Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T08:40:43.690Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Barles, G. and Souganidis, P. E., Convergence of approximation schemes for fully nonlinear second order equations, Asympt. Anal., 4 (1991), 271–283.Google Scholar
[2]Beckmann, M., McGuire, C. and Winsten, C., Studies in Economics of Transportation, Yale University Press, New Haven, 1956.Google Scholar
[3]Benmansour, F., Carlier, G., Peyré, G. and Santambrogio, F., Derivatives with respect to metrics and applications: subgradient marching algorithm, to appear in Numerische Mathematik.Google Scholar
[4]Benmansour, F., Carlier, G., Peyré, G. and Santambrogio, F., Numerical approximation of conti-nous traffic congestion equilibria, Network and Heterogeneous Media, 4(3) (2009), 605–623.CrossRefGoogle Scholar
[5]Bonnans, J. F., Gilbert, J. C., Lemaréchal, C. and Sagastizäbal, C., Numerical Optimization, 2nd Ed., Springer-Verlag, Heidelberg, 2006.Google Scholar
[6]Boué, M. and Dupuis, P., Markov chain approximations for deterministic control problems with affine dynamics and quadratic cost in the cotrol, SIAM J. Numer. Anal., 36(3) (1999), 667–695.Google Scholar
[7]Carlier, G., Jimenez, C. and Santambrogio, F., Optimal transportation with traffic congestion and Wardrop equilibria, SIAM J. Control Opt., 47(3) (2008), 1330–1350.CrossRefGoogle Scholar
[8]Correa, R. and Lemarèchal, C., Convergence of some algorithms for convex minimization, Math. Program., 62 (1993), 261–275.Google Scholar
[9]Crandall, M. G. and Lions, P.-L., Viscosity solutions of Hamilton-Jacobi equations, Trans. Am. Math. Soc., 277 (1983), 1–42.CrossRefGoogle Scholar
[10]Crandall, M. G. and Lions, P. L., Two approximations of solutions of Hamilton-Jacobi equations, Math. Comput., 43 (1984), 1–19.Google Scholar
[11]Fomel, S., Luo, S. and Zhao, H.-K., Fast sweeping method for the factored eikonal equation, J. Comput. Phys., 228(17) (2009), 6440–6455.CrossRefGoogle Scholar
[12]Knight, F., Some fallacies in the interpretation of social cost, Quart. J. Economics, 38 (1924), 582–606.Google Scholar
[13]Leung, S. and Qian, J., An adjoint state method for three-dimensional transmission traveltime tomography using first-arrivials, Comm. Math. Sci., 4(1) (2006), 249–266.CrossRefGoogle Scholar
[14]Lions, P.-L., Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, 1982.Google Scholar
[15]Luo, S., Numerical Methods for Static Hamilton-Jacobi Equations, Ph.D Thesis, University of California, Irvine, 2009.Google Scholar
[16]Dal Maso, G., Introduction to Γ-Convergence, Birkhauser, Basel, 1992.Google Scholar
[17]Qian, J., Zhang, Y.-T. and Zhao, H.-K., A fast sweeping methods for static convex Hamitlon-Jacobi equations, J. Sci. Comput., 31(1/2) (2007), 237–271.Google Scholar
[18]Qian, J. L., Zhang, Y. T. and Zhao, H. K., Fast sweeping methods for eiknonal equations on triangulated meshes, SIAM J. Numer. Anal., 45 (2007), 83–107.Google Scholar
[19]Robinson, S. M., Linear convergence of epsilon-subgradient descent methods for a class of convex functions, Math. Program., 86 (1999), 41–50.CrossRefGoogle Scholar
[20]Rouy, E. and Tourin, A., A viscosity solutions approach to shape-from-shading, SIAM J. Numer. Anal., 29 (1992), 867–884.Google Scholar
[21]Rudin, W., Functional Analysis, McGraw-Hill, 1991.Google Scholar
[22]Sethian, J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1999.Google Scholar
[23]Shor, N. Z., Minimization Methods for Non-Differentiable Functions, Springer Seriesin Computational Mathematics 3, Springer-Verlag, 1985.Google Scholar
[24]Wardrop, J. G., Some theoretical aspects of road traffic research, Proc. Inst. Civ. Eng., Part II, 1 (1952), 325–378.Google Scholar
[25]Zhao, H.-K., A fast sweeping method for eikonal equations, Math. Comput., 74 (2005), 603–627.Google Scholar