Abstract
An oriented transportation network can be modeled by a 1-dimensional chain whose boundary is the difference between the demand and supply distributions, represented by weighted sums of point masses. To accommodate efficiencies of scale into the model, one uses a suitable Mα norm for transportation cost for α ∈ (0, 1]. One then finds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this paper, we construct a continuous flow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this Mα mass reducing flow for real-valued flat chains by finding a higher dimensional real chain whose slices dictate the flow. Keeping the boundary fixed, this flow reduces the Mα mass of the initial chain and is Lipschitz continuous under the flat-α norm. To complete the paper, we apply this flow to transportation networks, showing that the flow indeed evolves branching transport networks to be more cost efficient.
Highlights
Transport theory as in the Monge-Kantorovich problem (1781; 1942) focuses on minimizing transportation cost in a way that depends only on optimally allocating or distributing goods [26, 28]
We explore applications to transportation networks
[40] The set of flat chains of finite generalized mass and boundary mass with coefficients in an abelian group G is compact with respect to the generalized flat norm if and only if G has the property that balls are compact in G; i.e. {g ∈ G : |g| ≤ r} is compact for all r > 0
Summary
Transport theory as in the Monge-Kantorovich problem (1781; 1942) focuses on minimizing transportation cost in a way that depends only on optimally allocating or distributing goods [26, 28]. Keywords and phrases: Optimal transport networks, mass reducing flows, flat chains. Santambrogio, and Xia in [31] numerically approximated minimal paths from one source via an adaption of phase-field approximation techniques Instead of solving this optimization problem outright or following the numerical approximation techniques above, we create a cost reducing flow to evolve transport networks. We flow an initial transport path to a local minimum by viewing transportation networks as 1-dimensional, real-valued flat chains and their transport costs as their Mα masses, as in Bernot, Caselles, and Morel’s book [5] and in the work of Xia [41,42,43,44,45,46,47,48,49,50]. The minimizing flow encourages overlapping structures, naturally changing the topology of the initial network through a “(un)zipping” process which creates or eliminates interior vertices. (See the motivating example in Fig. 1.) Because of this and the fact that the flow fixes a chain’s boundary, it is amenable to help find optimal transport networks
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