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June  2012, 7(2): 197-217. doi: 10.3934/nhm.2012.7.197

Iterative strategies for solving linearized discrete mean field games systems

Yves Achdou 1, and  Victor Perez 1,

1. 

Université Paris Diderot, UMR 7598, Laboratoire Jacques-Louis Lions, Paris, France, France

Received  November 2011 Revised  March 2012 Published  June 2012

Mean fields games (MFG) describe the asymptotic behavior of stochastic differential games in which the number of players tends to $+\infty$. Under suitable assumptions, they lead to a new kind of system of two partial differential equations: a forward Bellman equation coupled with a backward Fokker-Planck equation. In earlier articles, finite difference schemes preserving the structure of the system have been proposed and studied. They lead to large systems of nonlinear equations in finite dimension. A possible way of numerically solving the latter is to use inexact Newton methods: a Newton step consists of solving a linearized discrete MFG system. The forward-backward character of the MFG system makes it impossible to use time marching methods. In the present work, we propose three families of iterative strategies for solving the linearized discrete MFG systems, most of which involve suitable multigrid solvers or preconditioners.
Keywords: multigrid methods., numerical methods, Mean field games, iterative solvers.
Mathematics Subject Classification: Primary: 91-08, 91A23, 65N22; Secondary: 65M06, 65F1.
Citation: 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
References:
[1]

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

[3]

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

[4]

J.-D. Benamou and Y. Brenier, Mixed $L^2$-Wasserstein optimal mapping between prescribed density functions,, J. Optim. Theory Appl., 111 (2001), 255. doi: 10.1023/A:1011926116573. Google Scholar

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J.-D. Benamou, Y. Brenier and K. Guittet, The Monge-Kantorovitch mass transfer and its computational fluid mechanics formulation,, ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 40 (2002), 21. Google Scholar

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A. Brandt, Rigorous quantitative analysis of multigrid. I. Constant coefficients two-level cycle with $L_2$-norm,, SIAM J. Numer. Anal., 31 (1994), 1695. doi: 10.1137/0731087. Google Scholar

[7]

D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games,, J. Math. Pures Appl. (9), 93 (2010), 308. Google Scholar

[8]

O. Guéant, Mean field games equations with quadratic hamiltonian: A specific approach,, \arXiv{1106.3269}, (2011). Google Scholar

[9]

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

[10]

S. Henn, A multigrid method for a fourth-order diffusion equation with application to image processing,, SIAM J. Sci. Comput., 27 (2005), 831. doi: 10.1137/040611124. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229. Google Scholar

[15]

P.-L. Lions, Cours du Collège de France, 2007-2011., Available from: \url{http://www.college-de-france.fr/default/EN/all/equ_der/}., (). Google Scholar

[16]

U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid,", With contributions by A. Brandt, (2001). Google Scholar

[17]

H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,, SIAM J. Sci. Statist. Comput., 13 (1992), 631. doi: 10.1137/0913035. Google Scholar

show all references

References:
[1]

, UMFPACK., Available from: \url{http://www.cise.ufl.edu/research/sparse/umfpack/current/}., (). Google Scholar

[2]

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

[3]

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

[4]

J.-D. Benamou and Y. Brenier, Mixed $L^2$-Wasserstein optimal mapping between prescribed density functions,, J. Optim. Theory Appl., 111 (2001), 255. doi: 10.1023/A:1011926116573. Google Scholar

[5]

J.-D. Benamou, Y. Brenier and K. Guittet, The Monge-Kantorovitch mass transfer and its computational fluid mechanics formulation,, ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 40 (2002), 21. Google Scholar

[6]

A. Brandt, Rigorous quantitative analysis of multigrid. I. Constant coefficients two-level cycle with $L_2$-norm,, SIAM J. Numer. Anal., 31 (1994), 1695. doi: 10.1137/0731087. Google Scholar

[7]

D. A. Gomes, J. Mohr and R. R. Souza, Discrete time, finite state space mean field games,, J. Math. Pures Appl. (9), 93 (2010), 308. Google Scholar

[8]

O. Guéant, Mean field games equations with quadratic hamiltonian: A specific approach,, \arXiv{1106.3269}, (2011). Google Scholar

[9]

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

[10]

S. Henn, A multigrid method for a fourth-order diffusion equation with application to image processing,, SIAM J. Sci. Comput., 27 (2005), 831. doi: 10.1137/040611124. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

J.-M. Lasry and P.-L. Lions, Mean field games,, Jpn. J. Math., 2 (2007), 229. Google Scholar

[15]

P.-L. Lions, Cours du Collège de France, 2007-2011., Available from: \url{http://www.college-de-france.fr/default/EN/all/equ_der/}., (). Google Scholar

[16]

U. Trottenberg, C. W. Oosterlee and A. Schüller, "Multigrid,", With contributions by A. Brandt, (2001). Google Scholar

[17]

H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,, SIAM J. Sci. Statist. Comput., 13 (1992), 631. doi: 10.1137/0913035. Google Scholar

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