Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation

We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measu...

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Bibliographic Details
Published in:arXiv.org 2024-08
Main Authors: Ishida, Sadashige, Lavenant, Hugo
Format: Article
Language:English
Subjects:
Convergence
Discretization
Hamilton-Jacobi equation
Regularity
Velocity distribution
Online Access:Get full text
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Summary:We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates, we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton-Jacobi equation.
ISSN:2331-8422