Generalized Wasserstein Distance and its Application to Transport Equations with Source

By Benedetto Piccoli, Francesco Rossi. 2012

219 citations recorded by the source corpus.

In this article, we generalize the Wasserstein distance to measures with different masses. We study the properties of this distance. In particular, we show that it metrizes weak convergence for tight sequences. We use this generalized Wasserstein distance to study a transport equation with a source, in which both the vector field and the source depend on the measure itself. We prove the existence and uniqueness of the solution to the Cauchy problem when the vector field and the source are Lipschitzian with respect to the generalized Wasserstein distance.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/539980f31fd7547826990007f6a29178ea162fd3

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In Applied Mathematics

Citations in the corpus

Unbalanced optimal transport: Dynamic and Kantorovich formulations (2015, background): “1, optimal partial transport is tightly linked to the generalized transport proposed in [22, 23] which allows a dynamic formulation of the optimal partial transport problem.”
Computational Optimal Transport (2018, methodology): “Several different proposals have been introduced in the literature, for instance using a L2 cost Piccoli and Rossi [2014]. In order to avoid having “teleportation of mass” (mass which travels at infinite speed, and suddenly grows in region where there was no mass before), the associated cost should ”
Locally Private k-Means in One Round (2021, background): “Several works have defined similar but slightly different notions (see, e.g., Benamou, 2003; Piccoli and Rossi, 2014).”
Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures (2015, background): “9 The Piccoli-Rossi “generalized Wasserstein distance” [35, 36]: for a metric space (X, d), set X1 = X2 = X , let τ be induced by d, and consider F1(s) = F2(s) := V (s) = |s− 1| with c(x1, x2) := d(x1, x2).”
Mean-field and graph limits for collective dynamics models with time-varying weights (2020, background): “In Theorem 4, ρ denotes the Bounded Lipschitz distance, also known as the generalized Wasserstein distance W 1,1 1 [27, 29].”
A new optimal transport distance on the space of finite Radon measures (2015, methodology): “Recently, some efforts [23, 40, 41] were made to construct new optimal transport distances between measures with different masses, and some non-mass preserving reaction-diffusion systems were interpreted as gradient flows [14, 33].”