Commun. Comput. Phys., 10 (2011), pp. 1113-1131. An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria Songting Luo 1*, Shingyu Leung 2, Jianliang Qian 11 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA. 2 Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. Received 2 February 2010; Accepted (in revised version) 31 December 2010 Available online 2 August 2011 doi:10.4208/cicp.020210.311210a 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. AMS subject classifications: 65K10, 90C25, 49L25, 65N06 Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Traffic congestion, adjoint state method, gradient descent, fast sweeping method, factored eikonal equation. *Corresponding author. Email: luos@math.msu.edu (S. Luo), masyleung@ust.hk (S. Leung), qian@math.msu.edu (J. Qian)