In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric setting. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.
Keywords: Unbalanced optimal transport, Wasserstein-Fisher-Rao, Hellinger-Kantorovich, JKO scheme, reaction-diffusion-advection equations
Gallouët, Thomas 1 ; Laborde, Maxime 1 ; Monsaingeon, Léonard 1
@article{COCV_2019__25__A8_0,
author = {Gallou\"et, Thomas and Laborde, Maxime and Monsaingeon, L\'eonard},
title = {An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2019},
publisher = {EDP Sciences},
volume = {25},
doi = {10.1051/cocv/2018001},
zbl = {1444.35090},
mrnumber = {3943365},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2018001/}
}
TY - JOUR AU - Gallouët, Thomas AU - Laborde, Maxime AU - Monsaingeon, Léonard TI - An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2018001/ DO - 10.1051/cocv/2018001 LA - en ID - COCV_2019__25__A8_0 ER -
%0 Journal Article %A Gallouët, Thomas %A Laborde, Maxime %A Monsaingeon, Léonard %T An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2018001/ %R 10.1051/cocv/2018001 %G en %F COCV_2019__25__A8_0
Gallouët, Thomas; Laborde, Maxime; Monsaingeon, Léonard. An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 8. doi: 10.1051/cocv/2018001
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