-
Previous Article
Linear-quadratic zero-sum mean-field type games: Optimality conditions and policy optimization
- JDG Home
- This Issue
-
Next Article
Origin-to-destination network flow with path preferences and velocity controls: A mean field game-like approach
October
2021, 8(4): 381-402.
doi: 10.3934/jdg.2021013
Approximation of an optimal control problem for the time-fractional Fokker-Planck equation
| 1. | Dip. di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, via Scarpa 16, 00161 Roma, Italy |
| 2. | King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia |
| 3. | China University of Geosciences, Wuhan, China |
In this paper, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The system is composed of a backward time-fractional Hamilton-Jacobi-Bellman equation and a forward time-fractional Fokker-Planck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the well-posedness of the scheme and the convergence to the solution of the continuous problem.
Keywords:
Time-fractional Fokker-Planck equation,
Caputo derivative,
Mean Field Games,
finite-differences,
convergence.
Mathematics Subject Classification:
65N06, 91A16, 35R11.
Citation:
Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics and Games,
2021, 8
(4)
: 381-402.
doi: 10.3934/jdg.2021013
![]()
References:
| [1] |
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. |
| [2] |
Y. Achdou and I. Capuzzo-Dolcetta,
Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.
doi: 10.1137/090758477. |
| [3] |
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. |
| [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] |
M. Annunziato, A. Borzì, M. Magdziarz and A. Weron,
A fractional Fokker-Planck control framework for subdiffusion processes, Optimal Control Appl. Methods, 37 (2016), 290-304.
doi: 10.1002/oca.2168. |
| [6] |
J.-P. Bouchaud and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
| [7] |
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. |
| [8] |
F. Camilli and R. De Maio,
A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554.
|
| [9] |
F. Camilli and A. Goffi, Existence and regularity results for viscous Hamilton-Jacobi equations with Caputo time-fractional derivative, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 37pp.
doi: 10.1007/s00030-020-0624-0. |
| [10] |
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. |
| [11] |
R. Carmona and M. Lauriére, Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: Ⅱ – The finite horizon case, preprint, arXiv: 1908.01613v1. |
| [12] |
Q. Du, An invitation to nonlocal modeling, analysis and computation, Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018. Vol. Ⅳ, World Sci. Publ., Hackensack, NJ, 2018, 3541–3569.
doi: 10.1142/9789813272880_0191. |
| [13] |
Q. Du, J. Yang and Z. Zhou, Time-fractional Allen-Cahn equations: Analysis and numerical methods, J. Sci. Comput., 85 (2020), 30pp.
doi: 10.1007/s10915-020-01351-5. |
| [14] |
Y. Giga, Q. Liu and H. Mitake,
On a discrete scheme for time fractional fully nonlinear evolution equations, Asymptot. Anal., 120 (2020), 151-162.
doi: 10.3233/ASY-191583. |
| [15] |
R. Gorenflo, Y. Luchko and M. Yamamoto,
Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.
doi: 10.1515/fca-2015-0048. |
| [16] |
B. Jin, R. Lazarov, J. Pasciak and W. Rundell,
Variational formulation of problems involving fractional order differential operators, Math. Comp., 84 (2015), 2665-2700.
doi: 10.1090/mcom/2960. |
| [17] |
B. Jin, R. Lazarov and Z. Zhou,
An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221.
doi: 10.1093/imanum/dru063. |
| [18] |
B. Jin, R. Lazarov and Z. Zhou,
Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview, Comput. Methods Appl. Mech. Engrg., 346 (2019), 332-358.
doi: 10.1016/j.cma.2018.12.011. |
| [19] |
V. N. Kolokoltsov and M. A. Veretennikova, A fractional Hamilton Jacobi Bellman equation for scaled limits of controlled continuous time random walks, Commun. Appl. Ind. Math., 6 (2014), 18pp.
doi: 10.1685/journal.caim.484. |
| [20] |
J.-M. Lasry and P.-L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
| [21] |
D. Li, J. Wang and J. Zhang, Unconditionally convergent $L1$-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput., 39 (2017), A3067–A3088.
doi: 10.1137/16M1105700. |
| [22] |
Y. Lin and C. Xu,
Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.
doi: 10.1016/j.jcp.2007.02.001. |
| [23] |
M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, De Gruyter, Berlin, 2019.
doi: 10.1515/9783110560244. |
| [24] |
R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161–R208.
doi: 10.1088/0305-4470/37/31/R01. |
| [25] |
T. Namba, On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 39pp.
doi: 10.1007/s00030-018-0513-y. |
| [26] |
G. Pang, L. Lu and G. E. Karniadakis, fPINNs: Fractional physics-informed neural networks, SIAM J. Sci. Comput., 41 (2019), A2603–A2626.
doi: 10.1137/18M1229845. |
| [27] |
J. Shen and C.-T. Sheng,
An efficient space–time method for time fractional diffusion equation, J. Sci. Comput., 81 (2019), 1088-1110.
doi: 10.1007/s10915-019-01052-8. |
| [28] |
Q. Tang,
On an optimal control problem of time-fractional advection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 761-779.
doi: 10.3934/dcdsb.2019266. |
| [29] |
T. Tang, H. Yu and T. Zhou, On energy dissipation theory and numerical stability for time-fractional phase-field equations, SIAM J. Sci. Comput., 41 (2019), A3757–A3778.
doi: 10.1137/18M1203560. |
| [30] |
E. Topp and M. Yangari,
Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.
doi: 10.1016/j.jde.2017.02.024. |
| [31] |
G. M. Zaslavsky,
Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580.
doi: 10.1016/S0370-1573(02)00331-9. |
show all references
References:
| [1] |
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. |
| [2] |
Y. Achdou and I. Capuzzo-Dolcetta,
Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.
doi: 10.1137/090758477. |
| [3] |
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. |
| [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] |
M. Annunziato, A. Borzì, M. Magdziarz and A. Weron,
A fractional Fokker-Planck control framework for subdiffusion processes, Optimal Control Appl. Methods, 37 (2016), 290-304.
doi: 10.1002/oca.2168. |
| [6] |
J.-P. Bouchaud and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
| [7] |
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. |
| [8] |
F. Camilli and R. De Maio,
A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554.
|
| [9] |
F. Camilli and A. Goffi, Existence and regularity results for viscous Hamilton-Jacobi equations with Caputo time-fractional derivative, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 37pp.
doi: 10.1007/s00030-020-0624-0. |
| [10] |
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. |
| [11] |
R. Carmona and M. Lauriére, Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: Ⅱ – The finite horizon case, preprint, arXiv: 1908.01613v1. |
| [12] |
Q. Du, An invitation to nonlocal modeling, analysis and computation, Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018. Vol. Ⅳ, World Sci. Publ., Hackensack, NJ, 2018, 3541–3569.
doi: 10.1142/9789813272880_0191. |
| [13] |
Q. Du, J. Yang and Z. Zhou, Time-fractional Allen-Cahn equations: Analysis and numerical methods, J. Sci. Comput., 85 (2020), 30pp.
doi: 10.1007/s10915-020-01351-5. |
| [14] |
Y. Giga, Q. Liu and H. Mitake,
On a discrete scheme for time fractional fully nonlinear evolution equations, Asymptot. Anal., 120 (2020), 151-162.
doi: 10.3233/ASY-191583. |
| [15] |
R. Gorenflo, Y. Luchko and M. Yamamoto,
Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.
doi: 10.1515/fca-2015-0048. |
| [16] |
B. Jin, R. Lazarov, J. Pasciak and W. Rundell,
Variational formulation of problems involving fractional order differential operators, Math. Comp., 84 (2015), 2665-2700.
doi: 10.1090/mcom/2960. |
| [17] |
B. Jin, R. Lazarov and Z. Zhou,
An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221.
doi: 10.1093/imanum/dru063. |
| [18] |
B. Jin, R. Lazarov and Z. Zhou,
Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview, Comput. Methods Appl. Mech. Engrg., 346 (2019), 332-358.
doi: 10.1016/j.cma.2018.12.011. |
| [19] |
V. N. Kolokoltsov and M. A. Veretennikova, A fractional Hamilton Jacobi Bellman equation for scaled limits of controlled continuous time random walks, Commun. Appl. Ind. Math., 6 (2014), 18pp.
doi: 10.1685/journal.caim.484. |
| [20] |
J.-M. Lasry and P.-L. Lions,
Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8. |
| [21] |
D. Li, J. Wang and J. Zhang, Unconditionally convergent $L1$-Galerkin FEMs for nonlinear time-fractional Schrödinger equations, SIAM J. Sci. Comput., 39 (2017), A3067–A3088.
doi: 10.1137/16M1105700. |
| [22] |
Y. Lin and C. Xu,
Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.
doi: 10.1016/j.jcp.2007.02.001. |
| [23] |
M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, De Gruyter, Berlin, 2019.
doi: 10.1515/9783110560244. |
| [24] |
R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161–R208.
doi: 10.1088/0305-4470/37/31/R01. |
| [25] |
T. Namba, On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 39pp.
doi: 10.1007/s00030-018-0513-y. |
| [26] |
G. Pang, L. Lu and G. E. Karniadakis, fPINNs: Fractional physics-informed neural networks, SIAM J. Sci. Comput., 41 (2019), A2603–A2626.
doi: 10.1137/18M1229845. |
| [27] |
J. Shen and C.-T. Sheng,
An efficient space–time method for time fractional diffusion equation, J. Sci. Comput., 81 (2019), 1088-1110.
doi: 10.1007/s10915-019-01052-8. |
| [28] |
Q. Tang,
On an optimal control problem of time-fractional advection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 761-779.
doi: 10.3934/dcdsb.2019266. |
| [29] |
T. Tang, H. Yu and T. Zhou, On energy dissipation theory and numerical stability for time-fractional phase-field equations, SIAM J. Sci. Comput., 41 (2019), A3757–A3778.
doi: 10.1137/18M1203560. |
| [30] |
E. Topp and M. Yangari,
Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations, 262 (2017), 6018-6046.
doi: 10.1016/j.jde.2017.02.024. |
| [31] |
G. M. Zaslavsky,
Chaos, fractional kinetics, and anomalous transport, Phys. Rep., 371 (2002), 461-580.
doi: 10.1016/S0370-1573(02)00331-9. |
Figure 2.
Level sets of the mass evolution (A) for $ \alpha = 1 $ (B) for $ \alpha = 0.85 $ (C) for $ \alpha = 0.7 $
Figure 3.
Level sets of the mass evolution (A) for $ \alpha = 1 $ (B) for $ \alpha = 0.85 $ (C) for $ \alpha = 0.7 $
Figure 4.
Level sets of the mass evolution (A) for $ \alpha = 1 $ (B) for $ \alpha = 0.85 $ (C) for $ \alpha = 0.7 $
| [1] |
Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124 |
| [2] |
Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 |
| [3] |
Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 741-754. doi: 10.3934/dcdss.2020041 |
| [4] |
Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 |
| [5] |
Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016 |
| [6] |
Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485 |
| [7] |
José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 |
| [8] |
Anton Arnold, Beatrice Signorello. Optimal non-symmetric Fokker-Planck equation for the convergence to a given equilibrium. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022009 |
| [9] |
Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011 |
| [10] |
Guillaume Bal, Benjamin Palacios. Pencil-beam approximation of fractional Fokker-Planck. Kinetic and Related Models, 2021, 14 (5) : 767-817. doi: 10.3934/krm.2021024 |
| [11] |
Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056 |
| [12] |
Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008 |
| [13] |
Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028 |
| [14] |
Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028 |
| [15] |
Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure and Applied Analysis, 2021, 20 (2) : 583-621. doi: 10.3934/cpaa.2020282 |
| [16] |
Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The Numerical Solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 621-636. doi: 10.3934/naco.2021026 |
| [17] |
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 |
| [18] |
Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215 |
| [19] |
Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009 |
| [20] |
Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]