Volume 10, Issue 5
An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria

Songting Luo, Shingyu Leung & Jianliang Qian

Commun. Comput. Phys., 10 (2011), pp. 1113-1131.

Published online: 2011-10

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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.

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@Article{CiCP-10-1113, author = {Songting Luo, Shingyu Leung and Jianliang Qian}, title = {An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria}, journal = {Communications in Computational Physics}, year = {2011}, volume = {10}, number = {5}, pages = {1113--1131}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.020210.311210a}, url = {http://global-sci.org/intro/article_detail/cicp/7477.html} }
TY - JOUR T1 - An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria AU - Songting Luo, Shingyu Leung & Jianliang Qian JO - Communications in Computational Physics VL - 5 SP - 1113 EP - 1131 PY - 2011 DA - 2011/10 SN - 10 DO - http://dor.org/10.4208/cicp.020210.311210a UR - https://global-sci.org/intro/cicp/7477.html KW - AB -

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.

Songting Luo, Shingyu Leung & Jianliang Qian. (1970). An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria. Communications in Computational Physics. 10 (5). 1113-1131. doi:10.4208/cicp.020210.311210a
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