2017/04/29 by Alessio Brancolini, Brancolini, Alessio, Benedikt Wirth +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Engineering · Social Sciences · #49Q10 #49Q20 #90B10 #Classical Analysis and ODEs (math.CA) #Diffusion and Search Dynamics #FOS: Mathematics #Optimization and Control (math.OC) #Traffic control and management #Transportation Planning and Optimization
paper · pdf · doi:10.48550/arxiv.1705.00162
openalex publication_date 2017/04/29 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
A prominent model for transportation networks is branched transport, which\nseeks the optimal transportation scheme to move material from a given initial\nto a final distribution. The cost of the scheme encodes a higher transport\nefficiency the more mass is moved together, which automatically leads to\noptimal transportation networks with a hierarchical branching structure. The\ntwo major existing model formulations, either using mass fluxes (vector-valued\nmeasures) or patterns (probabilities on the space of particle paths), are\nrather different. Once their equivalence was established, the analysis of\noptimal networks could rest on both.\n The transportation cost of classical branched transport is a fractional power\nof the transported mass, and several model properties and proof techniques\nbuild on its strict concavity. We generalize the model and its analysis to the\nmost general class of reasonable transportation costs, essentially increasing,\nsubadditive functions. This requires several modifications or new approaches.\nIn particular, for the equivalence between mass flux and pattern formulation it\nturns out advantageous to resort to a description via 1-currents, an intuition\nwhich already Xia exploited. In addition, some already existing arguments are\ngiven a more concise and perhaps simpler form. The analysis includes the\nwell-posedness, a metrization and a length space property of the model cost,\nthe equivalence between the different model formulations, as well as a few\nnetwork properties.\n