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Optimal dynamic transport with mass consumption


The problem of optimal transport has remained very academic for a long time at least in the formulation that G. Monge made of it, and took on a more concrete aspect with the idea of L. Kantorovitch to formulate it in terms of optimal allocation of resources. More recently, works such as those by R. J. McCann [13], Y. Brenier and J.-D. Benamou [1] made it possible to give a dynamic formulation of this type of problem (mainly for quadratic Euclidean transport costs). This formulation open the way to easier numerical process since it allows to apply classical numerical methods in the domain of variational and convex optimization under constraint. For absolutely continuous measures with L 2-densities, we can cite in particular the augmented Lagrangian method [1], splitting-proximal or primal-dual[8, 14, 3]. Our present work will be based on the augmented Lagrangian formulation. An important tool from the Optimal Transportation theory is the Wasserstein distance, a metric allowing to estimate the difference between two probability measures (or more generally, of two measures with the same "mass"). This distance represents a "global transportation cost" between two measures, assuming the local mass displacement cost to depend on the distance and linearly on the local mass. One of the main limitation of Optimal Transport is that it can only be applied between measures with the same "total mass". This is why a lot of research works have proposed various extensions of these Wassertein distances to measure spaces with potentially different "masses" [2, 17, 16]. The use which is made of Optimal Transport for interpolation problems has also encouraged research for new optimal "unbalanced" mass transport models, and in particular dynamic models. Indeed, in addition to offering a variational framework more suited for numerical processing, those models provide a continuous time evolution (a "geodesic interpolation" for the Wasserstein metric in the classic case) between the source and target measures. The dynamic formulation proposed in [1] represents the constraint of mass conservation during the time-continuous transport with a conservative continuity equation. Their model uses absolutely continuous measures with L 2-densities: if ρ(t, x) represents the time evolution of the interpolated density, and v(t, x) represents the velocity field derived from the transportation map, we therefore have ∂ t ρ + div(ρv) = 0. Many models, which aim to generalize these interpolations for measures or densities with different mass, use a source term in the continuity equation (which become ∂ t ρ + div(ρv) = f) which has to be taken into account in the energy fonctional (see [9, 10], or [6, 5, 15, 11, 12] with also applications to images processing, or [7] for a use in data assimilation). * Research report produced within the team Optimization and Optimal Control (RICAM, Johannes Kepler University , 4040 Linz
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Dates and versions

hal-02516695 , version 1 (24-03-2020)
hal-02516695 , version 2 (08-04-2020)
hal-02516695 , version 3 (09-04-2020)


  • HAL Id : hal-02516695 , version 3


Romain Hug. Optimal dynamic transport with mass consumption. 2020. ⟨hal-02516695v3⟩
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