An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria (Communications in Computational Physics, 2011)
The equilibrium metric for minimizing a continuous congested traffic modelis the solution of a variational problem involving geodesic distances. The continuousequilibrium metric and its associated variational problem are closely related to theclassical discrete Wardrop's equilibrium. We propose an adjoint state method to numericallyapproximate continuous traffic congestion equilibria through the continuousformulation. The method formally derives an adjoint state equation to compute thegradient descent direction so as to minimize a nonlinear functional involving the equilibriummetric and the resulting geodesic distances. The geodesic distance needed forthe state equation is computed by solving a factored eikonal equation, and the adjointstate equation is solved by a fast sweeping method. Numerical examples demonstratethat the proposed adjoint state method produces desired equilibrium metrics and outperformsthe subgradient marching method for computing such equilibrium metrics.