Dynamic Optimal Transport with Optimal Preferential Paths

Technical Summary: Dynamic Optimal Transport with Optimal Preferential Paths

Problem Formulation
This work investigates a dynamic optimal transport problem defined on a domain $\Omega \subset \mathbb{R}^d$ comprising a bulk region and a lower-dimensional, non-intersecting, regular curve $\Gamma \subset \Omega$. The model describes the transport of mass where the population can move within the bulk $\Omega$ and along the curve $\Gamma$, with the possibility of exchanging mass between the two regions at a cost. The total mass of the system is conserved.

The dynamics are governed by a pair of coupled continuity equations. Let $\rho_t$ and $\mu_t$ denote the time-dependent mass distributions in the bulk and on the curve, respectively, with associated momenta $J_t$ and $V_t$. The exchange of mass is modeled by a flux term $f_t$ (normal exchange) and boundary terms $G^0_t, G^1_t$ (tangential exchange at the curve's endpoints). The system is subject to non-linear mobility functions $m_\Omega$ and $m_\Gamma$, which regulate the dynamics in the bulk and on the curve, respectively.

The objective is to minimize an action functional $A_\Gamma$ that aggregates the kinetic energy in the bulk, the kinetic energy on the curve, and the cost of mass transfer between them. The functional is defined as:
$$A_\Gamma = \int_\Omega \frac{|J_t|^2}{m_\Omega(\rho_t)} d\lambda_\Omega + \alpha_1 \int_\Gamma \frac{|V_t|^2}{m_\Gamma(\mu_t)} d\lambda_\Gamma + \alpha_2 \int_\Gamma \frac{|f_t|^2}{m_\Gamma(\mu_t)} d\lambda_\Gamma + \dots$$
where $\alpha_1, \alpha_2 > 0$ are parameters controlling the relative costs of transport on the curve versus the bulk, and the cost of entering or leaving the curve.

The paper addresses two distinct scenarios:

1. Fixed Curve: The geometry of $\Gamma$ is fixed, and the goal is to find the optimal evolution of measures $(\rho_t, \mu_t, J_t, V_t, f_t)$.
2. Varying Curve: The shape of $\Gamma$ is also a variable in the optimization problem. The goal is to find both the optimal curve and the optimal transport dynamics.

Methodology
The authors employ the direct method of the calculus of variations to establish the existence of minimizers.

• Existence for Fixed Curves: The authors define a space of admissible tuples satisfying the coupled continuity equations in a weak (distributional) sense. They prove that the action functional is convex and lower semi-continuous with respect to the narrow convergence of measures. By establishing Hölder estimates for the time-dependent measures and utilizing a generalized Arzelà-Ascoli theorem, they demonstrate the compactness of minimizing sequences, thereby proving the existence of a minimizer.
• Asymptotic Analysis: The paper analyzes the behavior of the system as the cost parameters $\alpha_2$ and $\alpha_3$ (controlling mass exchange) tend to zero or infinity.
  • As $\alpha \to \infty$, the exchange cost becomes prohibitive, and the problem decouples into two independent optimal transport problems on $\Omega$ and $\Gamma$ (provided mass compatibility conditions are met).
  • As $\alpha \to 0$, the exchange becomes free, and the problem converges to a coupled transport problem with zero exchange cost.
• Existence for Varying Curves: To handle the optimization over the curve's shape, the authors introduce a regularization term to the action functional to prevent self-intersections and ensure regularity. They utilize the Tangent-Point energy ($E_{tp}^p$), originally defined for closed curves, adapted here for open curves. This energy penalizes self-intersections and ensures that curves with finite energy are embedded one-dimensional submanifolds. The total regularization functional $R(\gamma)$ includes the Tangent-Point energy, a logarithmic term to prevent the curve from closing (ensuring well-defined boundary conditions), and a length penalty.
  • Compactness is established by showing that the Tangent-Point energy bounds the $W^{2-1/p, p}$ Sobolev-Slobodeckij norm of the curve, which compactly embeds into $C^{1,\alpha}$.
  • The existence of a minimizer for the joint problem (curve and measures) is proven by combining the compactness of the measures (Proposition 4.6) with the lower semi-continuity of the action and the regularization functional.

Numerical Scheme
The authors develop a numerical algorithm based on a primal-dual approach and an Augmented Lagrangian formulation (ALG2).

• Discretization: The space-time domain is discretized using a finite element method with piecewise linear elements for the curve and prism elements for the space-time domain.
• Algorithm: For a fixed curve, the algorithm solves a saddle-point problem by alternating updates of the dual variables (potentials) and the primal variables (densities and momenta).
• Varying Curve: To optimize the curve, the Augmented Lagrangian method is embedded within a projected gradient descent loop. The gradient of the objective functional with respect to the curve's control points is approximated using central differences. The Tangent-Point energy is discretized using a specific quadrature rule to handle singularities.

Key Contributions and Results

1. Rigorous Formulation: The paper provides a rigorous definition of coupled continuity equations for mass transport between a bulk domain and a lower-dimensional curve, including mass exchange mechanisms and non-linear mobilities.
2. Existence of Minimizers: It establishes the existence of minimizers for the dynamic transport problem for both fixed and varying preferential paths.
3. Asymptotic Limits: The study characterizes the limiting behavior of the coupled system as exchange costs vanish or become infinite, linking the model to classical uncoupled optimal transport with mobilities.
4. Regularization via Tangent-Point Energy: The work extends the application of Tangent-Point energy to open curves within an optimal transport framework, proving that this energy enforces the necessary regularity (non-self-intersection and $C^1$ regularity) for the existence of solutions in the varying curve setting.
5. Numerical Implementation: The authors present a functional numerical scheme capable of computing geodesics for both fixed and optimized curves, demonstrating the influence of cost parameters and geometry on transport dynamics.

Significance
The paper claims to contribute to the theory of dynamic optimal transport by generalizing the Benamou-Brenier framework to include lower-dimensional structures (curves) within the domain. This models scenarios where transport is faster or cheaper along specific paths (e.g., highways) but requires a cost to access. By allowing the path itself to be optimized, the work addresses problems in optimal planning where the infrastructure (the path) is determined by the initial and final distributions of the population. The use of non-linear mobilities and the rigorous handling of the curve's geometry via Tangent-Point energy distinguish this work from previous studies that often relied on convex duality or focused solely on fixed boundaries. The numerical results illustrate how the interplay between transport costs and path geometry dictates the evolution of mass, offering a tool for modeling complex transport networks.