-
Previous Article
Higher order discrete controllability and the approximation of the minimum time function
- DCDS Home
- This Issue
-
Next Article
A nonlinear model of opinion formation on the sphere
A semi-Lagrangian scheme for a degenerate second order mean field game system
1. | "Sapienza" Università di Roma, Dipartimento di Matematica, P.le A. Moro 5, 00185 Roma |
2. | XLIM - DMI UMR CNRS 7252, Faculté des Sciences et Techniques, Université de Limoges, 123 Avenue Albert Thomas, 87060-Limoges Cedes, France |
References:
[1] |
Y. Achdou, F. Camilli and L. Corrias, On numerical approximations of the Hamilton-Jacobi-transport system arising in high frequency, Discrete and Continuous Dynamical Systems- Series B, 19 (2014), 629-650.
doi: 10.3934/dcdsb.2014.19.629. |
[2] |
Y. Achdou, F. Camilli and I. C. 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. C. Dolcetta, Mean field games: Numerical methods, SIAM Journal of Numerical Analysis, 48 (2010), 1136-1162.
doi: 10.1137/090758477. |
[4] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition. Lecture notes in Mathematics ETH Zürich. Birkhäuser Verlag, Bassel, 2008. |
[5] |
G. Barles and P. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283. |
[6] |
B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions, SIAM Journal on Control and Optimization, 49 (2011), 948-962.
doi: 10.1137/090752328. |
[7] |
L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, MA, 1968. |
[8] |
F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Modél. Math. Anal. Numér., 29 (1995), 97-122. |
[9] |
F. Camilli and F. J. Silva, A semi-discrete in time approximation for a first order-finite mean field game problem, Network and Heterogeneous Media, 7 (2012), 263-277.
doi: 10.3934/nhm.2012.7.263. |
[10] |
P. Cardaliaguet, Notes on Mean Field Games: From P.-L. Lions' lectures at Collège de France,, Lecture Notes given at Tor Vergata., ().
|
[11] |
E. Carlini and F. J. Silva, Semi-lagrangian schemes for mean field game models, in Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, (2013), 3115-3120.
doi: 10.1109/CDC.2013.6760358. |
[12] |
E. Carlini and F. J. Silva, A fully discrete semi-lagrangian scheme for a first order mean field game problem, SIAM Journal on Numerical Analysis, 52 (2014), 45-67.
doi: 10.1137/120902987. |
[13] |
P. G. Ciarlet and J.-L. Lions (eds.), Handbook of Numerical Analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991, Finite element methods. Part 1. |
[14] |
M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. American Mathematical Society (New Series), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[15] |
F. Da Lio and O. Ley, Uniqueness results for second-order bellman-isaacs equations under quadratic growth assumptions and applications, SIAM Journal on Control and Optimization, 45 (2006), 74-106.
doi: 10.1137/S0363012904440897. |
[16] |
K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Math. Comp., 82 (2013), 1433-1462.
doi: 10.1090/S0025-5718-2012-02632-9. |
[17] |
M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, MOS-SIAM Series on Optimization, 2013.
doi: 10.1137/1.9781611973051. |
[18] |
A. Figalli, Existence and uniqueness of martingale solutions for sdes with rough or degenerate coefficients, J. Funct. Anal., 254 (2008), 109-153.
doi: 10.1016/j.jfa.2007.09.020. |
[19] |
W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1993. |
[20] |
D. Gomes and J. Saúde, Mean field models, a brief survey, Dynamic Games and Applications, 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
[21] |
O. Guéant, Mean field games equations with quadratic hamiltonian: A specific approach, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250022, 37pp.
doi: 10.1142/S0218202512500224. |
[22] |
O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, vol. 2003 of Lecture Notes in Math., Springer, Berlin, (2011), 205-266.
doi: 10.1007/978-3-642-14660-2_3. |
[23] |
M. Huang, P. Caines and R. Malhamé, Individual and mass behavior in large population stochastic wireless power control problems: Centralized and Nash equillibrium solutions,, Proc. 42nd IEEE-CDC., ().
|
[24] |
M. Huang, P. Caines and R. Malhamé, Large populations stochastic dynamics games: Closed-loop McKean-Vlasov systems and the Nash certainly equivalence principle, Comm. Inf. Syst., 6 (2006), 221-251.
doi: 10.4310/CIS.2006.v6.n3.a5. |
[25] |
H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, vol. 24 of Applications of mathematics, Springer, New York, 2001, Second edition.
doi: 10.1007/978-1-4613-0007-6. |
[26] |
A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Mathematical Models and Methods in Applied Sciences, 20 (2010), 567-588.
doi: 10.1142/S0218202510004349. |
[27] |
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. |
[28] |
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. |
[29] |
J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[30] |
A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics (Second Ed.), Springer, Berlin, 2007. |
[31] |
C. Villani, Topics in Optimal Transportation, Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. |
[32] |
J. Yong and X. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
show all references
References:
[1] |
Y. Achdou, F. Camilli and L. Corrias, On numerical approximations of the Hamilton-Jacobi-transport system arising in high frequency, Discrete and Continuous Dynamical Systems- Series B, 19 (2014), 629-650.
doi: 10.3934/dcdsb.2014.19.629. |
[2] |
Y. Achdou, F. Camilli and I. C. 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. C. Dolcetta, Mean field games: Numerical methods, SIAM Journal of Numerical Analysis, 48 (2010), 1136-1162.
doi: 10.1137/090758477. |
[4] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition. Lecture notes in Mathematics ETH Zürich. Birkhäuser Verlag, Bassel, 2008. |
[5] |
G. Barles and P. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283. |
[6] |
B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions, SIAM Journal on Control and Optimization, 49 (2011), 948-962.
doi: 10.1137/090752328. |
[7] |
L. Breiman, Probability, Addison-Wesley Publishing Company, Reading, MA, 1968. |
[8] |
F. Camilli and M. Falcone, An approximation scheme for the optimal control of diffusion processes, RAIRO Modél. Math. Anal. Numér., 29 (1995), 97-122. |
[9] |
F. Camilli and F. J. Silva, A semi-discrete in time approximation for a first order-finite mean field game problem, Network and Heterogeneous Media, 7 (2012), 263-277.
doi: 10.3934/nhm.2012.7.263. |
[10] |
P. Cardaliaguet, Notes on Mean Field Games: From P.-L. Lions' lectures at Collège de France,, Lecture Notes given at Tor Vergata., ().
|
[11] |
E. Carlini and F. J. Silva, Semi-lagrangian schemes for mean field game models, in Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, (2013), 3115-3120.
doi: 10.1109/CDC.2013.6760358. |
[12] |
E. Carlini and F. J. Silva, A fully discrete semi-lagrangian scheme for a first order mean field game problem, SIAM Journal on Numerical Analysis, 52 (2014), 45-67.
doi: 10.1137/120902987. |
[13] |
P. G. Ciarlet and J.-L. Lions (eds.), Handbook of Numerical Analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991, Finite element methods. Part 1. |
[14] |
M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. American Mathematical Society (New Series), 27 (1992), 1-67.
doi: 10.1090/S0273-0979-1992-00266-5. |
[15] |
F. Da Lio and O. Ley, Uniqueness results for second-order bellman-isaacs equations under quadratic growth assumptions and applications, SIAM Journal on Control and Optimization, 45 (2006), 74-106.
doi: 10.1137/S0363012904440897. |
[16] |
K. Debrabant and E. R. Jakobsen, Semi-Lagrangian schemes for linear and fully non-linear diffusion equations, Math. Comp., 82 (2013), 1433-1462.
doi: 10.1090/S0025-5718-2012-02632-9. |
[17] |
M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, MOS-SIAM Series on Optimization, 2013.
doi: 10.1137/1.9781611973051. |
[18] |
A. Figalli, Existence and uniqueness of martingale solutions for sdes with rough or degenerate coefficients, J. Funct. Anal., 254 (2008), 109-153.
doi: 10.1016/j.jfa.2007.09.020. |
[19] |
W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1993. |
[20] |
D. Gomes and J. Saúde, Mean field models, a brief survey, Dynamic Games and Applications, 4 (2014), 110-154.
doi: 10.1007/s13235-013-0099-2. |
[21] |
O. Guéant, Mean field games equations with quadratic hamiltonian: A specific approach, Mathematical Models and Methods in Applied Sciences, 22 (2012), 1250022, 37pp.
doi: 10.1142/S0218202512500224. |
[22] |
O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, vol. 2003 of Lecture Notes in Math., Springer, Berlin, (2011), 205-266.
doi: 10.1007/978-3-642-14660-2_3. |
[23] |
M. Huang, P. Caines and R. Malhamé, Individual and mass behavior in large population stochastic wireless power control problems: Centralized and Nash equillibrium solutions,, Proc. 42nd IEEE-CDC., ().
|
[24] |
M. Huang, P. Caines and R. Malhamé, Large populations stochastic dynamics games: Closed-loop McKean-Vlasov systems and the Nash certainly equivalence principle, Comm. Inf. Syst., 6 (2006), 221-251.
doi: 10.4310/CIS.2006.v6.n3.a5. |
[25] |
H. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, vol. 24 of Applications of mathematics, Springer, New York, 2001, Second edition.
doi: 10.1007/978-1-4613-0007-6. |
[26] |
A. Lachapelle, J. Salomon and G. Turinici, Computation of mean field equilibria in economics, Mathematical Models and Methods in Applied Sciences, 20 (2010), 567-588.
doi: 10.1142/S0218202510004349. |
[27] |
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. |
[28] |
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. |
[29] |
J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
[30] |
A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics (Second Ed.), Springer, Berlin, 2007. |
[31] |
C. Villani, Topics in Optimal Transportation, Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003. |
[32] |
J. Yong and X. Zhou, Stochastic Controls, Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
[1] |
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 |
[2] |
Holger Heumann, Ralf Hiptmair. Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1471-1495. doi: 10.3934/dcds.2011.29.1471 |
[3] |
Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics and Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016 |
[4] |
Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics and Games, 2021, 8 (4) : 467-486. doi: 10.3934/jdg.2021014 |
[5] |
Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375 |
[6] |
Fabio Camilli, Francisco Silva. A semi-discrete approximation for a first order mean field game problem. Networks and Heterogeneous Media, 2012, 7 (2) : 263-277. doi: 10.3934/nhm.2012.7.263 |
[7] |
Hyunjung Choi, Yanxiang Zhao. Second-order stabilized semi-implicit energy stable schemes for bubble assemblies in binary and ternary systems. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021246 |
[8] |
Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817 |
[9] |
Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. Conference Publications, 2005, 2005 (Special) : 355-364. doi: 10.3934/proc.2005.2005.355 |
[10] |
Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic and Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251 |
[11] |
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 |
[12] |
George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 |
[13] |
Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control and Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018 |
[14] |
Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks and Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279 |
[15] |
Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks and Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021 |
[16] |
Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173 |
[17] |
Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311 |
[18] |
Adriano Festa, Diogo Gomes, Francisco J. Silva, Daniela Tonon. Preface: Mean field games: New trends and applications. Journal of Dynamics and Games, 2021, 8 (4) : i-ii. doi: 10.3934/jdg.2021025 |
[19] |
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 |
[20] |
Lucio Boccardo, Luigi Orsina. The duality method for mean field games systems. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1343-1360. doi: 10.3934/cpaa.2022021 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]