Advertisement

An Adjoint-Based Approach for a Class of Nonlinear Fokker-Planck Equations and Related Systems

  • Adriano Festa
  • Diogo A. GomesEmail author
  • Roberto M. Velho
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 28)

Abstract

Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer properties of schemes for HJ equations to FP equations. Hence, we get numerical schemes with desirable features such as positivity and mass-preservation. We illustrate this approach in examples that include mean-field games and a crowd motion model.

Keywords

Numerical methods Hamilton-Jacobi equations Fokker-Planck equations Mean-field games Hughes model 

Notes

Acknowledgements

The author “D. Gomes” was partially supported by KAUST baseline and start-up funds and by KAUST OSR-CRG2017-3452. The author “A. Festa” was partially supported by the Haute-Normandie Regional Council via the M2NUM project.

References

  1. 1.
    Achdou, Y., Capuzzo Dolcetta, I.: Mean field games: numerical methods. SIAM J. Numer. Anal. 48(3), 1136–1162 (2010)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Achdou, Y., Camilli, F., Capuzzo Dolcetta, I.: Mean field games: convergence of a finite difference method. SIAM J. Numer. Anal. 51(5), 2585–2612 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bossy, M., Gobet, E., Talay, D.: A symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41(3), 877–889 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cacace, S., Cristiani, E., Falcone, M., Picarelli, A.: A patchy dynamic programming scheme for a class of Hamilton–Jacobi–Bellman equations. SIAM J. Sci. Comput. 34(5), A2625–A2649 (2012)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cagnetti, F., Gomes, D., Tran, H. V.: Aubry-Mather measures in the nonconvex setting. SIAM J. Math. Anal. 43(6), 2601–2629 (2011). ISSN 0036-1410. https://doi.org/10.1137/100817656 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cagnetti, F., Gomes, D., Tran, H. V.: Adjoint methods for obstacle problems and weakly coupled systems of PDE. ESAIM Control Optim. Calc. Var. 19(3), 754–779 (2013). ISSN 1292-8119.  https://doi.org/10.1051/cocv/2012032 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cagnetti, F., Gomes, D., Tran, H. V.: Convergence of a semi-discretization scheme for the Hamilton-Jacobi equation: a new approach with the adjoint method. Appl. Numer. Math. 73, 2–15 (2013). ISSN 0168-9274. https://doi.org/10.1016/j.apnum.2013.05.004 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cagnetti, F., Gomes, D., Mitake, H., Tran, H. V.: A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(1), 183–200 (2015). ISSN 0294-1449. https://doi.org/10.1016/j.anihpc.2013.10.005 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Carlini, E., Silva, F. J.: Semi-lagrangian schemes for mean field game models. In 2013 IEEE 52nd Annual Conference on Decision and Control (CDC), pp. 3115–3120. IEEE, Piscataway (2013)Google Scholar
  10. 10.
    Carlini, E., Silva, F. J.: A fully discrete semi-Lagrangian scheme for a first order mean field game problem. SIAM J. Numer. Anal. 52(1), 45–67 (2014). ISSN 0036-1429. https://doi.org/10.1137/120902987 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Carlini, E., Silva, F. J.: A semi-Lagrangian scheme for a degenerate second order mean field game system. Discrete Contin. Dyn. Syst. 35(9), 4269–4292 (2015). ISSN 1078-0947.  https://doi.org/10.3934/dcds.2015.35.4269 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Carlini, E., Silva, F. J.: A semi-Lagrangian scheme for the Fokker-Planck equation. IFAC-PapersOnLine 49(8), 272–277 (2016). ISSN 2405-8963. https://doi.org/10.1016/j.ifacol.2016.07.453. http://www.sciencedirect.com/science/article/pii/S2405896316306619 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Carlini, E., Festa, A., Silva, F. J., Wolfram, M.-T.: A semi-Lagrangian scheme for a modified version of the Hughes’ model for pedestrian flow. Dyn. Games Appl. 7, 683–705 (2016). https://doi.org/10.1007/s13235-016-0202-6 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chavanis, P.H.: Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations. Eur. Phys. J. B-Condensed Matter Complex Syst. 62(2), 179–208 (2008)zbMATHGoogle Scholar
  15. 15.
    Evans, L. C.: Adjoint and compensated compactness methods for Hamilton–Jacobi pde. Arch. Ration. Mech. Anal. 197(3), 1053–1088 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Falcone, M., Ferretti, R.: Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2014). ISBN 978-1-611973-04-4Google Scholar
  17. 17.
    Festa, A.: Reconstruction of independent sub-domains for a class of Hamilton–Jacobi equations and application to parallel computing. ESAIM: Math. Model. Numer. Anal. 50(4), 1223–1240 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Festa, A.: Domain decomposition based parallel Howard’s algorithm. Math. Comput. Simul. 147, 121–139 (2017). ISSN 0378-4754. https://doi.org/10.1016/j.matcom.2017.04.008. http://www.sciencedirect.com/science/article/pii/S0378475417301684 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gobet, E.: Weak approximation of killed diffusion using Euler schemes. Stoch. Process. Appl. 87(2), 167–197 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gomes, D. A., Sedjro, M.: One-dimensional forward-forward mean-field games with congestion. Preprint - Submitted to Discrete and Continuous Dynamical Systems-Series S. (2017)Google Scholar
  21. 21.
    Gomes, D., Pimentel, E., Voskanyan, V.: Regularity theory for mean-field game systems. SpringerBriefs in Mathematics (2016)Google Scholar
  22. 22.
    Gomes, D. A., Nurbekyan, L., Sedjro, M.: One-dimensional forward-forward mean-field games. Appl. Math. Optim. 74(3), 619–642 (2016). ISSN 0095-4616. https://doi.org/10.1007/s00245-016-9384-y MathSciNetCrossRefGoogle Scholar
  23. 23.
    Goudon, T., Saad, M.: On a Fokker-Planck equation arising in population dynamics. Rev. Mat. Comput. 11(2), 353–372 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Guéant, O.: Mean field games and applications to economics. PhD thesis, Université Paris-Dauphine (2009)Google Scholar
  25. 25.
    Hughes, R. L.: A continuum theory for the flow of pedestrians. Transp. Res. B Methodol. 36(6), 507–535 (2002)CrossRefGoogle Scholar
  26. 26.
    Jourdain, B., Méléard, S.: Propagation of chaos and fluctuations for a moderate model with smooth initial data. Ann. Inst. H. Poincaré Probab. Stat. 34(6), 727–766 (1998). ISSN 0246-0203. https://doi.org/10.1016/S0246-0203(99)80002-8.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lachapelle, A., Wolfram, M.-T.: On a mean field game approach modeling congestion and aversion in pedestrian crowds. Transp. Res. B Methodol. 45, 1572–1589 (2011)CrossRefGoogle Scholar
  28. 28.
    Lachapelle, A., Salomon, J., Turinici, G.: Computation of mean field equilibria in economics. Math. Models Methods Appl. Sci. 20(4), 567–588 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    McKean, H. P. Jr.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. U.S.A. 56, 1907–1911 (1966). ISSN 0027-8424MathSciNetCrossRefGoogle Scholar
  30. 30.
    McKean, H. P. Jr.: Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pp. 41–57. Air Force Office of Scientific Research, Arlington (1967)Google Scholar
  31. 31.
    Méléard, S.: Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Mathematics, , vol. 1627, pp. 42–95. Springer, Berlin (1996).  https://doi.org/10.1007/BFb0093177 Google Scholar
  32. 32.
    Oberman, A. M.: Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton-Jacobi equations and free boundary problems. SIAM J. Numer. Anal. 44(2), 879–895 (2006). ISSN 0036-1429. https://doi.org/10.1137/S0036142903435235.MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sznitman, A.-S.: Topics in propagation of chaos. In École d’Été de Probabilités de Saint-Flour XIX—1989. Lecture Notes in Mathematics, vol. 1464, pp. 165–251. Springer, Berlin (1991).  https://doi.org/10.1007/BFb0085169 Google Scholar
  34. 34.
    Tosin, A., Festa, A., Wolfram, M.-T.: Kinetic description of collision avoidance in pedestrian crowds by sidestepping (2016). arXiv preprint arXiv:1610.05056Google Scholar
  35. 35.
    Tran, H. V.: Adjoint methods for static Hamilton-Jacobi equations. Calc. Var. Partial Differ. Equ. 41(3-4), 301–319 (2011). ISSN 0944-2669. https://doi.org/10.1007/s00526-010-0363-x MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Adriano Festa
    • 1
  • Diogo A. Gomes
    • 2
    Email author
  • Roberto M. Velho
    • 2
  1. 1.Institut National de Sciences Appliquées, LMI labSaint-Étienne-du-RouvrayFrance
  2. 2.King Abdullah University of Science and Technology (KAUST)CEMSE DivisionThuwalSaudi Arabia

Personalised recommendations