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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 and 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.

[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.

[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]

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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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

[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.

[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].

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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].

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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]

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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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

[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.

[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].

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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].

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.
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