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An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria

Published online by Cambridge University Press:  20 August 2015

Songting Luo ,
Shingyu Leung  and
Jianliang Qian
Songting Luo*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Shingyu Leung*
Affiliation:
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
Jianliang Qian*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
*
Corresponding author.Email:luos@math.msu.edu
Email:masyleung@ust.hk
Email:qian@math.msu.edu
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Abstract

The equilibrium metric for minimizing a continuous congested traffic model is the solution of a variational problem involving geodesic distances. The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop’s equilibrium. We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation. The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances. The geodesic distance needed for the state equation is computed by solving a factored eikonal equation, and the adjoint state equation is solved by a fast sweeping method. Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.

Keywords

65K1090C2549L2565N06Traffic congestionadjoint state methodgradient descentfast sweeping methodfactored eikonal equation
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011 , pp. 1113 - 1131
Copyright
Copyright © Global Science Press Limited 2011

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