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Dynamic Optimal Transport with Optimal Preferential Paths

Marcello Carioni, Juliane Krautz, Jan-F. Pietschmann

TL;DR

This work extends dynamic optimal transport to a domain consisting of a bulk $\Omega$ and an embedded curve $\Gamma$, introducing a mass-exchange mechanism between the two with non-linear mobilities. The authors establish existence of minimizers for both fixed and varying curves using the direct method, and analyze parameter scaling limits that decouple bulk and curve dynamics. To regularize curve evolution and prevent self-intersections, they incorporate Tangent-Point energy and prove compactness and existence results for the varying-path problem, complemented by Γ-convergence-type arguments. Numerically, they develop a primal–dual augmented Lagrangian scheme to solve the coupled bulk–curve transport, implementing finite-element discretizations and demonstrating fixed-curve and varying-curve experiments that illustrate how transport prioritizes the curve or the bulk depending on parameters. Overall, the paper provides a rigorous analytical and computational framework for dynamic transport with mass exchange on lower-dimensional structures and evolving preferential paths, with potential applications to planning and traffic modeling on highways or similar networks.

Abstract

We study a dynamic optimal transport type problem on a domain that consists of two parts: a compact set $Ω\subset \mathbb{R}^d$ (bulk) and a non-intersecting and sufficiently regular curve $Γ\subset Ω$. On each of them, a Benamou-Brenier type dynamic optimal transport problem is considered, yet with an additional mechanism that allows the exchange (at a cost) of mass between bulk and curve. In the respective actions, we also allow for non-linear mobilities. We first ensure the existence of minimizers by relying on the direct method of calculus of variations and we study the asymptotic properties of the minimizers under changes in the parameters regulating the dynamics in $Ω$ and $Γ$. Then, we study the case when the curve $Γ$ is also allowed to change, being the main interest in this paper. To this end, the Tangent-Point energy is added to the action functional in order to preserve the regularity properties of the curve and prevent self-intersections. Also in this case, by relying on suitable compactness estimates both for the time-dependent measures and the curve $Γ$, the existence of optimizers is shown. We extend these analytical findings by numerical simulations based on a primal-dual approach that illustrate the behaviour of geodesics, for fixed and varying curves.

Dynamic Optimal Transport with Optimal Preferential Paths

TL;DR

This work extends dynamic optimal transport to a domain consisting of a bulk and an embedded curve , introducing a mass-exchange mechanism between the two with non-linear mobilities. The authors establish existence of minimizers for both fixed and varying curves using the direct method, and analyze parameter scaling limits that decouple bulk and curve dynamics. To regularize curve evolution and prevent self-intersections, they incorporate Tangent-Point energy and prove compactness and existence results for the varying-path problem, complemented by Γ-convergence-type arguments. Numerically, they develop a primal–dual augmented Lagrangian scheme to solve the coupled bulk–curve transport, implementing finite-element discretizations and demonstrating fixed-curve and varying-curve experiments that illustrate how transport prioritizes the curve or the bulk depending on parameters. Overall, the paper provides a rigorous analytical and computational framework for dynamic transport with mass exchange on lower-dimensional structures and evolving preferential paths, with potential applications to planning and traffic modeling on highways or similar networks.

Abstract

We study a dynamic optimal transport type problem on a domain that consists of two parts: a compact set (bulk) and a non-intersecting and sufficiently regular curve . On each of them, a Benamou-Brenier type dynamic optimal transport problem is considered, yet with an additional mechanism that allows the exchange (at a cost) of mass between bulk and curve. In the respective actions, we also allow for non-linear mobilities. We first ensure the existence of minimizers by relying on the direct method of calculus of variations and we study the asymptotic properties of the minimizers under changes in the parameters regulating the dynamics in and . Then, we study the case when the curve is also allowed to change, being the main interest in this paper. To this end, the Tangent-Point energy is added to the action functional in order to preserve the regularity properties of the curve and prevent self-intersections. Also in this case, by relying on suitable compactness estimates both for the time-dependent measures and the curve , the existence of optimizers is shown. We extend these analytical findings by numerical simulations based on a primal-dual approach that illustrate the behaviour of geodesics, for fixed and varying curves.

Paper Structure

This paper contains 20 sections, 13 theorems, 133 equations, 13 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Further related work
  3. Notation
  4. Sobolev-Slobodeckij spaces
  5. Fixed preferential path
  6. Continuity equations and mass exchange
  7. Dynamic formulation
  8. Well-posedness of the dynamic formulation
  9. Parameter scaling limits
  10. Optimization over the preferential path
  11. The Tangent-Point energy for non-closed curves
  12. Existence of minimizers for varying preferential path
  13. The case of linear mobilities and beyond
  14. Numerics
  15. Augmented Lagrangian formulation
  16. ...and 5 more sections

Key Result

Theorem 1.5

Let $U\subset{\mathbb R}^d$ be a given domain with Lipschitz boundary. For $p\in[1,+\infty)$, $k\in\mathbb{N}_0$ and $s\in(0,1)$ such that $kp, sp>d$ there exists a constant $C=C(k,s,p,\Omega)>0$ with for $\alpha = s-\frac{d}{p}$ and $f\in W^{k+s,p}(U)$. By the Arzelà-Ascoli theorem the embedding $W^{k+s,p}(U)\hookrightarrow C^{k}(U)$ is compact.

Figures (13)

  • Figure 1: Circle tangent to the image of $\gamma(t)=t^3-t$ at $t=0$ and passing through $(t,\gamma(t))$ for $t=1$
  • Figure 2: Comparison between distances for the projected vectors
  • Figure 3: Polygonal curve inscribed in $\Gamma$
  • Figure 4: Initial and final data for a u-shaped curve
  • Figure 5: Time-evolution for $\alpha_1=0.01=\alpha_2$ and a u-shaped curve
  • ...and 8 more figures

Theorems & Definitions (42)

  • Remark 1.1
  • Definition 1.2: Extension of measures
  • Definition 1.3: Sobolev-Slobodeckij spaces
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2: Coupled continuity equations
  • Remark 2.3
  • Proposition 2.4
  • proof
  • ...and 32 more