Mean field games of controls: Finite difference approximations

Yves Achdou; Ziad Kobeissi; Yves Achdou; Ziad Kobeissi

doi:10.3934/mine.2021024

Mathematics in Engineering

2021, Volume 3, Issue 3: 1-35. doi: 10.3934/mine.2021024

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Mean field games of controls: Finite difference approximations

Yves Achdou ^,,
Ziad Kobeissi

Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions, (LJLL), F-75006 Paris, France

Received: 30 January 2020 Accepted: 08 April 2020 Published: 17 July 2020

We consider a class of mean field games in which the agents interact through both their states and controls, and we focus on situations in which a generic agent tries to adjust her speed (control) to an average speed (the average is made in a neighborhood in the state space). In such cases, the monotonicity assumptions that are frequently made in the theory of mean field games do not hold, and uniqueness cannot be expected in general. Such model lead to systems of forward-backward nonlinear nonlocal parabolic equations; the latter are supplemented with various kinds of boundary conditions, in particular Neumann-like boundary conditions stemming from reflection conditions on the underlying controled stochastic processes. The present work deals with numerical approximations of the above megntioned systems. After describing the finite difference scheme, we propose an iterative method for solving the systems of nonlinear equations that arise in the discrete setting; it combines a continuation method, Newton iterations and inner loops of a bigradient like solver. The numerical method is used for simulating two examples. We also make experiments on the behaviour of the iterative algorithm when the parameters of the model vary.
- mean field games,
- interactions via controls,
- crowd motion,
- numerical simulations,
- finite difference method
Citation: Yves Achdou, Ziad Kobeissi. Mean field games of controls: Finite difference approximations[J]. Mathematics in Engineering, 2021, 3(3): 1-35. doi: 10.3934/mine.2021024

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Abstract

We consider a class of mean field games in which the agents interact through both their states and controls, and we focus on situations in which a generic agent tries to adjust her speed (control) to an average speed (the average is made in a neighborhood in the state space). In such cases, the monotonicity assumptions that are frequently made in the theory of mean field games do not hold, and uniqueness cannot be expected in general. Such model lead to systems of forward-backward nonlinear nonlocal parabolic equations; the latter are supplemented with various kinds of boundary conditions, in particular Neumann-like boundary conditions stemming from reflection conditions on the underlying controled stochastic processes. The present work deals with numerical approximations of the above megntioned systems. After describing the finite difference scheme, we propose an iterative method for solving the systems of nonlinear equations that arise in the discrete setting; it combines a continuation method, Newton iterations and inner loops of a bigradient like solver. The numerical method is used for simulating two examples. We also make experiments on the behaviour of the iterative algorithm when the parameters of the model vary.

References

[1]	UMFPACK. Available from: http://www.cise.ufl.edu/research/sparse/umfpack/current/.
[2]	Achdou Y (2013) Finite difference methods for mean field games, In: Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Heidelberg: Springer, 1-47.
[3]	Achdou Y, Capuzzo-Dolcetta I (2010) Mean field games: Numerical methods. SIAM J Numer Anal 48: 1136-1162.
[4]	Achdou Y, Lasry JM (2019) Mean field games for modeling crowd motion, In: Contributions to Partial Differential Equations and Applications, Cham: Springer, 17-42.
[5]	Achdou Y, Porretta A (2018) Mean field games with congestion. Ann I H Poincaré Anal non linéaire 35: 443-480,
[6]	Bensoussan A, Frehse J, Yam P (2013) Mean field games and mean field type control theory. New York: Springer.
[7]	Bonnans FJ, Hadikhanloo S, Pfeiffer L (2019) Schauder estimates for a class of potential mean field games of controls. arXiv: 1902.05461.
[8]	Cardaliaguet P, Lehalle CA (2018) Mean field game of controls and an application to trade crowding. Math Financ Econ 12: 335-363.
[9]	Carmona R, Delarue F (2015) Forward-backward stochastic differential equations and controlled McKean-Vlasov dynamics. Ann Probab 43: 2647-2700.
[10]	Carmona R, Delarue F (2018) Probabilistic theory of mean field games with applications I, Cham: Springer.
[11]	Carmona R, Lacker D (2015)) A probabilistic weak formulation of mean field games and applications. Ann Appl Probab 25: 1189-1231.
[12]	Gomes D, Voskanyan V (2013) Extended mean field games. Izv Nats Akad Nauk Armenii Mat 48: 63-76.
[13]	Gomes DA, Patrizi S, Voskanyan V (2014) On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal 99: 49-79.
[14]	Huang M, Caines PE, Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria. IEEE T Automat Contr 52: 1560-1571.
[15]	Huang M, Caines PE, Malhamé RP (2006) Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6: 221-251.
[16]	Kobeissi Z (2019) On Classical Solutions to the Mean Field Game System of Controls. arXiv: 1904.11292.
[17]	Lasry JM, Lions PL (2006) Jeux à champ moyen. I. Le cas stationnaire. C R Math Acad Sci Paris 343: 619-625.
[18]	Lasry JM, Lions PL (2006) Jeux à champ moyen. II. Horizon fini et contr?le optimal. C R Math Acad Sci Paris 343: 679-684.
[19]	Lasry JM, Lions PL (2007) Mean field games. JPN J Math 2: 229-260.
[20]	Lions PL, Théorie des jeux à champs moyen. Video lecture series at Collège de France, 2011-2019. Available from: https://www.college-de-france.fr/site/pierre-louis-lions/index.htm.
[21]	van der Vorst HA (1992) Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Sci Statist Comput 13: 631-644.

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