Optimal mass transportation aims to find a cost efficient strategy for moving some commodity between initial and target positions. The well-known Monge-Kantorovich paradigm has found significant applications in pure and applied math. Another paradigm is called ramified transportation, which aims to model branching structures found in many engineered or natural transport systems, such as road networks, irrigation, electric wiring, lightning, or vasculature---as in placentas or the veins of a leaf. The critical difference is that ramified transport costs depend on the actual transport routes used rather than given simply by couplings between start and ending positions. Due to an ‘economy of scale’ where transporting mass together is more cost effective than transport separately, branched structures tend to emerge for increased efficiency.
In literature on ramified transportation there are broadly two approaches. One uses an Eulerian framework focusing on the net mass and direction of flow at a point. It represents transport systems using weighted-directed graphs, or more generally rectifiable 1-currents. An edge with weight w and length l contributes an α-cost of wαl, for α in [0, 1). The concavity of tα incentivizes branching to reduce cost. Another approach is Lagrangian, with a focus on the distinct individual trajectories traveled by transiting mass. It represents transport systems as measures on the space of Lipschitz curves from start positions to target (with appropriate α-cost function suited for this formulation). Both types of framework feature in well developed ramified theory and applications.
An important problem is to investigate landscape functions on ramified systems, corresponding (roughly) to the first variations of a system’s α-cost, or to the marginal transportation costs between points in a system. Separate approaches arise for the different frameworks in the literature. Using a Lagrangian perspective, F. Santambrogio’s pioneering work on landscape functions considers optimal systems transporting a single point-source. In contrast, using an Eulerian framework Q. Xia studies landscape functions on any finite (not necessarily optimal) acyclic weighted-directed graph.
In this dissertation we use geometric measure theory to develop a unified definition for landscape functions, defined on the broad class of “α-balanced” systems. Analyzing α-cost on rectifiable 1- currents, a system is deemed α-balanced if its first variation vanishes when taken with respect to any appropriately contained cycle. It turns out this class includes not only α-optimal transport systems and finite acyclic weighted-directed graphs, but also some non-optimal systems containing positive mass cycles. An interesting subclass are “α-efficient” systems, defined by relaxing the optimality condition on α-cost. To better study properties of landscape functions on balanced systems we also demonstrate the compatibility between Eulerian and Lagrangian frameworks for not-necessarily-optimal systems. Throughout these analyses we use an interesting geometric result that we found: a countable collection of absolutely continuous (or Lipschitz) curves have at most countably many transverse intersection points, with parallel tangents almost everywhere (with respect to 1-dimensional Hausdorff measure) on the set of their pairwise intersections.
|Commitee:||De Loera, Jesús, Saito, Naoki|
|School:||University of California, Davis|
|School Location:||United States -- California|
|Source:||DAI-A 82/2(E), Dissertation Abstracts International|
|Subjects:||Mathematics, Transportation, Applied Mathematics|
|Keywords:||Branching structures, Eulerian - Lagrangian compatibility, Geometric measure theory, Landscape functions, Ramified optimal transportation, Variational analysis|
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