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Discrete adjoint gradient computation for multiclass traffic flow models on road networks

Paola Goatin, Axel Klar, Carmen Mezquita-Nieto

Abstract

This paper applies a discrete adjoint gradient computation method for a multi-class traffic flow model on road networks. Vehicle classes are characterized by their specific velocity functions, which depend on the total traffic density, resulting in a coupled hyperbolic system of conservation laws. The system is discretized using a Godunov-type finite volume scheme based on demand and supply functions, extended to handle complex junction coupling conditions -- such as merges and diverges -- and boundary conditions with buffer lengths to account for congestion spillback. The optimization of different travel-related performance metrics, including total travel time and total travel distance, is formulated as a constrained minimization problem and is accomplished through the use of an adjoint gradient approach, allowing for an efficient computation of sensitivities with respect to the chosen time-dependent control variables. Numerical simulations on a sample network demonstrate the efficiency of the proposed framework, particularly as the number of control parameters increases. This approach provides a robust and computationally efficient solution, making it suitable for large-scale traffic network optimization.

Discrete adjoint gradient computation for multiclass traffic flow models on road networks

Abstract

This paper applies a discrete adjoint gradient computation method for a multi-class traffic flow model on road networks. Vehicle classes are characterized by their specific velocity functions, which depend on the total traffic density, resulting in a coupled hyperbolic system of conservation laws. The system is discretized using a Godunov-type finite volume scheme based on demand and supply functions, extended to handle complex junction coupling conditions -- such as merges and diverges -- and boundary conditions with buffer lengths to account for congestion spillback. The optimization of different travel-related performance metrics, including total travel time and total travel distance, is formulated as a constrained minimization problem and is accomplished through the use of an adjoint gradient approach, allowing for an efficient computation of sensitivities with respect to the chosen time-dependent control variables. Numerical simulations on a sample network demonstrate the efficiency of the proposed framework, particularly as the number of control parameters increases. This approach provides a robust and computationally efficient solution, making it suitable for large-scale traffic network optimization.

Paper Structure

This paper contains 21 sections, 1 theorem, 86 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. A multi-class traffic flow model on networks with general speed function
  3. Coupling conditions at junctions
  4. Solution for 1x1 junctions
  5. Solution for Mx1 (merge) junctions
  6. Solution for 1xM (diverge) junctions
  7. Boundary conditions
  8. Cost functionals
  9. Discrete adjoint gradient computation
  10. Numerical tests
  11. Gradient computation: discrete adjoint VS finite differences
  12. Pareto fronts
  13. Conclusions
  14. Details on DAG computation
  15. Update equation
  16. ...and 6 more sections

Key Result

Lemma 1

Under the CFL condition for any initial data $\boldsymbol{\rho}_0\in\mathcal{S}$, the approximate solutions computed by the Godunov scheme eq:FV - eq:MPgodunov satisfy the following uniform bounds: $\blacktriangleleft$$\blacktriangleleft$

Figures (11)

  • Figure 1: Structure of a general $n\times m$ junction $e$,
  • Figure 2: Structure of a $M\times 1$ junction $e$.
  • Figure 3: Structure of a $1\times M$ junction $e$.
  • Figure 4: Structure of the row vector $\partial_y J\in\mathbb{R}^{NMT}$.
  • Figure 5: Partial derivative terms in $\partial_y E$ are ordered first by time and then by cell index. Blocks where time iteration $k\notin\{\nu-1,\nu\}$ are 0.
  • ...and 6 more figures

Theorems & Definitions (1)

  • Lemma 1