American Institute of Mathematical Sciences

Advanced
June  2012, 7(2): 315-336. doi: 10.3934/nhm.2012.7.315

New numerical methods for mean field games with quadratic costs

Olivier Guéant 1,

1. 

UFR de Math, Universit, 175, rue du Chevaleret, 75013 Paris, France

Received  November 2011 Revised  March 2012 Published  June 2012

Mean field games have been introduced by J.-M. Lasry and P.-L. Lions in [13, 14, 15] as the limit case of stochastic differential games when the number of players goes to $+\infty$. In the case of quadratic costs, we present two changes of variables that allow to transform the mean field games (MFG) equations into two simpler systems of equations. The first change of variables, introduced in [11], leads to two heat equations with nonlinear source terms. The second change of variables, which is introduced for the first time in this paper, leads to two Hamilton-Jacobi-Bellman equations. Numerical methods based on these equations are presented and numerical experiments are carried out.
Keywords: Hopf-Cole transformation., finite difference methods, numerical methods, constructive scheme, Mean field games.
Mathematics Subject Classification: Primary: 35K91, 65M12; Secondary: 91A1.
Citation: Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315
References:
[1]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem,, SIAM J. Control Opt., 50 (2012), 77.  doi: 10.1137/100790069.  Google Scholar

[2]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM Journal on Numerical Analysis, 48 (2010), 1136.  doi: 10.1137/090758477.  Google Scholar

[3]

P. Cardaliaguet, Notes on mean field games,, from P.-L. Lions' lectures at Collège de France, (2010).   Google Scholar

[4]

M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations,, Mathematics of Computation, 43 (1984), 1.  doi: 10.1090/S0025-5718-1984-0744921-8.  Google Scholar

[5]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (2010).   Google Scholar

[6]

D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games,, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308.   Google Scholar

[7]

O. Guéant, Mean field games equations with quadratic hamiltonian: A specifc approach,, to appear in Mathematical Models and Methods in Applied Sciences (M3AS)., ().   Google Scholar

[8]

O. Guéant, Mean field games with quadratic hamiltonian: A constructive scheme,, to appear in the Annals of ISDG., ().   Google Scholar

[9]

O. Guéant, "Mean Field Games and Applications to Economics,", Ph.D thesis, (2009).   Google Scholar

[10]

O. Guéant, A reference case for mean field games models,, Journal de Mathématiques Pures et Appliquées (9), 92 (2009), 276.   Google Scholar

[11]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in, 2003 (2011), 205.   Google Scholar

[12]

A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics,, Mathematical Models and Methods in Applied Sciences, 20 (2010), 567.  doi: 10.1142/S0218202510004349.  Google Scholar

[13]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Acad. Sci. Paris, 343 (2006), 619.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[14]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal,, C. R. Acad. Sci. Paris, 343 (2006), 679.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[15]

J.-M. Lasry and P.-L. Lions, Mean field games,, Japanese Journal of Mathematics, 2 (2007), 229.   Google Scholar

[16]

P.-L. Lions, Théorie des jeux à champs moyens, Cours au Collège de France., Available from: \url{http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp}., ().   Google Scholar

show all references

References:
[1]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem,, SIAM J. Control Opt., 50 (2012), 77.  doi: 10.1137/100790069.  Google Scholar

[2]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods,, SIAM Journal on Numerical Analysis, 48 (2010), 1136.  doi: 10.1137/090758477.  Google Scholar

[3]

P. Cardaliaguet, Notes on mean field games,, from P.-L. Lions' lectures at Collège de France, (2010).   Google Scholar

[4]

M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton-Jacobi equations,, Mathematics of Computation, 43 (1984), 1.  doi: 10.1090/S0025-5718-1984-0744921-8.  Google Scholar

[5]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (2010).   Google Scholar

[6]

D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games,, Journal de Mathématiques Pures et Appliquées (9), 93 (2010), 308.   Google Scholar

[7]

O. Guéant, Mean field games equations with quadratic hamiltonian: A specifc approach,, to appear in Mathematical Models and Methods in Applied Sciences (M3AS)., ().   Google Scholar

[8]

O. Guéant, Mean field games with quadratic hamiltonian: A constructive scheme,, to appear in the Annals of ISDG., ().   Google Scholar

[9]

O. Guéant, "Mean Field Games and Applications to Economics,", Ph.D thesis, (2009).   Google Scholar

[10]

O. Guéant, A reference case for mean field games models,, Journal de Mathématiques Pures et Appliquées (9), 92 (2009), 276.   Google Scholar

[11]

O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications,, in, 2003 (2011), 205.   Google Scholar

[12]

A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics,, Mathematical Models and Methods in Applied Sciences, 20 (2010), 567.  doi: 10.1142/S0218202510004349.  Google Scholar

[13]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. I. Le cas stationnaire,, C. R. Acad. Sci. Paris, 343 (2006), 619.  doi: 10.1016/j.crma.2006.09.019.  Google Scholar

[14]

J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. II. Horizon fini et contrôle optimal,, C. R. Acad. Sci. Paris, 343 (2006), 679.  doi: 10.1016/j.crma.2006.09.018.  Google Scholar

[15]

J.-M. Lasry and P.-L. Lions, Mean field games,, Japanese Journal of Mathematics, 2 (2007), 229.   Google Scholar

[16]

P.-L. Lions, Théorie des jeux à champs moyens, Cours au Collège de France., Available from: \url{http://www.college-de-france.fr/default/EN/all/equ_der/audio_video.jsp}., ().   Google Scholar

[1]

Z. Jackiewicz, B. Zubik-Kowal, B. Basse. Finite-difference and pseudo-spectral methods for the numerical simulations of in vitro human tumor cell population kinetics. Mathematical Biosciences & Engineering, 2009, 6 (3) : 561-572. doi: 10.3934/mbe.2009.6.561
[2]

Xiaohai Wan, Zhilin Li. Some new finite difference methods for Helmholtz equations on irregular domains or with interfaces. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1155-1174. doi: 10.3934/dcdsb.2012.17.1155
[3]

Jitraj Saha, Nilima Das, Jitendra Kumar, Andreas Bück. Numerical solutions for multidimensional fragmentation problems using finite volume methods. Kinetic & Related Models, 2019, 12 (1) : 79-103. doi: 10.3934/krm.2019004
[4]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644
[5]

Emmanuel Frénod. Homogenization-based numerical methods. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : i-ix. doi: 10.3934/dcdss.201605i
[6]

Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279
[7]

Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173
[8]

Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311
[9]

Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks & Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021
[10]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics & Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016
[11]

Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133
[12]

Ching-Shan Chou, Yong-Tao Zhang, Rui Zhao, Qing Nie. Numerical methods for stiff reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 515-525. doi: 10.3934/dcdsb.2007.7.515
[13]

Emmanuel Frénod. An attempt at classifying homogenization-based numerical methods. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : i-vi. doi: 10.3934/dcdss.2015.8.1i
[14]

Sebastián J. Ferraro, David Iglesias-Ponte, D. Martín de Diego. Numerical and geometric aspects of the nonholonomic SHAKE and RATTLE methods. Conference Publications, 2009, 2009 (Special) : 220-229. doi: 10.3934/proc.2009.2009.220
[15]

Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks & Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241
[16]

Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289
[17]

Abdon Atangana, José Francisco Gómez-Aguilar, Jordan Y. Hristov. Preface on "New trends of numerical and analytical methods". Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : ⅰ-ⅱ. doi: 10.3934/dcdss.20203i
[18]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243
[19]

Yves Achdou, Victor Perez. Iterative strategies for solving linearized discrete mean field games systems. Networks & Heterogeneous Media, 2012, 7 (2) : 197-217. doi: 10.3934/nhm.2012.7.197
[20]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences