Issue |
ESAIM: M2AN
Volume 58, Number 3, May-June 2024
|
|
---|---|---|
Page(s) | 957 - 992 | |
DOI | https://doi.org/10.1051/m2an/2024024 | |
Published online | 10 June 2024 |
On the convergence of discrete dynamic unbalanced transport models
1
Department of Mathematics, Duke University, Durham, NC 27708, USA
2
Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
* Corresponding author: bowen.li200@duke.edu
Received:
19
November
2023
Accepted:
4
April
2024
A generalized unbalanced optimal transport distance WBΛ on matrix-valued measures M(Ω, 𝕊n+) was defined in Li and Zou (arXiv:2011.05845) à la Benamou–Brenier, which extends the Kantorovich–Bures and the Wasserstein–Fisher–Rao distances. In this work, we investigate the convergence properties of the discrete transport problems associated with WBΛ. We first present a convergence framework for abstract discretization. Then, we propose a specific discretization scheme that aligns with this framework, whose convergence relies on the assumption that the initial and final distributions are absolutely continuous with respect to the Lebesgue measure. Further, in the case of the Wasserstein–Fisher–Rao distance, thanks to the static formulation, we show that such an assumption can be removed.
Mathematics Subject Classification: 47N10 / 49Q22 / 65K10
Key words: Unbalanced optimal transport / discrete transportation metric / convergence analysis
© The authors. Published by EDP Sciences, SMAI 2024
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