Equivalent formulations for the branched transport and urban planning problems
We consider two variational models for transport networks, an urban planning and a branched transport model, in both of which there is a preference for networks that collect and transport lots of mass together rather than transporting all mass particles independently. The strength of this preference determines the ramification patterns and the degree of complexity of optimal networks. Traditionally, the models are formulated in very different ways, via cost functionals of the network in case of urban planning or via cost functionals of irrigation patterns or of mass fluxes in case of branched transport. We show here that actually both models can be described by all three types of formulations; in particular, the urban planning can be cast into a Eulerian (flux-based) or a Lagrangian (pattern-based) framework.
Keywords: optimal transport, optimal networks, branched transport, irrigation, urban planning, Wasserstein distance
2010 MSC: 49Q20, 49Q10, 90B10
The object of this paper are two transport problems, namely the branched transport problem and the urban planning problem. In general terms, a transport problem asks how to move mass from a given initial spatial distribution to a specific desired final mass distribution at the lowest possible cost. Different cost functionals now lead to quite different optimisation problems.
The prototype of all these problem is Monge’s problem. In a rather general setting, let be finite positive Borel measures on with the same total mass. The transportation of onto is modelled by a measurable map such that for all Borel sets . The cost to move an infinitesimal mass in the point to the the point is given by , and the total cost is then given by the formula
Usually, for some .
Branched transport problem.
Monge’s cost functional is linear in the transported mass and thus does not penalise spread out particle movement. Each particle is allowed to travel independently of the others. This feature makes the Monge’s problem unable to model systems which naturally show ramifications (e. g., root systems, leaves, the cardiovascular or bronchial system, etc.). For this reason, the branched transport problem has been introduced by Maddalena, Morel, and Solimini in [MSM03] and by Xia in [Xia03]. It involves a functional which forces the mass to be gathered as much as possible during the transportation. This is achieved using a cost which is strictly subadditive in the moved mass so that the marginal transport cost per mass decreases the more mass is transported together (cf. Figure 1). We will formally introduce branched transport later in Section 1.1.
Urban planning problem.
The second problem we are interested in is the urban planning problem, introduced in [BB05]. Here, the measures have the interpretation of the population and workplace densities, respectively. In this case the cost depends on the public transportation network , which is the object of optimisation. In fact, one part of the cost associated with a network is the optimal value of (1.1), where the cost depends on and is chosen in such a way that transportation on the network is cheaper than outside the network. The other part models the cost for building and maintaining the network. A detailed, rigorous description is given in Section 1.2.
Patterns and graphs.
Branched transport has been studied extensively and has several formulations. Maddalena, Morel, and, Solimini in [MSM03] and Bernot, Caselles, and Morel in [BCM05] proposed a Lagrangian formulation based on irrigation patterns that describe the position of each mass particle at time by . The difference between both articles is that in the latter the particle trajectories cannot be reparameterised without changing the transport cost. The viewpoint introduced by Xia in [Xia03] is instead Eulerian, where only the flux of particles is described, discarding its dependence on the time variable . A very interesting aspect of branched transport is its regularity theory as studied in several articles, among them [BCM08] and [Xia04] for the geometric regularity, [MS10] for the regularity of the tangents to the branched structure, [San07] and [BS11a] for the regularity of the landscape function, [BS14] for the fractal regularity of the minimisers. Equivalence of the different models and formulations are instead the topic of [MS09], [MS13]. Branched transport can also be modelled with curves in the Wasserstein space as in [BBS06], [BB10], [BS11b].
Main result of the paper.
The main result of this paper is a unified viewpoint of the branched transport problem and the urban planning problem. Indeed, we show that also the urban planning problem can be cast in the Eulerian or flux-based and in the Lagrangian or pattern-based framework. This involves the consideration of new functionals which are still subadditive in the moved mass, but not strictly so (cf. Figure 1), introducing several technical difficulties. The main theorem, Theorem 3.3.3, proves the equivalence between the original, the Lagrangian, and the Eulerian formulation of urban planning. One advantage of these equivalences is that now one can consider regularity questions in the most convenient formulation; as an example we show a single path property of optimal urban planning networks in Proposition 3.4.1. For the sake of symmetry we also introduce an additional formulation of branched transport which is based on the transport network as in the original urban planning formulation. Its equivalence to the pattern- and flux-based formulations is stated in Theorem 2.4.2.
The following paragraphs introduce branched transport and urban planning more formally via cost functionals of the transport network . Section 2 then recalls the Eulerian and Lagrangian formulation of branched transport and proves their equivalence to the formulation based on . Section 3 puts forward the novel Eulerian and Lagrangian formulation of urban planning and states their equivalence, the proof of which is deferred to the separate Section 4. Section 3 also states a regularity result for (a subset of all) minimisers, the single path property, based on the model equivalences.
1.1 Branched transport
Branched transport is the first network optimisation problem we consider. As already mentioned, it was introduced in [MSM03, Xia03] to model natural and artificial structures which show branching. Since it is simpler to state, we here take the viewpoint of network optimisation and therefore introduce a new formulation based on the network . The original flux- and pattern-based formulations will be introduced in Section 2.4.2, and their equivalence to the new formulation will be shown in Theorem 2.4.2.
The network is modelled as a one-dimensional set , thought of as a collection of pipes through which a quantity with initial distribution is transported to achieve the distribution . The transport cost scales linearly with transport distance, but is strictly subadditive in the transported mass, which models scale effects (i. e., transporting an amount of mass in bulk is cheaper than splitting it up and transporting the portions separately). Precisely, the cost of moving the mass from to via a rectifiable pipe network is given by
As it will be explained in more detail in Section 2, the vector measure describes the mass flux, and denotes its Radon–Nikodym derivative with respect to , the one-dimensional Hausdorff measure on . The divergence is taken in the distributional sense, and identifies as a flux source and as a sink. The parameter governs how strong the scale effect is, i. e., how much cost can be saved by transporting the mass in bulk. An optimal transport network is a minimizer of the functional .
1.2 Urban planning
The second energy functional we consider has been introduced as an urban planning model in [BB05]. Here, has the interpretation of a population density in an urban region, and represents the density of workplaces. The aim is to develop a public transportation network such that the population’s daily commute to work is most efficient.
The transportation network is described as a collection of one-dimensional curves; more precisely, it is a set with finite one-dimensional Hausdorff measure . An employee can travel part of the commute via own means, which is associated with a cost per distance, or via the transportation network at the distance-specific cost with . Hence, if a travelling path is represented by a curve , its cost is given by (for ease of notation we here identify the path with its image )
The minimum cost to get from a point to a point is given by the metric
denotes Lipschitz curves connecting and . The Wasserstein distance induced by this metric describes the minimum cost to connect the population density and the workplace density and is given by
It is the infimum of formula (1.1) where we choose . Here, denotes the set of transport plans, i. e., the set of non-negative finite Borel measures on the product space whose marginals are and , respectively,
where denotes the pushforward of under the projection .
The urban planning problem is the task to find an optimal network with respect to the transport cost and an additional penalty , the building and maintaining cost for the network. This leads to the energy functional
to be minimised among all sets . Existence of minimisers has been shown among all closed connected (see [BB05] or [BPSS09, Chap. 3]). Without requiring connectedness, existence is proved in [BPSS09, Chap. 4].
We will actually set and study without loss of generality, since .
1.3 Notation and useful notions
Let us briefly fix some frequently used basic notation.
Lebesgue measure. denotes the -dimensional Lebesgue measure.
Hausdorff measure. denotes the -dimensional Hausdorff measure.
Non-negative finite Borel measures. denotes the set of non-negative finite Borel measures on . Notice that these measures are countably additive and also regular by [Rud87, Thm. 2.18]. The corresponding total variation norm is denoted by .
(Signed or vector-valued) regular countably additive measures. denotes the set of (signed or vector-valued) regular countably additive measures on . The corresponding total variation norm is denoted by .
Weak- convergence. The weak- convergence on or is indicated by .
Restriction of a measure to a set. Let be a measure space and with . The measure is the measure defined by
Pushforward of a measure. For a measure space , a measurable space , and a measurable map , the pushforward of under is the measure on defined by
Continuous and smooth functions. and denote the set of continuous and smooth functions, respectively, with compact support on .
Absolutely continuous functions. denotes the set of absolutely continuous functions on the interval .
Lipschitz functions. denotes the set of Lipschitz functions on the compact domain .
Characteristic function of a set. Let be a set and . The characteristic function of the set is defined as
Dirac mass. Let . The Dirac mass in is the distribution defined by
The Dirac distribution is the measure if and else.
Finally, for the reader’s convenience we compile here a list of the most important symbols with references to the corresponding definitions.
: The unit interval.
: Lipschitz paths connecting and (see (1.3)).
: Flux associated with a discrete graph (see (2.1)).
: Reference space of all particles (Definition 2.2.1).
: Irrigation pattern of all particles (Definition 2.2.3).
: Solidarity class of and its mass (Definition 2.2.4).
: Irrigating and irrigated measure (Definition 2.2.6).
: Lipschitz curves on , . This notation is introduced in the framework of transport path measures (Definition 4.1.1).
: Transport path measures moving onto (Definition 4.1.1).
2 Branched transport formulations
In this section we will present the Eulerian or flux-based and the Lagrangian or pattern-based formulations of the branched transport problem and state their equivalence to the formulation from Section 1.1. We begin with the Eulerian formulation.
2.1 Flux-based formulation
We start considering the formulation given by Xia in [Xia03].
Let , be discrete finite non-negative measures with , . Suppose also that they have the same mass,
The object of the next definition is called transport path in [Xia03], and this is the commonly used term in the branched transport literature. We deliberately employ the term mass flux instead, since it does not only encode a path, but also the amount of mass transported. This way we avoid confusion when referring to actual paths as one-dimensional curves.
Definition 2.1.2 (Discrete mass flux and cost function).
A discrete mass flux between and is a weighted directed graph with vertices , straight edges , and edge weight function such that the following conditions are satisfied. Denoting and the initial and final point of edge , we require the following mass preserving conditions,
Given a parameter , we define the transport cost per transport length as (cf. Figure 1). The cost function associated with a mass flux is defined as
where is the length of edge .
In order to state the branched transport problem in the case of non-discrete finite Borel measures, we need to replace graphs with measures.
Definition 2.1.3 (Graphs as vectorial measures).
Consider a weighted oriented graph . Every edge with direction can be identified with the vector measure , and the graph can be identified with the vector measure
All mass preserving conditions satisfied by a discrete mass flux between and summarise as (in the distributional sense).
The identification of graphs with vector measures motivates the definition of a sum operation between graphs that we state here for later usage.
Definition 2.1.4 (Sums of graphs).
If and are weighted oriented graphs, then is unique graph such that
Definition 2.1.5 (Continuous mass flux and cost function).
Let of equal mass. A vector measure is a mass flux between and , if there exist sequences of discrete measures , with , , and a sequence of vector measures with , . Note that follows by continuity w.r.t. the weak- topology.
A sequence satisfying the previous properties is called approximating graph sequence, and we write .
If is a mass flux between and , the transport cost is defined as
Problem 2.1.6 (Branched transport problem, flux formulation).
Given , the branched transport problem is
Remark 2.1.7 (Existence of minimisers).
2.2 Pattern-based formulation
Definition 2.2.1 (Reference space).
Here we consider a separable uncountable metric space endowed with the -algebra of its Borel sets and a positive finite Borel measure with no atoms. We refer to as the reference space.
The reference space can be interpreted as the space of all particles that will be transported from a distribution to a distribution .
Let and be measure spaces. A map is said to be an isomorphism of measure spaces, if
for every , and ,
for every , and .
Recall that if is a complete separable metric space and is a positive Borel measure with no atoms (hence is uncountable), then is isomorphic to the standard space with (for a proof see [Roy88, Prop. 12 or Thm. 16 in Sec. 5 of Chap. 15] or [Vil09, Chap. 1]). As a consequence, the following definitions and results are independent of the particular choice of the reference space, and we may assume it to be the standard space without loss of generality.
Definition 2.2.3 (Irrigation pattern).
Let and be our reference space. An irrigation pattern is a measurable function such that for almost all we have .
A pattern is equivalent to if the images of through the maps are the same. Because of that, a pattern can be regarded as a map .
For intuition, can be viewed as the path followed by the particle . The image of , that is , is called a fibre and will frequently be identified with the particle .
Here we follow the setting recently introduced in [MS13].
Definition 2.2.4 (Solidarity class).
For every we consider the set
of all particles flowing through . The total mass of those particles is given by
Definition 2.2.5 (Cost density, cost functional).
Definition 2.2.6 (Irrigating and irrigated measure).
Let be an irrigation pattern. Let be defined as and . The irrigating measure and the irrigated measure are defined as the pushforward of via and , respectively,
Problem 2.2.7 (Branched transport problem, pattern formulation).
Given , the branched transport problem is
In Definition 2.2.3 one may equivalently require for almost all , instead of , which is the content of Propositions 2.3.3 and 2.3.4 below. This becomes necessary as we will later refer to results from works using either one or the other formulation. In addition, it allows us to assume Lipschitz continuous fibres throughout the remainder of the article.
Let us first recall the following result, whose proof can be found in [AGS08, Lem. 1.1.4].
Lemma 2.3.1 (Arc-length reparameterisation for ).
Let and let be its length. Let
then the following holds true,
with , ,
satisfies , , and a. e. in .
Proposition 2.3.2 (Arc-length reparameterisation of patterns).
Let be an irrigation pattern. Suppose has finite cost , and define
where for notational simplicity we define the infimum of the empty set as . Then, for almost all and all , is arc-length parameterised, and is measurable.
The fact that is arc-length parameterised for all follows from Lemma 2.3.1.
Since , its measurability properties are a consequence of the measurability of and and of the fact that for every null set the set is a null set in .
The measurability of is proved as in [BCM05] and follows from the measurability of the map . We now show that the set is measurable for any . Let be a dense sequence in . Since is nondecreasing and lower semicontinuous in the variable , we have
Since is measurable, we obtain that is measurable, too.
Finally, let be a null set, and let be a Borel set such that and . For almost all we have
Integrating over , we obtain
Due to , for every there exists such that for every set with we have
Since we have that , it follows that
Choosing arbitrarily small gives as desired. ∎
We may further reparameterise the irrigation pattern.
Proposition 2.3.3 (Constant speed reparameterisation of patterns).
Let be an irrigation pattern with finite cost , let be its fibre length, and let be as in Proposition 2.3.2. Then , , is an irrigation pattern which reparameterises the fibres of with and constant velocity for almost all .
This follows from the properties of . ∎
Proposition 2.3.4 (Reparameterised patterns have the same cost).
Let be an irrigation pattern with finite cost and let be its Lipschitz reparameterisation. Then .
The proof is straightforward, once one notices that the solidarity classes (2.3) do not depend on the parameterisation. ∎
2.4 Equivalence between the formulations
It has been proved by Bernot, Caselles, and Morel in [BCM08, Sec. 6] that the pattern-based formulation is equivalent to the formulation by Xia, even though Xia’s formulation does not include the particle motion, while in the pattern-based formulation by Maddalena, Morel, and Solimini the speed of particles occurs in the functional. In particular, minimisers exist for both models, and they can be identified with each other.
Definition 2.4.1 (Branched transport energies).
Given two measures of equal mass, for an irrigation pattern , a mass flux , and a rectifiable set we define
Theorem 2.4.2 (Equivalence of branched transport energies).
The minimisation problems associated with Definition 2.4.1 are equivalent in the sense that
The optima can be identified with each other via
where is the tangent unit vector to . Moreover,
is rectifiable [BCM05, Lem. 6.3], and thus for -a. e. point has a tangent unit vector . Defining a multiplicity via (see Definition 2.2.4) we obtain a flux as in Remark 2.1.8, and the proof of [BCM09, Prop. 9.8] implies that this flux is optimal.
The equality follows by choosing as the rectifiable set from Remark 2.1.8 corresponding to the optimal .
Finally, using the relation between the optimal and , for we have
This formula follows noting that if two fibres and coincide in an interval, then their tangents coincide -a. e., too. ∎
2.5 Regularity properties
Due to proof of equivalence one can examine regularity properties of minimisers in the most convenient formulation. The following is based on patterns.
Definition 2.5.1 (Loop-free paths and patterns).
Let and let be an irrigation pattern. Following [MS13, Def. 4.5], we say that has a loop if there exist such that
else we say that is loop-free. is said to be loop-free if is loop-free for almost all .
Definition 2.5.2 (Single path property).
Let be a loop-free irrigation pattern and let
Note that under the single path property, almost all trajectories from to coincide, but they need not coincide as functions of time (since the time variable can be reparameterised).
3 Urban planning formulations
Here we will employ the same notions as in the previous section to provide the Eulerian or flux-based and the Lagrangian or pattern-based formulations of urban transport. These will then be proved equivalent to the original definition, e. g. from [BPSS09].
3.1 Flux-based formulation
Let be a discrete mass flux between discrete measures . Let be a subgraph of ; is not required to be connected. Given parameters , , the cost function is defined as
where is the length of edge . is the cost for employees to travel from an initial distribution of homes to a distribution of workplaces via the network using public transport on . We wish to minimise among admissible pairs . For a pair to be optimal one must have
if , since otherwise the pair has a lower cost, and
if , since else has a lower cost.
As a result, the cost of an edge for an optimal is given by so that the problem can be the restated with just the mass flux variable .
Definition 3.1.1 (Cost function, flux formulation).
Let be a discrete mass flux between discrete measures . Given parameters , , we define the transport cost per transport length as (cf. Figure 1). The cost function associated with a mass flux is defined as
where is the length of edge .
If is a general mass flux between general measures , the cost function is defined as