on the continuous framework, see also 111.

  7. 7.

    Note that, even for Γ = {γ} (which is the most restrictive case), assuming \(\mu,\nu \in L^{\infty }\), considerations from incompressible fluid mechanics in [83] allow to build a Q such that \(i_{Q} \in L^{\infty }\).

  8. 8.

    Note that the same duality trick is also used in the discrete problem over networks, where solving the dual problem is much more convenient than the original one.

  9. 9.

    This procedure is just a particular case of what is usually called forward automatic differentiation.

  10. 10.

    We also observe that this reduction to a divergence-constrained convex minimization problem allows to provide alternative numerical approaches, as it is done in [35], in the same spirit of the Benamou-Brenier Augmented Lagrangian technique; see also Section 6.1

  11. 11.

    Observe that the same result can also be directly obtained from duality arguments, as it is done in Theorem 2.1 in [80].

  12. 12.

    To have an exact equivalence between the Steiner minimal connection problem and Gilbert problem with α = 0, one needs to consider a measure μ composed of a unique Dirac mass, so that every point in \(\mathop{\mathrm{spt}}\nolimits (\nu )\) must be connected to it, hence getting connectedness of the graph.

  13. 13.

    Unfortunately, to cope with the language usually adopted in branched transport, we cannot be completely coherent with the rest of the chapter, where we called “cycles” what we call here “strong cycles.”

  14. 14.

    For triple junctions, which are the most common ones, this gives interesting results: when α = 0, we get the well-known condition about Steiner trees, with three 120∘ angles, and for α = 0. 5, we have a 90∘ angle (see Ex(30)); yet, in general, the angles depend on the masses \(\theta _{k}\).

  15. 15.

    We already met rectifiable sets when studying the differentiability of convex functions, in Chapter 1: 1-rectifiable sets are defined as those sets which are covered, \(\mathcal{H}^{1}\)-a.e., by a countable union of Lipschitz curves. Anyway, the reader can easily pretend that “1-rectifiable” is a synonym of “1D” and it should be enough to follow the rest of the discussion.

  16. 16.

    We define rectifiable vector measures as those which can be expressed in the form \([E,\tau,\theta ]\) with E rectifiable. This language is borrowed from that of rectifiable currents, but currents can be considered vector measures. The reader can look at [161] for more details.

  17. 17.

    Indeed, even if not completely evident, it can be proven that, at least in the case where \(\mathop{\mathrm{spt}}\nolimits (\mu ) \cap \mathop{\mathrm{spt}}\nolimits (\nu ) =\emptyset\), the condition \(d_{\alpha }(\mu,\nu ) < +\infty \) is actually equivalent to \(d_{\alpha }(\mu,\delta _{0}),d_{\alpha }(\nu,\delta _{0}) < +\infty \).

  18. 18.

    What we provide here is just a translation into the language of this chapter of the model proposed in [48, 221], which uses “parametrized traffic plans” instead of measures on paths, but it is just a matter of language.

  19. 19.

    The terminology has been introduced for this very purpose by the authors of [48].

  20. 20.

    We can give a precise definition in the following way: for x,y ∈Ω, define \([x,y] =\{\omega \in \mathcal{C}: \exists \,s < t\text{ such that }\omega (s) = x\text{ and }\omega (t) = y\}\) ; we say that Q is cycle-free if there are not x 1 ,…,x n with x 1 = x n such that Q([x i ,x i+1 ]) > 0 or Q([x i+1 ,x i ]) > 0 for all i = 1,…,n − 1.

  21. 21.

    But it does not seem to come from a dual problem.

  22. 22.

    Other Holder results exist under different assumptions, if ν admits different dimensional lower bounds, i.e., it is Ahlfors regular of another exponent k < d, thus obtaining \(\beta = d(\alpha -(1 - \frac{1} {k}))\) ; see [78].

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  1. Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Orsay, France

    Filippo Santambrogio

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  1. Filippo Santambrogio
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Santambrogio, F. (2015). Minimal flows, divergence constraints, and transport density. In: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol 87. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20828-2_4

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