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

TL;DR

This work develops a dual-based discretization for the dynamic optimal transport problem and establishes a quantitative convergence rate for the transport cost of order √h as the mesh is refined. The approach discretizes the Hamilton–Jacobi equation using monotone finite-difference schemes and imposes a Lipschitz constraint on the initial potential to control the discrete dual solution on a periodic domain. Under mild regularity assumptions on the dual problem, the authors also derive convergence rates for the gradient of the optimal potentials and the velocity field via a duality-gap analysis, with explicit bounds when L and H satisfy a strengthened inequality. Numerically, the method—implemented via ADMM on 1D test cases—shows the expected behavior and compares favorably with standard finite-difference schemes, while revealing that the sqrt(h) rate may not be sharp in practice. Overall, the results provide a robust, quantitative convergence theory for discretizations of dynamic OT that require only bounded-support input measures and leverage dual formulation and HJ discretizations.

Abstract

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.

Figures (3)

• Figure 1: Densities of $\mu$ and $\nu$, the initial and final measures, with respect to the Lebesgue measure.
• Figure 2: Discrete and continuous optimal measures. From left to right: the discrete optimal measures $\overline{\Lambda}_\rho$ for $h=1/16$ and $h=1/16\times 2^5$, and the continuous optimal measure $\overline{\rho}$.
• Figure 3: Plots of the grid resolutions and the errors in the log-log domain. PPO denotes the discretization by papadakis2014optimal and HJ denotes our proposed one. The $\alpha$ values are the approximate rates of convergence.

Theorems & Definitions (39)

• Theorem 2.1: Existence of a solution in the primal problem
• Proposition 2.2: Existence of a solution in the dual problem
• Theorem 2.3: Existence and uniqueness of viscosity solution
• proof : Proof of Proposition \ref{['prop:bound_of_optimal_potential']}
• Theorem 2.4: Convergence of discrete Hamilton-Jacobi equation
• Remark 2.5
• Remark 2.6: Lipschitz in each coordinate and why we restrict to a periodic setting
• Remark 3.1: Choice of discrete measure
• Definition 3.2: Discrete dynamic optimal transport
• Remark 3.3: Break of the symmetry between $\mu$ and $\nu$