Understand the Back-and-Forth method Deeper

In the undergraduate independent study on Optimal Transport theory, Shirley and Wenxuan explore the back-and-forth method [1] in greater depth, with the goal of extending it to more versatile applications.

They implemented the method entirely in Python [2], handling both 1D and 2D datasets, and visualized some key steps in this algorithm, particularly the Jacobian computations related with the pushforward map. Moreover, a recent proposed metric (WOP distance [3]) between positive measures integrates naturally into this framework. They also developed a numerical solver for computing this metric.

References:
  1. Jacobs and Leger, The Back-and-Forth Method, 2020.
  2. Bao, Xu and Zhou, BFOT python package, 2025.
  3. Leblanc, Le Gouic, Liandrat and Tournus, Extending the Wasserstein metric to positive measures, 2023