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 M$$ norm for transportation cost for α ∈ (0, 1]. One then finds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this paper, we construct a continuous flow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this M$$ 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 M$$ 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.
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Keywords: Optimal transport networks, mass reducing flows, flat chains
@article{COCV_2021__27_1_A79_0,
author = {Downes, Carol Ann},
title = {A mass reducing flow for real-valued flat chains with applications to transport networks},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021075},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021075/}
}
TY - JOUR AU - Downes, Carol Ann TI - A mass reducing flow for real-valued flat chains with applications to transport networks JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021075/ DO - 10.1051/cocv/2021075 LA - en ID - COCV_2021__27_1_A79_0 ER -
%0 Journal Article %A Downes, Carol Ann %T A mass reducing flow for real-valued flat chains with applications to transport networks %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021075/ %R 10.1051/cocv/2021075 %G en %F COCV_2021__27_1_A79_0
Downes, Carol Ann. A mass reducing flow for real-valued flat chains with applications to transport networks. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 77. doi: 10.1051/cocv/2021075
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Cité par Sources :
I wish to express my gratitude to Rice University for their support during much of this research, and specifically, Dr. Robert Hardt for his guidance. I’d also like to thank Dr. Christopher Camfield for his thoughts during the writing of this paper. Finally, I thank the referees for their insights and corrections.
