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Gradient enhanced ADMM Algorithm for dynamic optimal transport on surfaces

Guozhi Dong, Hailong Guo, Chengrun Jiang, Zuoqiang Shi

TL;DR

This work tackles computing dynamic optimal transport on general surfaces by recasting OT in the Benamou–Brenier dynamical form and solving it with a gradient-enhanced ALG2 ADMM. It introduces gradient recovery on manifolds, notably temporal $G_{\tau}$ and surface PPPR $G_h$, to improve gradient/divergence accuracy and enable a single-mesh formulation without stagger grids. A fast solver based on spectral decomposition decouples the time dimension, allowing efficient, robust solves of the time-space Poisson equations that arise in the dual updates. Numerical experiments on flat domains and curved surfaces demonstrate optimal convergence rates, substantial speedups, and stability under high curvature and complex topology. The method offers a practical, geometry-aware tool for dynamic OT on surfaces with potential applications in imaging and geometric data analysis.

Abstract

A gradient enhanced ADMM algorithm for optimal transport on general surfaces is proposed in this paper. Based on Benamou and Brenier's dynamical formulation, we combine gradient recovery techniques on surfaces with the ADMM algorithm, not only improving the computational accuracy, but also providing a novel method to deal with dual variables in the algorithm. This method avoids the use of stagger grids, has better accuracy and is more robust comparing to other averaging techniques.

Gradient enhanced ADMM Algorithm for dynamic optimal transport on surfaces

TL;DR

This work tackles computing dynamic optimal transport on general surfaces by recasting OT in the Benamou–Brenier dynamical form and solving it with a gradient-enhanced ALG2 ADMM. It introduces gradient recovery on manifolds, notably temporal and surface PPPR , to improve gradient/divergence accuracy and enable a single-mesh formulation without stagger grids. A fast solver based on spectral decomposition decouples the time dimension, allowing efficient, robust solves of the time-space Poisson equations that arise in the dual updates. Numerical experiments on flat domains and curved surfaces demonstrate optimal convergence rates, substantial speedups, and stability under high curvature and complex topology. The method offers a practical, geometry-aware tool for dynamic OT on surfaces with potential applications in imaging and geometric data analysis.

Abstract

A gradient enhanced ADMM algorithm for optimal transport on general surfaces is proposed in this paper. Based on Benamou and Brenier's dynamical formulation, we combine gradient recovery techniques on surfaces with the ADMM algorithm, not only improving the computational accuracy, but also providing a novel method to deal with dual variables in the algorithm. This method avoids the use of stagger grids, has better accuracy and is more robust comparing to other averaging techniques.
Paper Structure (18 sections, 2 theorems, 61 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 18 sections, 2 theorems, 61 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Benamou--Brenier formulation of OT
  4. Gradient recovery on manifolds
  5. Polynomial preserving recovery
  6. Parametric polynomial preserving recovery
  7. Gradient enhanced ADMM algorithm
  8. ADMM algorithm for dynamical OT
  9. Discretization of the time-space Poisson equation
  10. Gradient enhanced ADMM
  11. Fast solver of linear system
  12. Numerical examples
  13. Numerical convergence rate
  14. Computational efficiency
  15. Numerical simulation of dynamical optimal transport on general surfaces
  16. ...and 3 more sections

Key Result

Lemma 1

For any $\rho\in \mathbb{R}^+$, and $\mathbf{m}\in \mathbb{R}^d$, we have where $A$ is defined as

Figures (5)

  • Figure 1: Numerical error of the optimal transport on the unit square: (a) discrete $L^2$ error; (b) discrete $L^1$ error.
  • Figure 2: Evolution of mass distribution on the unit sphere.
  • Figure 3: Effect of the regularizing parameter $\alpha_r$. The first row is the numerical result for $\alpha_r = 1$ and the second row is the numerical result of $\alpha_r = 0.0001$.
  • Figure 4: Evolution of mass distribution on Enzensberger-Stern star algebraic surface.
  • Figure 5: Evolution of mass distribution on a teapot.

Theorems & Definitions (3)

  • Lemma 1
  • Lemma 2
  • proof