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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.
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Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal., 51 (2013), 2585-2612.
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Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), 1136-1162.
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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.
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Two numerical approaches to stationary mean-field games, Dyn. Games Appl., 7 (2017), 657-682.
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A fractional Fokker-Planck control framework for subdiffusion processes, Optimal Control Appl. Methods, 37 (2016), 290-304.
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F. Camilli and R. De Maio,
A time-fractional mean field game, Adv. Differential Equations, 24 (2019), 531-554.
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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.
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Mean field games, Jpn. J. Math., 2 (2007), 229-260.
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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. |
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Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.
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M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics, 43, De Gruyter, Berlin, 2019.
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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.
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[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. Google Scholar |
[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. |



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