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Minimal flows, divergence constraints, and transport density

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

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 | xy | , 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 | xy | 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].

References

  1. L. Ambrosio, Lecture notes on optimal transport problems, in Mathematical Aspects of Evolving Interfaces. Lecture Notes in Mathematics (1812) (Springer, New York, 2003), pp. 1–52

    Google Scholar 

  2. M. Beckmann, A continuous model of transportation. Econometrica 20, 643–660 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Beckmann, C. McGuire, C. Winsten, Studies in Economics of Transportation (Yale University Press, New Haven, 1956)

    Google Scholar 

  4. J.-D. Benamou, G. Carlier, Augmented Lagrangian methods for transport optimization, mean-field games and degenerate PDEs. (2014) https://hal.inria.fr/hal-01073143

  5. F. Benmansour, G. Carlier, G. Peyré, F. Santambrogio, Numerical approximation of continuous traffic congestion equilibria. Net. Het. Media 4(3), 605–623 (2009)

    Article  MATH  Google Scholar 

  6. F. Benmansour, G. Carlier, G. Peyré, F. Santambrogio, Fast marching derivatives with respect to metrics and applications. Numer. Math. 116(3), 357–381 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Bernot, Optimal transport and irrigation. Ph.D. Thesis, ENS Cachan (2005). Available at http://perso.crans.org/bernot

  8. M. Bernot, A. Figalli, Synchronized traffic plans and stability of optima. ESAIM Control Optim. Calc. Var. 14, 864–878 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Bernot, V. Caselles, J.-M. Morel, Traffic plans. Publ. Math. 49(2), 417–451 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Bernot, V. Caselles, J.-M. Morel, The structure of branched transportation networks. Calc. Var. Part. Differ. Equat. 32(3), 279–317 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Bernot, V. Caselles, J.-M. Morel, Optimal Transportation Networks, Models and Theory. Lecture Notes in Mathematics, vol. 1955 (Springer, New York, 2008)

    Google Scholar 

  12. M. Bernot, A. Figalli, F. Santambrogio, Generalized solutions for the Euler equations in one and two dimensions. J. Math. Pures et Appl. 91(2), 137–155 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Bianchini, M. Gloyer, On the Euler-Lagrange equation for a variational problem: the general case II. Math. Zeit. 265(4), 889–923 (2009)

    Article  MathSciNet  Google Scholar 

  14. G. Bouchitté, G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. 3(2), 139–168 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. G. Bouchitté, G. Buttazzo, P. Seppecher, Shape optimization solutions via Monge-Kantorovich equation. C. R. Acad. Sci. Paris Sér. I Math. 324(10), 1185–1191 (1997)

    Article  MATH  Google Scholar 

  16. D. Braess, Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1969)

    MathSciNet  Google Scholar 

  17. A. Brancolini, S. Solimini, On the holder regularity of the landscape function. Interfaces Free Boundaries 13(2), 191–222 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. L. Brasco, G. Carlier, F. Santambrogio, Congested traffic dynamics, weak flows and very degenerate elliptic equations. J. Math. Pures et Appl. 93(6), 652–671 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Mat. Soc. 2, 225–255 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Y. Brenier, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Commun. Pure Appl. Math. 52, 411–452 (1999)

    Article  MathSciNet  Google Scholar 

  21. G. Carlier, F. Santambrogio, A variational model for urban planning with traffic congestion. ESAIM Control Optim. Calc. Var. 11(4), 595–613 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. G. Carlier, F. Santambrogio, A continuous theory of traffic congestion and Wardrop equilibria, proceedings of the conference, proceedings of optimization and stochastic methods for spatially distributed information, St Petersburg, 2010, published (English version). J. Math. Sci. 181(6), 792–804 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. G. Carlier, C. Jimenez, F. Santambrogio, Optimal transportation with traffic congestion and Wardrop equilibria. SIAM J. Control Optim. 47, 1330–1350 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Colombo, A. Figalli, Regularity results for very degenerate elliptic equations. J. Math. Pures Appl. 101(1), 94–117 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  25. B. Dacorogna, J. Moser, On a partial differential equation involving the Jacobian determinant. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(1), 1–26 (1990)

    MATH  MathSciNet  Google Scholar 

  26. G. Dal Maso, An Introduction to Γ-Convergence (Birkhauser, Basel, 1992)

    Google Scholar 

  27. E. De Giorgi, T. Franzoni, Su un tipo di convergenza variazionale. Atti Acc. Naz. Lincei Rend. 58(8), 842–850 (1975)

    MATH  Google Scholar 

  28. L. De Pascale, A. Pratelli, Regularity properties for monge transport density and for solutions of some shape optimization problem. Calc. Var. Part. Differ. Equat. 14(3), 249–274 (2002)

    Article  MATH  Google Scholar 

  29. L. De Pascale, L.C. Evans, A. Pratelli, Integral estimates for transport densities. Bull. Lond. Math. Soc. 36(3), 383–385 (2004)

    Article  MATH  Google Scholar 

  30. G. Devillanova, S. Solimini, On the dimension of an irrigable measure. Rend. Semin. Mat. Univ. Padova 117, 1–49 (2007)

    MATH  MathSciNet  Google Scholar 

  31. G. Devillanova, S. Solimini, Elementary properties of optimal irrigation patterns. Calc. Var. Part. Differ. Equat. 28(3), 317–349 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. R.J. DiPerna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  33. L.C. Evans, W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Am. Math. Soc. 137(653) (1999)

    Google Scholar 

  34. H. Federer, Geometric Measure Theory. Classics in Mathematics (Springer, New York, 1996 (reprint of the 1st edn. Berlin, Heidelberg, New York 1969 edition)

    Google Scholar 

  35. M. Feldman, R. McCann, Uniqueness and transport density in Monge’s mass transportation problem. Calc. Var. Par. Differ. Equat. 15(1), 81–113 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. A. Figalli, N. Juillet, Absolute continuity of Wasserstein geodesics in the Heisenberg group. J. Funct. Anal. 255(1), 133–141 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. I. Fragalà, M.S. Gelli, A. Pratelli, Continuity of an optimal transport in Monge problem. J. Math. Pures Appl. 84(9), 1261–1294 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. E.N. Gilbert, Minimum cost communication networks. Bell Syst. Tech. J. 46, 2209–2227 (1967)

    Article  Google Scholar 

  39. E.N. Gilbert, H.O. Pollak, Steiner minimal trees. SIAM J. Appl. Math. 16, 1–29 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  40. C. Jimenez, Optimisation de Problèmes de Transport. Ph.D. thesis of Université du Sud-Toulon-Var, 2005

    Google Scholar 

  41. C. Jimenez, Dynamic formulation of optimal transport problems. J. Convex Anal. 15(3), 593–622 (2008)

    MATH  MathSciNet  Google Scholar 

  42. J. Lellmann, D.A. Lorenz, C. Schönlieb, T. Valkonen, Imaging with Kantorovich-Rubinstein discrepancy. SIAM J. Imag. Sci. 7(4), 2833–2859 (2014)

    Article  MATH  Google Scholar 

  43. F. Maddalena, S. Solimini, Transport distances and irrigation models. J. Conv. Ann. 16(1), 121–152 (2009)

    MATH  MathSciNet  Google Scholar 

  44. F. Maddalena, S. Solimini, J.-M. Morel, A variational model of irrigation patterns. Interfaces Free Boundaries 5, 391–416 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  45. L. Modica, S. Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14(3), 526–529 (1977)

    Google Scholar 

  46. J.-M. Morel, F. Santambrogio, The regularity of optimal irrigation patterns. Arch. Ration. Mech. Ann. 195(2), 499–531 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  47. É. Oudet, F. Santambrogio, A Modica-Mortola approximation for branched transport and applications. Arch. Ration. Mech. Ann. 201(1), 115–142 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  48. I. Rodríguez-Iturbe, A. Rinaldo, Fractal River Basins: Chance and Self-Organization (Cambridge University Press, Cambridge, 2001)

    Google Scholar 

  49. T. Roughgarden, Selfish Routing and the Price of Anarchy (MIT, Cambridge, 2005)

    Google Scholar 

  50. E. Rouy, A. Tourin, A viscosity solution approach to shape from shading. SIAM J. Numer. Anal. 29, 867–884 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  51. F. Santambrogio, Variational problems in transport theory with mass concentration. Ph.D. thesis, Edizioni della Normale, Birkhäuser, 2007

    Google Scholar 

  52. F. Santambrogio, Optimal channel networks, landscape function and branched transport. Interfaces and Free Boundaries 9, 149–169 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  53. F. Santambrogio, Absolute continuity and summability of transport densities: simpler proofs and new estimates. Calc. Var. Par. Differ. Equat. 36(3), 343–354 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  54. F. Santambrogio, A Dacorogna-Moser approach to flow decomposition and minimal flow problems. ESAIM: Proc. Surv. (SMAI 2013) 45, 265–174 (2014)

    Google Scholar 

  55. F. Santambrogio, V. Vespri, Continuity for a very degenerate elliptic equation in two dimensions. Nonlinear Anal.: Theory Methods Appl. 73, 3832–3841 (2010)

    Google Scholar 

  56. J.A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge University Press, Cambridge, 1999)

    MATH  Google Scholar 

  57. S.K. Smirnov, Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows. Algebra i Analiz 5(4), 206–238 (1993). Later translated into English in St. Petersburg Math. J. 5(4), 841–867 (1994)

    Google Scholar 

  58. J.G. Wardrop, Some theoretical aspects of road traffic research. Proc. Inst. Civ. Eng. 2, 325–378 (1952)

    Google Scholar 

  59. B. White, Rectifiability of flat chains. Ann. Math. (2) 150(1), 165–184 (1999)

    Google Scholar 

  60. Q. Xia, Optimal paths related to transport problems. Commun. Contemp. Math. 5(2), 251–279 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  61. Q. Xia, Interior regularity of optimal transport paths. Calc. Var. Part. Differ. Equat. 20(3), 283–299 (2004)

    Article  MATH  Google Scholar 

  62. Q. Xia, Numerical simulation of optimal transport paths, in Proceedings of the Second International Conference on Computer Modeling and Simulation (ICCMS 2010), vol. 1 (2010), pp. 521–525

    Google Scholar 

  63. Q. Xia, Boundary regularity of optimal transport paths. Adv. Calc. Var. 4(2), 153–174 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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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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