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## Numerical methods for mean field games - Lecture 2: Monotone finite difference schemes Achdou, Yves | CIRM H

Post-edited

Research schools;Computer Science;Control Theory and Optimization;Partial Differential Equations;Numerical Analysis and Scientific Computing

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
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 ...

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## Control and optimization with PDE constraints.Based on the international workshop on control and optimization of PDEsMariatrost # october 10-14, 2011 Bredies, Kristian ; Clason, Christian ; Kunisch, Karl ; von Winckel, Gregory | Birkhäuser 2013

Congrès

- x; 215 p.
ISBN 978-3-0348-0630-5

International series of numerical mathematics , 0164

Localisation : Colloque 1er étage (MARI)

EDP # équation aux dérivées partielles # optimisation mathématique # méthode numérique pour le calcul des variations

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## Evolution equations of hyperbolic and Schrödinger type:asymptotics, estimates and nonlinearities.Based on a workshop on asymptotic properties of solutions to hyperbolic equationsLondon # March 2011 Ruzhansky, Michael ; Sugimoto, Mitsuru ; Wirth, Jens | Birkhäuser 2012

Congrès

- viii; 324 p.
ISBN 978-3-0348-0453-0

Progress in mathematics , 0301

Localisation : Collection 1er étage

équation d'évolution # équation différentielle hyperbolique # équation de Schrödinger

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## Fast solution of discretized optimization problems :workshoop held at the Weierstrass institute for applied analysis and stochastics#May 8-12 Hoffmann, Karl-Heinz ; Hoppe, Ronald H. W. ; Schulz, Volker | Birkhäuser Verlag 2001

Congrès

- 283 p.
ISBN 978-3-7643-6599-8

I.S.N.M. , 0138

Localisation : Colloque 1er étage (BERL)

équation différentielle aux dérivées partielles # EDP # problème # méthode d'approximation successive # algorithme de programmation mathématique # problème inverse # solution d'équation # discrétisation d'équation # problème à grande échelle # problème quadratique # méthode de type programmation successive quadratique

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## Modern developments in multivariate approximation :5th international conference on ...#Sept. 22-27 Haussmann, Werner ; Jetter, Kurt ; Reimer, Manfred ; Stöckler, Joachim | Birkhäuser Verlag 2003

Congrès

- 319 p.
ISBN 978-3-7643-2195-6

I.S.N.M. , 0145

Localisation : Colloque 1er étage (WITT)

approximation # approximation multivariée # EDP # intégration numérique # finance # reconstitution d'image # compression de données # spline multivariée # interpolation de Lagrange # quasi-interpolation # ensemble sphérique de points # subdivision non stationnaire # ondelette # formule de courbure # singularité des fonctions harmoniques

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## Optimal transportation and applications :lectures given at the CIME summer school#Sept. 2-8 Ambrosio, L. ; Caffarelli, Luis A. ; Brenier, Y. | Springer 2003

Congrès

- 169 p.
ISBN 978-3-540-40192-6

Lecture notes in mathematics , 1813

Localisation : Collection 1er étage

calcul de variations # équation de Monge-Ampère # trajectoire optimale # déplacement optimal # inégalité fonctionnelle # EDP # théorie de Monge-Kantorovich # transport de masse

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## Regularity estimates for nonlinear elliptic and parabolic problems.Notes of the CIME courseCetraro # june 22-27, 2009 Gianazza, Ugo ; Lewis, John | Springer;Fondazione CIME 2012

Congrès

- xi; 247 p.
ISBN 978-3-642-27144-1

Lecture notes in mathematics , 2045

Localisation : Collection 1er étage

EDP # controle optimal

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## Séminaire équations aux dérivées partielles 2009-2010 | Centre de Mathématiques Ecole polytechnique U.R.A. 169 du C.N.R.S. 2012

Congrès

- pag. mult.
ISBN 978-2-7302-1613-55

Localisation : Salle de manutention

équation aux dérivées partielles # EDP

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## Mean field type control with congestion Laurière, Mathieu | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations;Numerical Analysis and Scientific Computing

The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation. They describe respectively the evolution of the distribution of the agents' states and the evolution of the value function. Since it comes from a control problem, this PDE system differs in general from the one arising in mean field games.
Recently, this kind of model has been applied to crowd dynamics. More precisely, in this talk we will be interested in modeling congestion effects: the agents move but try to avoid very crowded regions. One way to take into account such effects is to let the cost of displacement increase in the regions where the density of agents is large. The cost may depend on the density in a non-local or in a local way. We will present one class of models for each case and study the associated PDE systems. The first one has classical solutions whereas the second one has weak solutions. Numerical results based on the Newton algorithm and the Augmented Lagrangian method will be presented.
This is joint work with Yves Achdou.
The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilto...

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## Numerical methods for mean field games - Lecture 1: Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions Achdou, Yves | CIRM H

Multi angle

Research schools;Computer Science;Control Theory and Optimization;Partial Differential Equations;Numerical Analysis and Scientific Computing

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
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 ...

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## Numerical methods for mean field games - Lecture 3: Variational MFG and related algorithms for solving the discrete system of nonlinear equations Achdou, Yves | CIRM H

Multi angle

Research schools;Computer Science;Control Theory and Optimization;Partial Differential Equations;Numerical Analysis and Scientific Computing

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
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 ...

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## On the stability of the Bossel-Daners inequality Trombetti, Cristina | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations

The Bossel-Daners is a Faber-Krahn type inequality for the first Laplacian eigenvalue with Robin boundary conditions. We prove a stability result for such inequality.

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## Analysis and control of nonlinear infinite dimensional systems Barbu, Viorel | Academic Press Inc. 1993

Ouvrage

- 477 p.
ISBN 978-0-12-078145-4

Mathematics on science and engineering , 190

Localisation : Ouvrage RdC (BARB)

contrôle optimal en temps réel # infinie # inégalité variationelle elliptique controlée # inégalité variationelle parabolique # optimisation # opérateur non linéaire # système d'opérateur # système non linéaire de dimension # théorie de contrôle # équation différentielle non linéaire

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## Constrained optimization and optimal control for partial differential equations Leugering, Günter ; Engell, Sebastian ; Griewank, Andreas ; Hinze, Michael ; Rannacher, Rolf ; Schulz, Volker ; Ulbrich, Michael ; Ulbrich, Stefan | Birkhäuser 2012

Ouvrage

- xi; 622 p.
ISBN 978-3-0348-0132-4

International series of numerical mathematics , 0160

Localisation : Ouvrage RdC (CONS)

analyse numérique # calcul de variations # EDP # méthode aux différences finies # programmation non-linéaire

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## Cours d'analyse de l'école polytechnique. Tome III :équations différentielles Jordan, C. | Gauthier-Villars 1915

Ouvrage

- 631 p.

Localisation : Ouvrage RdC (JORD)

analyse # calcul des variations # système d'équation linéaire # équation au dérivées partielles # équation différentielles

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## Domain decomposotion methods in optimal control of partial differential equations Lagnese, John E. ; Leugering, Günter | Birkhäuser Verlag 2004

Ouvrage

- 443 p.
ISBN 978-3-7643-2194-9

I.S.N.M. , 0148

Localisation : Ouvrage RdC (LEGN)

calcul de variation # control optimal # EDP # programmation quadratique # méthode de décomposition # technique d'optimisation # domaine de décomposition # problème de valeur limite # équation hyperbolique # équation elliptique # système dynamique

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## Elementary feedback stabilization of the linear reaction-convection-diffusion equation and the wave equation Liu, Weijiu | Springer;Société de Mathématiques Appliquées et Industrielles 2010

Ouvrage

- x;294 p.
ISBN 978-3-642-04612-4

Mathématiques & applications , 0066

Localisation : Collection 1er étage

équation aux dérivées partielles # système à réaction # théorie de la commande # équation d'onde

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## Extensions of Moser-Bangert theory:locally minimal solutions Rabinowitz, Paul H. ; Stredulinsky, Edward W. | Birkhäuser 2011

Ouvrage

- viii; 208 p.
ISBN 978-0-8176-8116-6

Progress in nonlinear differential equations and their applications , 0081

Localisation : Ouvrage RdC (RABI)

équation aux dérivées partielles # équation différentielles non linéaires # modèle non-linéaire # fonction renormalisée # solution hétérocline # solution homocline # équation de Allen-Cahn # méthode variationnelle

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## Methods of nonlinear analysis :applications to differential equations Dràbek, Pavel ; Milota, Jaroslav | Birkhäuser 2007

Ouvrage

- 568 p.
ISBN 978-3-7643-8146-2

Localisation : Ouvrage RdC (DRAB)

analyse non-linéaire # équation différentielle # problème de valeur aux limites # méthode variationnelle # méthode topologique # calcul différentiel # intégrale abstraite # opérateur linéaire # opérateur non-linéaire

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## Optimal control of coupled systems of partial differential equations Kunisch, Karl ; Leugering, Günter ; Sprekels, Jürgen ; Troltzsch, Fredi | Birkhäuser 2009

Ouvrage

- vi; 345 p.
ISBN 978-3-7643-8922-2

International series of numerical mathematics , 0158

Localisation : Ouvrage RdC (OPTI)

contrôle optimal # EDP # feedback

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