A Benamou–Brenier Approach to Branched Transport

Abstract

The problem of branched transportation aims to describe the movement of masses when, due to concavity effects, they have the interest to travel together as much as possible, because the cost for a path of length ℓ covered by a mass m is proportional to m α ℓ with 0 < α < 1. The optimization of this criterion let branched structures appear and is suitable to applications like road systems, blood vessels, river networks. . . Several models have been employed in the literature to present this transport problem, and the present paper looks at a dynamical one, similar to the celebrated Benamou-Brenier formulation of Kantorovitch optimal transport. The movement is represented by a path ρ t of probabilities, connecting an initial state µ 0 to a final state µ 1 , satisfying the continuity equation ∂ t ρ + div x q = 0 together with a velocity field v (with q = ρv being the momentum). The transportation cost to be minimized is non-convex and finite on atomic measures:

A User’s Guide to Optimal Transport

“…The interior regularity result (Theorem 5.3) has been proved By Q. Xia in [82] and M. Bernot, V. Caselles and J.-M. Morel in [14]. Also, we remark that L. Brasco, G. Buttazzo and F. Santambrogio proved a kind of Benamou-Brenier formula for branched transport in [17]. The content of Section 5.2 comes from J. Dolbeault, B. Nazaret and G. Savaré [33] and [26] of J. Carrillo, S. Lisini, G. Savaré and D. Slepcev.…”

Variational Analysis of Landscape Elevation and Drainage Networks

“…The intertwined channel and ridge networks that emerge from this type of evolution (see Fig. 1 for example) are suggestive of an optimal transport strategy as observed in the branched regime of Monge-Kantorovich problem where the cost function favors mass aggregation [16,17,18]. The optimality principle behind river networks was first studied within the context of so-called Optimal Channel Networks (OCNs) and, more generally in the context of optimal transportation theory [19,20,21,22].…”

Variational analysis of landscape elevation and drainage networks

“…For γ < 0, loops are preferred and, if one excludes loops, spiral networks emerge [26] with configurations very different from those observed in the natural river networks. The role of γ is reminiscent of the behaviour of the Monge–Kantorovich optimal transport, where congested (similar to figure 3 a ) or branched (similar to figure 3 b ) transport may occur depending on the exponent of the Wasserstein distance used to penalize the transport of mass [17,18]. The extended dynamic Monge–Kantorovich formulation proposed by [35] also exhibits a similar transition from congested to branched transport depending on the sub- or super-linear growth of the transport density with respect to the transport flux.…”

“…The intertwined channel and ridge networks that emerge from this type of evolution (see figure 1, for example) are suggestive of an optimal transport strategy as observed in the branched regime of the Monge–Kantorovich problem where the cost function favours mass aggregation [16–18]. The optimality principle behind land-surface formation and river networks has been studied under transport-limited conditions and for the area of unchannelled slopes [19,20], and more commonly within the context of so-called optimal channel networks (OCNs) and optimal transportation theory [21–24].…”