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

New numerical methods for mean field games with quadratic costs

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.
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]

Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021006

[2]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565

[3]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021011

[4]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[5]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234

[6]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[7]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265

[8]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208

[9]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006

[10]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[11]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025

[12]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027

[13]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

[14]

Raghda A. M. Attia, Dumitru Baleanu, Dianchen Lu, Mostafa M. A. Khater, El-Sayed Ahmed. Computational and numerical simulations for the deoxyribonucleic acid (DNA) model. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021018

[15]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

[16]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

[17]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[18]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205

[19]

Vieri Benci, Marco Cococcioni. The algorithmic numbers in non-archimedean numerical computing environments. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1673-1692. doi: 10.3934/dcdss.2020449

[20]

Vakhtang Putkaradze, Stuart Rogers. Numerical simulations of a rolling ball robot actuated by internal point masses. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 143-207. doi: 10.3934/naco.2020021

2019 Impact Factor: 1.053

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (24)

Other articles
by authors

[Back to Top]