• Previous Article
    Long-time behavior of the one-phase Stefan problem in periodic and random media
  • DCDS-S Home
  • This Issue
  • Next Article
    A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations
October  2018, 11(5): 963-990. doi: 10.3934/dcdss.2018057

One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach

4700 King Abdullah University of Science and Technology, CEMSE Division, Thuwal, 23955-6900, KSA

Received  March 2017 Revised  September 2017 Published  June 2018

Fund Project: The author is supported by KAUST baseline and start-up funds and KAUST SRI, Uncertainty Quantification Center in Computational Science and Engineering.

Here, we study a one-dimensional, non-local mean-field game model with congestion. When the kernel in the non-local coupling is a trigonometric polynomial we reduce the problem to a finite dimensional system. Furthermore, we treat the general case by approximating the kernel with trigonometric polynomials. Our technique is based on Fourier expansion methods.

Citation: Levon Nurbekyan. One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 963-990. doi: 10.3934/dcdss.2018057
References:
[1]

Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, vol. 2074 of Lecture Notes in Math., Springer, Heidelberg, 2013, 1–47. doi: 10.1007/978-3-642-36433-4_1. Google Scholar

[2]

Y. AchdouF. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109. doi: 10.1137/100790069. Google Scholar

[3]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477. Google Scholar

[4]

Y. Achdou and V. Perez, Iterative strategies for solving linearized discrete mean field games systems, Netw. Heterog. Media, 7 (2012), 197-217. doi: 10.3934/nhm.2012.7.197. Google Scholar

[5]

Y. Achdou and A. Porretta, Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games, SIAM J. Numer. Anal., 54 (2016), 161-186. doi: 10.1137/15M1015455. Google Scholar

[6]

N. AlmullaR. Ferreira and D. Gomes, Two numerical approaches to stationary mean-field games, Dynamic Games and Applications, 7 (2017), 657-682. doi: 10.1007/s13235-016-0203-5. Google Scholar

[7]

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. Google Scholar

[8]

P. Cardaliaguet, Notes on Mean Field Games, 2013, URL https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.Google Scholar

[9]

P. Cardaliaguet, Weak solutions for first order mean field games with local coupling, in Analysis and Geometry in Control Theory and Its Applications, vol. 11 of Springer INdAM Ser., Springer, Cham, 2015, 111–158. doi: 10.1007/978-3-319-06917-3_5. Google Scholar

[10]

P. CardaliaguetP.J. GraberA. 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. Google Scholar

[11]

P. CardaliaguetA. R. Mészáros and F. Santambrogio, First order mean field games with density constraints: pressure equals price, SIAM J. Control Optim., 54 (2016), 2672-2709. doi: 10.1137/15M1029849. Google Scholar

[12]

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. Google Scholar

[13]

M. Cirant, Stationary focusing mean-field games, Comm. Partial Differential Equations, 41 (2016), 1324-1346. doi: 10.1080/03605302.2016.1192647. Google Scholar

[14]

J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001, Translated and revised from the 1995 Spanish original by David Cruz-Uribe Google Scholar

[15]

D. Evangelista and D. Gomes, On the existence of solutions for stationary mean-field games with congestion, Journal of Dynamics and Differential Equations, (2017), 1-24. doi: 10.1007/s10884-017-9615-1. Google Scholar

[16]

R. Ferreira and D. Gomes, Existence of weak solutions for stationary mean-field games through variational inequalities, arXiv preprint, arXiv: 1512.05828, [math. AP].Google Scholar

[17]

D. Gomes and H. Mitake, Existence for stationary mean-field games with congestion and quadratic Hamiltonians, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1897-1910. doi: 10.1007/s00030-015-0349-7. Google Scholar

[18]

D. Gomes, L. Nurbekyan and M. Prazeres, Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion, in 2016 IEEE 55th Conference on Decision and Control (CDC), 2016, 4534–4539. doi: 10.1109/CDC.2016.7798959. Google Scholar

[19]

D. GomesL. Nurbekyan and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dynamic Games and Applications, 8 (2018), 315-351. doi: 10.1007/s13235-017-0223-9. Google Scholar

[20]

D. Gomes and E. Pimentel, Time-dependent mean-field games with logarithmic nonlinearities, SIAM Journal on Mathematical Analysis, 47 (2015), 3798-3812. doi: 10.1137/140984622. Google Scholar

[21]

D. Gomes and E. Pimentel, Local regularity for mean-field games in the whole space, Minimax Theory and its Applications, 1 (2016), 65-82. Google Scholar

[22]

D. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case, Comm. Partial Differential Equations, 40 (2015), 40-76. doi: 10.1080/03605302.2014.903574. Google Scholar

[23]

D. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the superquadratic case, ESAIM: COCV, 22 (2016), 562-580. doi: 10.1051/cocv/2015029. Google Scholar

[24]

D. Gomes, E. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics, Springer, [Cham], 2016. doi: 10.1007/978-3-319-38934-9. Google Scholar

[25]

D. GomesG. E. Pires and H. Sánchez-Morgado, A-priori estimates for stationary mean-field games, Netw. Heterog. Media, 7 (2012), 303-314. doi: 10.3934/nhm.2012.7.303. Google Scholar

[26]

D. Gomes and H. Sánchez-Morgado, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc., 366 (2014), 903-929. doi: 10.1090/S0002-9947-2013-05936-3. Google Scholar

[27]

D. 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. Google Scholar

[28]

D. Gomes and V. Voskanyan, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2), 92 (2015), 778-799. doi: 10.1112/jlms/jdv052. Google Scholar

[29]

J. Graber, Weak solutions for mean field games with congestion, arXiv preprint, URL https://arXiv.org/abs/1503.04733, arXiv: 1503.04733v3 [math. AP].Google Scholar

[30]

O. Guéant, New numerical methods for mean field games with quadratic costs, Netw. Heterog. Media, 7 (2012), 315-336. doi: 10.3934/nhm.2012.7.315. Google Scholar

[31]

O. Guéant, Existence and uniqueness result for mean field games with congestion effect on graphs, Appl. Math. Optim., 72 (2015), 291-303. doi: 10.1007/s00245-014-9280-2. Google Scholar

[32]

O. Guéant, J. -M. Lasry and P. -L. Lions, Mean field games and applications, in ParisPrinceton 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. Google Scholar

[33]

M. HuangP. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ε-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571. doi: 10.1109/TAC.2007.904450. Google Scholar

[34]

M. HuangR. 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. Google Scholar

[35]

J. -M. Lasry and P. -L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229–260, URL http://www.ifd.dauphine.fr/fileadmin/mediatheque/recherche_et_valo/FDD/Cahier_Chaire_2.pdf. doi: 10.1007/s11537-007-0657-8. Google Scholar

[36]

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. Google Scholar

[37]

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. Google Scholar

[38]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8. Google Scholar

[39]

P. -L. Lions, College de France course on mean-field games, URL https://www.college-de-france.fr/site/en-pierre-louis-lions/_course.htm.Google Scholar

[40]

A. R. Mészáros and F. J. Silva, A variational approach to second order mean field games with density constraints: The stationary case, J. Math. Pures Appl. (9), 104 (2015), 1135-1159. doi: 10.1016/j.matpur.2015.07.008. Google Scholar

[41]

E. Pimentel and V. Voskanyan, Regularity for second-order stationary mean-field games, Indiana Univ. Math. J., 66 (2017), 1-22. doi: 10.1512/iumj.2017.66.5944. Google Scholar

[42]

A. Porretta, On the planning problem for the mean field games system, Dyn. Games Appl., 4 (2014), 231-256. doi: 10.1007/s13235-013-0080-0. Google Scholar

[43]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Ration. Mech. Anal., 216 (2015), 1-62. doi: 10.1007/s00205-014-0799-9. Google Scholar

show all references

References:
[1]

Y. Achdou, Finite difference methods for mean field games, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, vol. 2074 of Lecture Notes in Math., Springer, Heidelberg, 2013, 1–47. doi: 10.1007/978-3-642-36433-4_1. Google Scholar

[2]

Y. AchdouF. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109. doi: 10.1137/100790069. Google Scholar

[3]

Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162. doi: 10.1137/090758477. Google Scholar

[4]

Y. Achdou and V. Perez, Iterative strategies for solving linearized discrete mean field games systems, Netw. Heterog. Media, 7 (2012), 197-217. doi: 10.3934/nhm.2012.7.197. Google Scholar

[5]

Y. Achdou and A. Porretta, Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games, SIAM J. Numer. Anal., 54 (2016), 161-186. doi: 10.1137/15M1015455. Google Scholar

[6]

N. AlmullaR. Ferreira and D. Gomes, Two numerical approaches to stationary mean-field games, Dynamic Games and Applications, 7 (2017), 657-682. doi: 10.1007/s13235-016-0203-5. Google Scholar

[7]

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. Google Scholar

[8]

P. Cardaliaguet, Notes on Mean Field Games, 2013, URL https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.Google Scholar

[9]

P. Cardaliaguet, Weak solutions for first order mean field games with local coupling, in Analysis and Geometry in Control Theory and Its Applications, vol. 11 of Springer INdAM Ser., Springer, Cham, 2015, 111–158. doi: 10.1007/978-3-319-06917-3_5. Google Scholar

[10]

P. CardaliaguetP.J. GraberA. 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. Google Scholar

[11]

P. CardaliaguetA. R. Mészáros and F. Santambrogio, First order mean field games with density constraints: pressure equals price, SIAM J. Control Optim., 54 (2016), 2672-2709. doi: 10.1137/15M1029849. Google Scholar

[12]

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. Google Scholar

[13]

M. Cirant, Stationary focusing mean-field games, Comm. Partial Differential Equations, 41 (2016), 1324-1346. doi: 10.1080/03605302.2016.1192647. Google Scholar

[14]

J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001, Translated and revised from the 1995 Spanish original by David Cruz-Uribe Google Scholar

[15]

D. Evangelista and D. Gomes, On the existence of solutions for stationary mean-field games with congestion, Journal of Dynamics and Differential Equations, (2017), 1-24. doi: 10.1007/s10884-017-9615-1. Google Scholar

[16]

R. Ferreira and D. Gomes, Existence of weak solutions for stationary mean-field games through variational inequalities, arXiv preprint, arXiv: 1512.05828, [math. AP].Google Scholar

[17]

D. Gomes and H. Mitake, Existence for stationary mean-field games with congestion and quadratic Hamiltonians, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1897-1910. doi: 10.1007/s00030-015-0349-7. Google Scholar

[18]

D. Gomes, L. Nurbekyan and M. Prazeres, Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion, in 2016 IEEE 55th Conference on Decision and Control (CDC), 2016, 4534–4539. doi: 10.1109/CDC.2016.7798959. Google Scholar

[19]

D. GomesL. Nurbekyan and M. Prazeres, One-dimensional stationary mean-field games with local coupling, Dynamic Games and Applications, 8 (2018), 315-351. doi: 10.1007/s13235-017-0223-9. Google Scholar

[20]

D. Gomes and E. Pimentel, Time-dependent mean-field games with logarithmic nonlinearities, SIAM Journal on Mathematical Analysis, 47 (2015), 3798-3812. doi: 10.1137/140984622. Google Scholar

[21]

D. Gomes and E. Pimentel, Local regularity for mean-field games in the whole space, Minimax Theory and its Applications, 1 (2016), 65-82. Google Scholar

[22]

D. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the subquadratic case, Comm. Partial Differential Equations, 40 (2015), 40-76. doi: 10.1080/03605302.2014.903574. Google Scholar

[23]

D. GomesE. Pimentel and H. Sánchez-Morgado, Time-dependent mean-field games in the superquadratic case, ESAIM: COCV, 22 (2016), 562-580. doi: 10.1051/cocv/2015029. Google Scholar

[24]

D. Gomes, E. Pimentel and V. Voskanyan, Regularity Theory for Mean-Field Game Systems, SpringerBriefs in Mathematics, Springer, [Cham], 2016. doi: 10.1007/978-3-319-38934-9. Google Scholar

[25]

D. GomesG. E. Pires and H. Sánchez-Morgado, A-priori estimates for stationary mean-field games, Netw. Heterog. Media, 7 (2012), 303-314. doi: 10.3934/nhm.2012.7.303. Google Scholar

[26]

D. Gomes and H. Sánchez-Morgado, A stochastic Evans-Aronsson problem, Trans. Amer. Math. Soc., 366 (2014), 903-929. doi: 10.1090/S0002-9947-2013-05936-3. Google Scholar

[27]

D. 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. Google Scholar

[28]

D. Gomes and V. Voskanyan, Short-time existence of solutions for mean-field games with congestion, J. Lond. Math. Soc. (2), 92 (2015), 778-799. doi: 10.1112/jlms/jdv052. Google Scholar

[29]

J. Graber, Weak solutions for mean field games with congestion, arXiv preprint, URL https://arXiv.org/abs/1503.04733, arXiv: 1503.04733v3 [math. AP].Google Scholar

[30]

O. Guéant, New numerical methods for mean field games with quadratic costs, Netw. Heterog. Media, 7 (2012), 315-336. doi: 10.3934/nhm.2012.7.315. Google Scholar

[31]

O. Guéant, Existence and uniqueness result for mean field games with congestion effect on graphs, Appl. Math. Optim., 72 (2015), 291-303. doi: 10.1007/s00245-014-9280-2. Google Scholar

[32]

O. Guéant, J. -M. Lasry and P. -L. Lions, Mean field games and applications, in ParisPrinceton 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. Google Scholar

[33]

M. HuangP. E. Caines and R. P. Malhamé, Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized ε-Nash equilibria, IEEE Trans. Automat. Control, 52 (2007), 1560-1571. doi: 10.1109/TAC.2007.904450. Google Scholar

[34]

M. HuangR. 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. Google Scholar

[35]

J. -M. Lasry and P. -L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229–260, URL http://www.ifd.dauphine.fr/fileadmin/mediatheque/recherche_et_valo/FDD/Cahier_Chaire_2.pdf. doi: 10.1007/s11537-007-0657-8. Google Scholar

[36]

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. Google Scholar

[37]

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. Google Scholar

[38]

J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260. doi: 10.1007/s11537-007-0657-8. Google Scholar

[39]

P. -L. Lions, College de France course on mean-field games, URL https://www.college-de-france.fr/site/en-pierre-louis-lions/_course.htm.Google Scholar

[40]

A. R. Mészáros and F. J. Silva, A variational approach to second order mean field games with density constraints: The stationary case, J. Math. Pures Appl. (9), 104 (2015), 1135-1159. doi: 10.1016/j.matpur.2015.07.008. Google Scholar

[41]

E. Pimentel and V. Voskanyan, Regularity for second-order stationary mean-field games, Indiana Univ. Math. J., 66 (2017), 1-22. doi: 10.1512/iumj.2017.66.5944. Google Scholar

[42]

A. Porretta, On the planning problem for the mean field games system, Dyn. Games Appl., 4 (2014), 231-256. doi: 10.1007/s13235-013-0080-0. Google Scholar

[43]

A. Porretta, Weak solutions to Fokker-Planck equations and mean field games, Arch. Ration. Mech. Anal., 216 (2015), 1-62. doi: 10.1007/s00205-014-0799-9. Google Scholar

Figure 1.  The kernel G1 and the potential V.
Figure 2.  The approximate solutions $\widetilde{m}_1$ and $\widetilde{u}_1$.
Figure 3.  The error Er1.
Figure 4.  The kernel G2 and the potential V.
Figure 5.  The approximate solutions $\widetilde{m}_2$ and $\widetilde{u}_2$.
Figure 6.  The error Er2.
Figure 7.  The kernel G3 and the potential V.
Figure 8.  The approximate solutions $\widetilde{m}_3$ and $\widetilde{u}_3$.
Figure 9.  The error Er3.
[1]

Diogo Gomes, Marc Sedjro. One-dimensional, forward-forward mean-field games with congestion. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 901-914. doi: 10.3934/dcdss.2018054

[2]

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

[3]

Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115

[4]

Salah Eddine Choutri, Boualem Djehiche, Hamidou Tembine. Optimal control and zero-sum games for Markov chains of mean-field type. Mathematical Control & Related Fields, 2019, 9 (3) : 571-605. doi: 10.3934/mcrf.2019026

[5]

Stig-Olof Londen, Hana Petzeltová. Convergence of solutions of a non-local phase-field system. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 653-670. doi: 10.3934/dcdss.2011.4.653

[6]

Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299

[7]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929

[14]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385

[15]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013

[16]

Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086

[17]

Franco Flandoli, Enrico Priola, Giovanni Zanco. A mean-field model with discontinuous coefficients for neurons with spatial interaction. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3037-3067. doi: 10.3934/dcds.2019126

[18]

Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036

[19]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511

[20]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (33)
  • HTML views (77)
  • Cited by (0)

Other articles
by authors

[Back to Top]