
- Previous Article
- JDG Home
- This Issue
-
Next Article
On some singular mean-field games
Splitting methods for a class of non-potential mean field games
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, USA |
We extend the methods from [
References:
[1] |
Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math., 2074, Springer, Heidelberg, 2013, 1-47.
doi: 10.1007/978-3-642-36433-4_1. |
[2] |
Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta,
Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.
doi: 10.1137/120882421. |
[3] |
Y. Achdou and I. Capuzzo-Dolcetta,
Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.
doi: 10.1137/090758477. |
[4] |
N. Almulla, R. Ferreira and D. Gomes,
Two numerical approaches to stationary mean-field games, Dyn. Games Appl., 7 (2017), 657-682.
doi: 10.1007/s13235-016-0203-5. |
[5] |
J.-D. Benamou and G. Carlier,
Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl., 167 (2015), 1-26.
doi: 10.1007/s10957-015-0725-9. |
[6] |
J. -D. Benamou, G. Carlier and F. Santambrogio, Variational mean field games, in Active Particles. Vol. 1. Advances in Theory, Models, and Applications
doi: 10.1007/978-3-319-49996-3_4. |
[7] |
L. M. Briceño-Arias and P. L. Combettes, Monotone operator methods for Nash equilibria in non-potential games, in Computational and Analytical Mathematics, Springer Proc. Math. Stat., 50, Springer, New York, 2013, 143-159.
doi: 10.1007/978-1-4614-7621-4_9. |
[8] |
L. Briceño Arias, D. Kalise, Z. Kobeissi, M. Laurière, A. Mateos González and F. J. Silva, On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings, in CEMRACS 2017-Numerical Methods for Stochastic Models: Control, Uncertainty Quantification, Mean-Field, ESAIM Proc. Surveys, 65, EDP Sci., Les Ulis, 2019, 330-348.
doi: 10.1051/proc/201965330. |
[9] |
L. M. Briceño Arias, D. Kalise and F. J. Silva,
Proximal methods for stationary mean field games with local couplings, SIAM J. Control Optim., 56 (2018), 801-836.
doi: 10.1137/16M1095615. |
[10] |
F. Camilli and F. Silva,
A semi-discrete approximation for a first order mean field game problem, Netw. Heterog. Media, 7 (2012), 263-277.
doi: 10.3934/nhm.2012.7.263. |
[11] |
P. Cardaliaguet,
Long time average of first order mean field games and weak KAM theory, Dyn. Games Appl., 3 (2013), 473-488.
doi: 10.1007/s13235-013-0091-x. |
[12] |
P. Cardaliaguet, Notes on Mean Field Games, (2013)., Available from: Https://www.ceremade.dauphine.fr/ cardaliaguet/. |
[13] |
P. Cardaliaguet, P. J. Graber, A. Porretta and D. Tonon,
Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1287-1317.
doi: 10.1007/s00030-015-0323-4. |
[14] |
P. Cardaliaguet and S. Hadikhanloo,
Learning in mean field games: The fictitious play, ESAIM Control Optim. Calc. Var., 23 (2017), 569-591.
doi: 10.1051/cocv/2016004. |
[15] |
E. Carlini and F. J. Silva,
A fully discrete semi-Lagrangian scheme for a first order mean field game problem, SIAM J. Numer. Anal., 52 (2014), 45-67.
doi: 10.1137/120902987. |
[16] |
E. Carlini and F. J. Silva,
A semi-Lagrangian scheme for a degenerate second order mean field game system, Discrete Contin. Dyn. Syst., 35 (2015), 4269-4292.
doi: 10.3934/dcds.2015.35.4269. |
[17] |
E. Carlini and F. J. Silva,
On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications, SIAM J. Numer. Anal., 56 (2018), 2148-2177.
doi: 10.1137/17M1143022. |
[18] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[19] |
A. Chambolle and T. Pock,
On the ergodic convergence rates of a first-order primal-dual algorithm, Math. Program., 159 (2016), 253-287.
doi: 10.1007/s10107-015-0957-3. |
[20] |
R. Ferreira and D. Gomes,
Existence of weak solutions to stationary mean-field games through variational inequalities, SIAM J. Math. Anal., 50 (2018), 5969-6006.
doi: 10.1137/16M1106705. |
[21] |
R. Ferreira, D. Gomes and T. Tada, Existence of weak solutions to time-dependent mean-field games, preprint, arXiv: 2001.03928. |
[22] |
T. Goldstein, M. Li, X. Yuan, E. Esser and R. Baraniuk, Adaptive primal-dual hybrid gradient methods for saddle-point problems, preprint, arXiv: 1305.0546. |
[23] |
D. A. Gomes and J. Saúde,
Mean field games models-A brief survey, Dyn. Games Appl., 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
[24] |
D. A. Gomes and J. Saúde,
Numerical methods for finite-state mean-field games satisfying a monotonicity condition, Appl. Math. Optim., 83 (2021), 51-82.
doi: 10.1007/s00245-018-9510-0. |
[25] |
P. J. Graber and A. R. Mészáros,
Sobolev regularity for first order mean field games, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1557-1576.
doi: 10.1016/j.anihpc.2018.01.002. |
[26] |
P. J. Graber, A. R. Mészáros, F. J. Silva and D. Tonon, The planning problem in mean field games as regularized mass transport, Calc. Var. Partial Differential Equations, 58 (2019), 28pp.
doi: 10.1007/s00526-019-1561-9. |
[27] |
O. Guéant, J. -M. Lasry and P. -L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., 2003, Springer, Berlin, 2011, 205-266.
doi: 10.1007/978-3-642-14660-2_3. |
[28] |
S. Hadikhanloo, Learning in anonymous nonatomic games with applications to first-order mean field games, preprint, arXiv: 1704.00378. |
[29] |
S. Hadikhanloo and F. J. Silva,
Finite mean field games: Fictitious play and convergence to a first order continuous mean field game, J. Math. Pures Appl. (9), 132 (2019), 369-397.
doi: 10.1016/j.matpur.2019.02.006. |
[30] |
M. Huang, P. E. Caines and R. P. Malhamé,
Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.
doi: 10.1109/TAC.2007.904450. |
[31] |
M. Huang, R. P. Malhamé and P. E. Caines,
Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.
doi: 10.4310/CIS.2006.v6.n3.a5. |
[32] |
M. Jacobs and F. Léger, A fast approach to optimal transport: The back-and-forth method, Numer. Math., 146 (2020), 513-544.
doi: 10.1007/s00211-020-01154-8. |
[33] |
M. Jacobs, F. Léger, W. Li and S. Osher,
Solving large-scale optimization problems with a convergence rate independent of grid size, SIAM J. Numer. Anal., 57 (2019), 1100-1123.
doi: 10.1137/18M118640X. |
[34] |
J.-M. Lasry and P.-L. Lions,
Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.
doi: 10.1016/j.crma.2006.09.019. |
[35] |
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), 679-684.
doi: 10.1016/j.crma.2006.09.018. |
[36] |
J.-M. Lasry and P.-L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[37] |
S. Liu, M. Jacobs, W. Li, L. Nurbekyan and S. J. Osher, Computational methods for nonlocal mean field games with applications, preprint, arXiv: 2004.12210. |
[38] |
L. Nurbekyan,
One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 963-990.
doi: 10.3934/dcdss.2018057. |
[39] |
L. Nurbekyan and J. Saúde,
Fourier approximation methods for first-order nonlocal mean-field games, Port. Math., 75 (2018), 367-396.
doi: 10.4171/PM/2023. |
[40] |
B. C. Vũ,
A variable metric extension of the forward-backward-forward algorithm for monotone operators, Numer. Funct. Anal. Optim., 34 (2013), 1050-1065.
doi: 10.1080/01630563.2013.763825. |
show all references
References:
[1] |
Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math., 2074, Springer, Heidelberg, 2013, 1-47.
doi: 10.1007/978-3-642-36433-4_1. |
[2] |
Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta,
Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.
doi: 10.1137/120882421. |
[3] |
Y. Achdou and I. Capuzzo-Dolcetta,
Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.
doi: 10.1137/090758477. |
[4] |
N. Almulla, R. Ferreira and D. Gomes,
Two numerical approaches to stationary mean-field games, Dyn. Games Appl., 7 (2017), 657-682.
doi: 10.1007/s13235-016-0203-5. |
[5] |
J.-D. Benamou and G. Carlier,
Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl., 167 (2015), 1-26.
doi: 10.1007/s10957-015-0725-9. |
[6] |
J. -D. Benamou, G. Carlier and F. Santambrogio, Variational mean field games, in Active Particles. Vol. 1. Advances in Theory, Models, and Applications
doi: 10.1007/978-3-319-49996-3_4. |
[7] |
L. M. Briceño-Arias and P. L. Combettes, Monotone operator methods for Nash equilibria in non-potential games, in Computational and Analytical Mathematics, Springer Proc. Math. Stat., 50, Springer, New York, 2013, 143-159.
doi: 10.1007/978-1-4614-7621-4_9. |
[8] |
L. Briceño Arias, D. Kalise, Z. Kobeissi, M. Laurière, A. Mateos González and F. J. Silva, On the implementation of a primal-dual algorithm for second order time-dependent mean field games with local couplings, in CEMRACS 2017-Numerical Methods for Stochastic Models: Control, Uncertainty Quantification, Mean-Field, ESAIM Proc. Surveys, 65, EDP Sci., Les Ulis, 2019, 330-348.
doi: 10.1051/proc/201965330. |
[9] |
L. M. Briceño Arias, D. Kalise and F. J. Silva,
Proximal methods for stationary mean field games with local couplings, SIAM J. Control Optim., 56 (2018), 801-836.
doi: 10.1137/16M1095615. |
[10] |
F. Camilli and F. Silva,
A semi-discrete approximation for a first order mean field game problem, Netw. Heterog. Media, 7 (2012), 263-277.
doi: 10.3934/nhm.2012.7.263. |
[11] |
P. Cardaliaguet,
Long time average of first order mean field games and weak KAM theory, Dyn. Games Appl., 3 (2013), 473-488.
doi: 10.1007/s13235-013-0091-x. |
[12] |
P. Cardaliaguet, Notes on Mean Field Games, (2013)., Available from: Https://www.ceremade.dauphine.fr/ cardaliaguet/. |
[13] |
P. Cardaliaguet, P. J. Graber, A. Porretta and D. Tonon,
Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1287-1317.
doi: 10.1007/s00030-015-0323-4. |
[14] |
P. Cardaliaguet and S. Hadikhanloo,
Learning in mean field games: The fictitious play, ESAIM Control Optim. Calc. Var., 23 (2017), 569-591.
doi: 10.1051/cocv/2016004. |
[15] |
E. Carlini and F. J. Silva,
A fully discrete semi-Lagrangian scheme for a first order mean field game problem, SIAM J. Numer. Anal., 52 (2014), 45-67.
doi: 10.1137/120902987. |
[16] |
E. Carlini and F. J. Silva,
A semi-Lagrangian scheme for a degenerate second order mean field game system, Discrete Contin. Dyn. Syst., 35 (2015), 4269-4292.
doi: 10.3934/dcds.2015.35.4269. |
[17] |
E. Carlini and F. J. Silva,
On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications, SIAM J. Numer. Anal., 56 (2018), 2148-2177.
doi: 10.1137/17M1143022. |
[18] |
A. Chambolle and T. Pock,
A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.
doi: 10.1007/s10851-010-0251-1. |
[19] |
A. Chambolle and T. Pock,
On the ergodic convergence rates of a first-order primal-dual algorithm, Math. Program., 159 (2016), 253-287.
doi: 10.1007/s10107-015-0957-3. |
[20] |
R. Ferreira and D. Gomes,
Existence of weak solutions to stationary mean-field games through variational inequalities, SIAM J. Math. Anal., 50 (2018), 5969-6006.
doi: 10.1137/16M1106705. |
[21] |
R. Ferreira, D. Gomes and T. Tada, Existence of weak solutions to time-dependent mean-field games, preprint, arXiv: 2001.03928. |
[22] |
T. Goldstein, M. Li, X. Yuan, E. Esser and R. Baraniuk, Adaptive primal-dual hybrid gradient methods for saddle-point problems, preprint, arXiv: 1305.0546. |
[23] |
D. A. Gomes and J. Saúde,
Mean field games models-A brief survey, Dyn. Games Appl., 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
[24] |
D. A. Gomes and J. Saúde,
Numerical methods for finite-state mean-field games satisfying a monotonicity condition, Appl. Math. Optim., 83 (2021), 51-82.
doi: 10.1007/s00245-018-9510-0. |
[25] |
P. J. Graber and A. R. Mészáros,
Sobolev regularity for first order mean field games, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1557-1576.
doi: 10.1016/j.anihpc.2018.01.002. |
[26] |
P. J. Graber, A. R. Mészáros, F. J. Silva and D. Tonon, The planning problem in mean field games as regularized mass transport, Calc. Var. Partial Differential Equations, 58 (2019), 28pp.
doi: 10.1007/s00526-019-1561-9. |
[27] |
O. Guéant, J. -M. Lasry and P. -L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, Lecture Notes in Math., 2003, Springer, Berlin, 2011, 205-266.
doi: 10.1007/978-3-642-14660-2_3. |
[28] |
S. Hadikhanloo, Learning in anonymous nonatomic games with applications to first-order mean field games, preprint, arXiv: 1704.00378. |
[29] |
S. Hadikhanloo and F. J. Silva,
Finite mean field games: Fictitious play and convergence to a first order continuous mean field game, J. Math. Pures Appl. (9), 132 (2019), 369-397.
doi: 10.1016/j.matpur.2019.02.006. |
[30] |
M. Huang, P. E. Caines and R. P. Malhamé,
Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized $\epsilon$-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571.
doi: 10.1109/TAC.2007.904450. |
[31] |
M. Huang, R. P. Malhamé and P. E. Caines,
Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006), 221-251.
doi: 10.4310/CIS.2006.v6.n3.a5. |
[32] |
M. Jacobs and F. Léger, A fast approach to optimal transport: The back-and-forth method, Numer. Math., 146 (2020), 513-544.
doi: 10.1007/s00211-020-01154-8. |
[33] |
M. Jacobs, F. Léger, W. Li and S. Osher,
Solving large-scale optimization problems with a convergence rate independent of grid size, SIAM J. Numer. Anal., 57 (2019), 1100-1123.
doi: 10.1137/18M118640X. |
[34] |
J.-M. Lasry and P.-L. Lions,
Jeux à champ moyen. I. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.
doi: 10.1016/j.crma.2006.09.019. |
[35] |
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), 679-684.
doi: 10.1016/j.crma.2006.09.018. |
[36] |
J.-M. Lasry and P.-L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[37] |
S. Liu, M. Jacobs, W. Li, L. Nurbekyan and S. J. Osher, Computational methods for nonlocal mean field games with applications, preprint, arXiv: 2004.12210. |
[38] |
L. Nurbekyan,
One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 963-990.
doi: 10.3934/dcdss.2018057. |
[39] |
L. Nurbekyan and J. Saúde,
Fourier approximation methods for first-order nonlocal mean-field games, Port. Math., 75 (2018), 367-396.
doi: 10.4171/PM/2023. |
[40] |
B. C. Vũ,
A variable metric extension of the forward-backward-forward algorithm for monotone operators, Numer. Funct. Anal. Optim., 34 (2013), 1050-1065.
doi: 10.1080/01630563.2013.763825. |





1.00e-3 | 1.00e-4 | 1.00e-5 | 1.00e-6 | 1.00e-7 | |
PDHG | 393 | 955 | 3030 | 9108 | |
adaptive PDHG | 295 | 701 | 2033 | 6200 | 17842 |
1.00e-3 | 1.00e-4 | 1.00e-5 | 1.00e-6 | 1.00e-7 | |
PDHG | 393 | 955 | 3030 | 9108 | |
adaptive PDHG | 295 | 701 | 2033 | 6200 | 17842 |
[1] |
Tigran Bakaryan, Rita Ferreira, Diogo Gomes. A potential approach for planning mean-field games in one dimension. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2147-2187. doi: 10.3934/cpaa.2022054 |
[2] |
Levon Nurbekyan. One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 963-990. doi: 10.3934/dcdss.2018057 |
[3] |
Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006 |
[4] |
Marco Cirant, Diogo A. Gomes, Edgard A. Pimentel, Héctor Sánchez-Morgado. On some singular mean-field games. Journal of Dynamics and Games, 2021, 8 (4) : 445-465. doi: 10.3934/jdg.2021006 |
[5] |
Max-Olivier Hongler. Mean-field games and swarms dynamics in Gaussian and non-Gaussian environments. Journal of Dynamics and Games, 2020, 7 (1) : 1-20. doi: 10.3934/jdg.2020001 |
[6] |
Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks and Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303 |
[7] |
Jorge J. Betancor, Alejandro J. Castro, Marta De León-Contreras. Variation operators for semigroups associated with Fourier-Bessel expansions. Communications on Pure and Applied Analysis, 2022, 21 (1) : 239-273. doi: 10.3934/cpaa.2021176 |
[8] |
Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks and Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315 |
[9] |
Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1 |
[10] |
Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control and Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026 |
[11] |
Diogo Gomes, Marc Sedjro. One-dimensional, forward-forward mean-field games with congestion. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 901-914. doi: 10.3934/dcdss.2018054 |
[12] |
Diogo A. Gomes, Hiroyoshi Mitake, Kengo Terai. The selection problem for some first-order stationary Mean-field games. Networks and Heterogeneous Media, 2020, 15 (4) : 681-710. doi: 10.3934/nhm.2020019 |
[13] |
Jianhui Huang, Shujun Wang, Zhen Wu. Backward-forward linear-quadratic mean-field games with major and minor agents. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 8-. doi: 10.1186/s41546-016-0009-9 |
[14] |
René Carmona, Kenza Hamidouche, Mathieu Laurière, Zongjun Tan. Linear-quadratic zero-sum mean-field type games: Optimality conditions and policy optimization. Journal of Dynamics and Games, 2021, 8 (4) : 403-443. doi: 10.3934/jdg.2021023 |
[15] |
Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control and Related Fields, 2022, 12 (2) : 371-404. doi: 10.3934/mcrf.2021026 |
[16] |
Anna Chiara Lai, Paola Loreti. Robot's finger and expansions in non-integer bases. Networks and Heterogeneous Media, 2012, 7 (1) : 71-111. doi: 10.3934/nhm.2012.7.71 |
[17] |
Pieter C. Allaart. An algebraic approach to entropy plateaus in non-integer base expansions. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6507-6522. doi: 10.3934/dcds.2019282 |
[18] |
Karma Dajani, Charlene Kalle. Random β-expansions with deleted digits. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 199-217. doi: 10.3934/dcds.2007.18.199 |
[19] |
Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic and Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299 |
[20] |
Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]