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A Fully-Discrete Scheme for Systems of Nonlinear Fokker-Planck-Kolmogorov Equations

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PDE Models for Multi-Agent Phenomena

Part of the book series: Springer INdAM Series ((SINDAMS,volume 28))

Abstract

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied in Carlini and Silva (SIAM J. Numer. Anal., 2018, To appear) for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games.

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References

  1. Achdou, Y., Bardi, M., Cirant, M.: Mean field games models of segregation. Math. Models Methods Appl. Sci. 27(01), 75–113 (2017)

    Article  MathSciNet  Google Scholar 

  2. Ambrosio, L., Gigli, N., Savaré, G.: Gradient flows in metric spaces and in the space of probability measures, 2nd edn. Lecture Notes in Mathematics ETH Zürich. Birkhäuser Verlag, Bassel (2008)

    Google Scholar 

  3. Benamou, J.-D., Carlier, G., Laborde, M.: An augmented Lagrangian approach to Wasserstein gradient flows and applications. ESAIM: Proc. Surv. 54, 1–17 (2016)

    Article  MathSciNet  Google Scholar 

  4. Bogachev, V. I., Krylov, N. V., Röckner, M., Shaposhnikov, S. V.: Fokker-Planck-Kolmogorov Equations. Mathematical Surveys and Monographs, vol. 207. American Mathematical Society, Providence (2015)

    Google Scholar 

  5. Cardaliaguet, P., Hadikhanloo, S.: Learning in mean field games: the fictitious play. ESAIM Control Optim. Calc. Var. 23(2), 569–591 (2017)

    Article  MathSciNet  Google Scholar 

  6. Carlier, G., Laborde, M.: On systems of continuity equations with nonlinear diffusion and nonlocal drifts, pp. 1–26 (2015). arXiv:1505.01304

    Google Scholar 

  7. 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)

    Article  MathSciNet  Google Scholar 

  8. Carlini, E., Silva, F. J.: A semi-Lagrangian scheme for a degenerate second order mean field game system. Discrete Contin. Dynam. Systems 35(9), 4269–4292 (2015)

    Article  MathSciNet  Google Scholar 

  9. Carlini, E., Silva, F. J.: On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications. SIAM J. Numer. Anal. 56(4), 2148–2177 (2018)

    Article  MathSciNet  Google Scholar 

  10. Cirant, M.: Multi-population mean field games systems with Neumann boundary conditions. J. Math. Pures Appl. (9) 103(5), 1294–1315 (2015)

    Article  MathSciNet  Google Scholar 

  11. Costantini, C., Pacchiarotti, B., Sartoretto, F.: Numerical approximation for functionals of reflecting diffusion processes. SIAM J. Appl. Math. 58(1), 73–102 (1998)

    Article  MathSciNet  Google Scholar 

  12. Di Francesco, M., Fagioli, S.: Measure solutions for non-local interaction PDEs with two species. Nonlinearity 26(10), 2777–2808 (2013)

    Article  MathSciNet  Google Scholar 

  13. Fleming, W. H., Soner, H. M.: Controlled Markov Processes and Viscosity Solutions. Stochastic Modelling and Applied Probability, vol. 25, 2nd edn. Springer, New York (2006)

    Google Scholar 

  14. Kushner, H. J., Dupuis, P.: Numerical Methods for Stochastic Control Problems in Continuous Time. Applications of Mathematics (New York), vol. 24, 2nd edn. Stochastic Modelling and Applied Probability. Springer, New York (2001)

    Book  Google Scholar 

  15. Manita, O. A., Shaposhnikov, S. V.: Nonlinear parabolic equations for measures. Dokl. Akad. Nauk 447(6), 610–614 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Manita, O. A., Shaposhnikov, S. V.: Nonlinear parabolic equations for measures. Algebra i Analiz 25(1), 64–93 (2013)

    MathSciNet  Google Scholar 

  17. Schelling, T. C.: Dynamic models of segregation. J. Math. Sociol. 1(2), 143–186 (1971)

    Article  Google Scholar 

Download references

Acknowledgements

The first author acknowledges financial support by the Indam GNCS project “Metodi numerici per equazioni iperboliche e cinetiche e applicazioni”. The second author is partially supported by the ANR project MFG ANR-16-CE40-0015-01 and the PEPS-INSMI Jeunes project “Some open problems in Mean Field Games” for the years 2016 and 2017.

Both authors acknowledge financial support by the PGMO project VarPDEMFG.

Author information

Authors and Affiliations

  1. “Sapienza”, Università di Roma, Dipartimento di Matematica Guido Castelnuovo, Rome, Italy

    Elisabetta Carlini

  2. Institut de recherche XLIM-DMI, UMR-CNRS 7252 Faculté des sciences et techniques Université de Limoges, Limoges, France

    Francisco J. Silva

  3. TSE-R, Université Toulouse I Capitole, Toulouse, France

    Francisco J. Silva

Authors
  1. Elisabetta Carlini
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  2. Francisco J. Silva
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Corresponding author

Correspondence to Francisco J. Silva .

Editor information

Editors and Affiliations

  1. Ceremade, Université Paris-Dauphine, Paris, France

    Pierre Cardaliaguet

  2. Dept. of Mathematics, Universitá di Roma “Tor Vergata”, Roma, Italy

    Alessio Porretta

  3. Dept. of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Francesco Salvarani

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Carlini, E., Silva, F.J. (2018). A Fully-Discrete Scheme for Systems of Nonlinear Fokker-Planck-Kolmogorov Equations. In: Cardaliaguet, P., Porretta, A., Salvarani, F. (eds) PDE Models for Multi-Agent Phenomena. Springer INdAM Series, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-030-01947-1_9

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