- Connaître la Bibliothèque
- Trouver un livre
- Trouver une revue
- Faire une recherche bibliographique
- Trouver un document électronique
0
Multi angle
Numerical methods for mean field games - Lecture 3: Variational MFG and related algorithms for solving the discrete system of nonlinear equations
Auteurs : Achdou, Yves (Auteur de la Conférence)
CIRM (Editeur )
CIRM (Editeur )
- [Multi angle] Forward and backward simulation of Euler scheme / Auteur de la Conférence Gobet, Emmanuel.
- [Multi angle] Stochastic variational inequalities for random mechanics / Auteur de la Conférence Mertz, Laurent.
- [Multi angle] Mean field type control with congestion / Auteur de la Conférence Laurière, Mathieu.
- [Multi angle] On the discretization of some nonlinear Fokker-Planck-Kolmogorov equations and applications / Auteur de la Conférence Silva Álvarez, Francisco José.
- [Multi angle] Project evaluation under uncertainty / Auteur de la Conférence Zubelli, Jorge P..
- [Multi angle] Some asymptotic results about American options and volativity / Auteur de la Conférence De Marco, Stefano.
- [Multi angle] Projected particle methods for solving McKean Vlasov SDEs / Auteur de la Conférence Belomestny, Denis.
- [Multi angle] Splitting algorithm for nested events / Auteur de la Conférence Goudenège, Ludovic.
- [Multi angle] Lecture 2: Introduction to HPC - MPI: design of parallel program and MPI / Auteur de la Conférence Lelong, Jérôme.
- [Multi angle] Lecture 1: Introduction to HPC, random generation, and OpenMP / Auteur de la Conférence Lelong, Jérôme.
- [ Post-edited] Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes / Auteur de la Conférence Achdou, Yves.
- [Multi angle] Numerical methods for mean field games - Lecture 1: Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions / Auteur de la Conférence Achdou, Yves.
- [Multi angle] Metamodels for uncertainty quantification and reliability analysis / Auteur de la Conférence Marelli, Stefano.
- [Multi angle] Global sensitivity analysis in stochastic systems / Auteur de la Conférence Le Maître, Olivier.
- [Multi angle] Subsurface flow with uncertainty : applications and numerical analysis issues / Auteur de la Conférence Charrier, Julia.
- [Multi angle] Dynamic formulations of optimal transportation and variational MFGs / Auteur de la Conférence Benamou, Jean-David.
- [Multi angle] Least squares regression Monte Carlo for approximating BSDES and semilinear PDES / Auteur de la Conférence Turkedjiev, Plamen.
- [Multi angle] Bandits in auctions (& more) / Auteur de la Conférence Perchet, Vianney.
- [Multi angle] Multilevel and multi-index sampling methods with applications - Lecture 2: Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation / Auteur de la Conférence Tempone, Raul.
- [Multi angle] Multilevel and multi-index sampling methods with applications - Lecture 1: Adaptive strategies for Multilevel Monte Carlo / Auteur de la Conférence Tempone, Raul.
- [Multi angle] Model-free control and deep learning / Auteur de la Conférence Bellemare, Marc.
- [Multi angle] Optimal vector quantization: from signal processing to clustering and numerical probability / Auteur de la Conférence Pagès, Gilles.
- [Multi angle] The Metropolis Hastings algorithm: introduction and optimal scaling of the transient phase / Auteur de la Conférence Jourdain, Benjamin.
- [Multi angle] Branching for PDEs / Auteur de la Conférence Warin, Xavier.
- [Multi angle] On the interplay between kinetic theory and game theory / Auteur de la Conférence Degond, Pierre.
- [Multi angle] Particle algorithm for McKean SDE: a short review on numerical analysis / Auteur de la Conférence Bossy, Mireille.
- [Multi angle] Cubature methods and applications / Auteur de la Conférence Crisan, Dan.
- [Multi angle] Capacity expansion games with application to competition in power generation investments / Auteur de la Conférence Aïd, René.
- [Multi angle] An introduction to BSDE / Auteur de la Conférence Imkeller, Peter.
- [Multi angle] Mean field games with major and minor players / Auteur de la Conférence Carmona, René.
- Achdou, Y., & Porretta, A. (2016). Convergence of a finite difference scheme to weak solutions of the system of partial differential equation arising in mean field games. SIAM Journal on Numerical Analysis, 54(1), 161-186 - http://dx.doi.org/10.1137/15M1015455
- Achdou, Y., & Capuzzo-Dolcetta, I. (2010). Mean field games: numerical methods. SIAM Journal on Numerical Analysis, 48(3), 1136-1162 - http://dx.doi.org/10.1137/090758477
- Benamou, J.-D., & Carlier, G. (2015). Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations. Journal of Optimization Theory and Applications, 167(1), 1-26 - http://dx.doi.org/10.1007/s10957-015-0725-9
- Cardaliaguet, P., Graber, P.J., Porretta, A., & Tonon, D. (2015). Second order mean field games with degenerate diffusion and local coupling. NoDEA. Nonlinear Differential Equations and Applications, 22(5), 1287-1317 - http://dx.doi.org/10.1007/s00030-015-0323-4
- Cardaliaguet, P., Delarue, F., Lasry, J.-M., & Lions, P.-L. (2015). The master equation and the convergence problem in mean field games.
- Cardaliaguet, P. (2013). Notes on mean field games. Preprint - https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf
- Lasry, J.-M., & Lions, P.-L. (2007). Mean field games. Japanese Journal of Mathematics, 2(1), 229-260 - http://dx.doi.org/10.1007/s11537-007-0657-8
- Porretta, A. (2015). Weak solutions to Fokker-Planck equations and mean field games. Archive for Rational Mechanics and Analysis, 216(1), 1-62 - http://dx.doi.org/10.1007/s00205-014-0799-9
Loading the player...
Résumé :
Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:
- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments
- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit
- convergence to classical or weak solutions can be proved
Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows
1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.
2. Monotone finite difference schemes
3. Examples of applications
4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations
Codes MSC : In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:
- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments
- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit
- convergence to classical or weak solutions can be proved
Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows
1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.
2. Monotone finite difference schemes
3. Examples of applications
4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations
35K40 - Second-order parabolic systems
35K55 - Nonlinear parabolic equations
49K20 - Optimal control problems with PDE (optimality conditions)
65K10 - Optimization and variational techniques
65M06 - Finite difference methods (IVP of PDE)
65M12 - Stability and convergence of numerical methods (IVP of PDE)
91A15 - Stochastic games
91A23 - Differential games
35F21 35Q84 - Fokker-Planck equations
49N70 - Differential games in calculus of variations
Ressources complémentaires :
http://smai.emath.fr/cemracs/cemracs17/Slides/achdou.pdf
|
Informations sur la Vidéo
Langue : AnglaisDate de publication : 26/07/17 Date de captation : 18/07/17 Collection : Research schools ; Computer Science ; Control Theory and Optimization ; Partial Differential Equations Format : MP4 Durée : 01:11:56 Domaine : Numerical Analysis & Scientific Computing ; Computer Science ; Control Theory & Optimization ; PDE Audience : Chercheurs ; Etudiants Science Cycle 2 ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2017-07-18_Achdou_part3.mp4 
|
Informations sur la rencontre
Nom de la rencontre : CEMRACS - Summer school: Numerical methods for stochastic models: control, uncertainty quantification, mean-field / CEMRACS - École d'été : Méthodes numériques pour équations stochastiques : contrôle, incertitude, champ moyenOrganisateurs de la rencontre : Bouchard, Bruno ; Chassagneux, Jean-François ; Delarue, François ; Gobet, Emmanuel ; Lelong, Jérôme Dates : 17/07/17 - 25/08/17 Année de la rencontre : 2017 URL Congrès : http://conferences.cirm-math.fr/1556.html Citation Data
DOI : 10.24350/CIRM.V.19202203Cite this video as: Achdou, Yves (2017). Numerical methods for mean field games - Lecture 3: Variational MFG and related algorithms for solving the discrete system of nonlinear equations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19202203 URI : http://dx.doi.org/10.24350/CIRM.V.19202203 |
Voir aussi
Bibliographie
Z