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Numerical computation of generalized Wasserstein distances with applications to traffic model analysis

Maya Briani, Emiliano Cristiani, Giovanni Franzina, Francesca L. Ignoto

TL;DR

The paper tackles the challenge of comparing unbalanced mass distributions in traffic models by developing and comparing four numerical approaches to generalized Wasserstein distances (GWDs). It recaps four theoretical GWD formulations (Figalli–Gigli, Piccoli–Rossi, Gaussian Hellinger–Kantorovich, and Savaré–Sodini) and introduces discrete and computational schemes to approximate them, followed by a systematic comparison on academic tests and four traffic-focused scenarios. The study finds that the Figalli–Gigli approach offers the best balance of computational efficiency and modeling insight for traffic forecasting, while Piccoli–Rossi provides favorable performance for calibration tasks; HK-related methods exhibit plateau behavior near optima, and exhaustive discrete-search schemes are impractical for large problems. These results bridge optimal transport theory with traffic-flow sensitivity analysis, enabling robust model calibration, validation, and scenario analysis under mass imbalance due to boundary effects and inputs. The practical impact lies in providing concrete, implementable tools to quantify model sensitivity and guide parameter estimation in first- and second-order traffic models.

Abstract

Generalized Wasserstein distances allow to quantitatively compare two continuous or atomic mass distributions with equal or different total mass. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the last years, giving some insights about their physical meaning. After that, we explore their usage in the context of the sensitivity analysis of differential models for traffic flow. The quantification of models sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.

Numerical computation of generalized Wasserstein distances with applications to traffic model analysis

TL;DR

The paper tackles the challenge of comparing unbalanced mass distributions in traffic models by developing and comparing four numerical approaches to generalized Wasserstein distances (GWDs). It recaps four theoretical GWD formulations (Figalli–Gigli, Piccoli–Rossi, Gaussian Hellinger–Kantorovich, and Savaré–Sodini) and introduces discrete and computational schemes to approximate them, followed by a systematic comparison on academic tests and four traffic-focused scenarios. The study finds that the Figalli–Gigli approach offers the best balance of computational efficiency and modeling insight for traffic forecasting, while Piccoli–Rossi provides favorable performance for calibration tasks; HK-related methods exhibit plateau behavior near optima, and exhaustive discrete-search schemes are impractical for large problems. These results bridge optimal transport theory with traffic-flow sensitivity analysis, enabling robust model calibration, validation, and scenario analysis under mass imbalance due to boundary effects and inputs. The practical impact lies in providing concrete, implementable tools to quantify model sensitivity and guide parameter estimation in first- and second-order traffic models.

Abstract

Generalized Wasserstein distances allow to quantitatively compare two continuous or atomic mass distributions with equal or different total mass. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the last years, giving some insights about their physical meaning. After that, we explore their usage in the context of the sensitivity analysis of differential models for traffic flow. The quantification of models sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.

Paper Structure

This paper contains 43 sections, 68 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Context.
  3. Relevant literature.
  4. Main contribution.
  5. Plan of the paper.
  6. The Wasserstein distance in the context of vehicular traffic
  7. Motivations
  8. Basic theory
  9. Numerical setting
  10. Basic theory for Generalized Wasserstein distances
  11. Figalli & Gigli approach figalli2010JMPA
  12. Piccoli & Rossi approach piccoli2014ARMA
  13. Gaussian Hellinger--Kantorovich approach LieMieSav2018InventMath
  14. Savaré & Sodini approach savare2024JMPA
  15. Discrete and computational approaches for Generalized Wasserstein distances
  16. ...and 28 more sections

Figures (21)

  • Figure 1: Three density functions with disjoint supports.
  • Figure 2: Property of monotone rearrangement.
  • Figure 3: Numerical setting for a generic one-dimensional problem. Two copies of the domain $X$ with supply mass only (left) and demand mass only (right).
  • Figure 4: Additional discretization of the concentrated masses.
  • Figure 5: Moving distributions $\rho^\textsc{s}$ and $\rho^\textsc{d}$ as defined in \ref{['def:distributions_cfr']}.
  • ...and 16 more figures

Theorems & Definitions (7)

  • Definition 1: Wasserstein distance
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6