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# Documents  65M06 | enregistrements trouvés : 42

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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

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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## Inhomogeneities and temperature effects in Bose-Einstein condensates de Bouard, Anne | CIRM H

Post-edited

Research talks;Partial Differential Equations;Probability and Statistics

We will review in this talk some mathematical results concerning stochastic models used by physicist to describe BEC in the presence of fluctuations (that may arise from inhomogeneities in the confinement parameters), or BEC at finite temperature. The results describe the effect of those fluctuations on the structures - e.g. vortices - which are present in the deterministic model, or the convergence to equilibrium in the models at finite temperature. We will also describe the numerical methods which have been developed for those models in the framework of the ANR project Becasim. These are joint works with Reika Fukuizumi, Arnaud Debussche, and Romain Poncet. We will review in this talk some mathematical results concerning stochastic models used by physicist to describe BEC in the presence of fluctuations (that may arise from inhomogeneities in the confinement parameters), or BEC at finite temperature. The results describe the effect of those fluctuations on the structures - e.g. vortices - which are present in the deterministic model, or the convergence to equilibrium in the models at finite ...

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## Hyperbolic problems :theory, numerics, applicationsproceedings of the XIth international conference on...#July, 17-21 Benzoni-Gavage, Sylvie ; Serre, Denis | Springer-Verlag 2008

Congrès

- 1123 p.
ISBN 978-3-540-75711-5

Localisation : Colloque 1er étage (LYON)

EDP # mécanique des fluides # équations hyperboliques # lois de conservation # singularités # EDP pour la relativité # généralisation de maillage # méthode numérique

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## Hyperbolic systems of balance laws :lectures given at the C.I.M.E. summer school held in Cetraro#July 14-21 Bressan, Alberto ; Serre, Denis ; Williams, Mark ; Zumbrun, Kevin ; Marcati, Pierangelo | Springer 2007

Congrès

- 346 p.
ISBN 978-3-540-72186-4

Lecture notes in mathematics , 1911

Localisation : Collection 1er étage

EDP # lois de conservation hyperboliques # viscosité de disparition # onde # profils discrets de choc # choc visqueux # linéarisation # fonction de Evans # déterminants de Lopatinski

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## Finite-difference methods theory and applications cfdm98. vol. 3Proceedings of the 2nd international conference on ... | National Academy of Sciences of Belarus 1998

Congrès

ISBN 978-985-6499-06-0

Localisation : Colloque 1er étage (BELA)

EDP # analyse numérique # modèle mathématique # méthode de différence finie # physique mathématique # problème aux limites de type multi-dimensionnel # problème de stabilité des méthodes numériques # problème à la frontière

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## Finite-difference methods theory and applications cfdm98. vol. 2Proceedings of the 2nd international conference on ... | National Academy of Sciences of Belarus 1998

Congrès

ISBN 978-985-6499-05-3

Localisation : Colloque 1er étage (BELA)

EDP # analyse numérique # modèle mathématique # méthode de différence finie # physique mathématique # problème aux limites de type multi-dimensionnel # problème de stabilité des méthodes numériques # problème à la frontière

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## Finite-difference methods theory and applications CFDM98. Vol. 1proceedings of the 2nd international conference on ... | National Academy of Sciences of Belarus 1998

Congrès

ISBN 978-985-6499-04-6

Localisation : Colloque 1er étage (BELA)

EDP # analyse numérique # modèle mathématique # méthode de différence finie # physique mathématique # problème aux limites de type multi-dimensionnel # problème de stabilité des méthodes numériques # problème à la frontière

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## Computer algebra in scientific computing casc'99Proceedings of the second workshop on computer algebra in scientific computing, munich, may 31-june 4, 1999 Ganzha Victor G. ; Mayr, Ernst W. ; Vorozhtsov Evgenii V. | Springer-Verlag 1999

Congrès

ISBN 978-3-540-66047-7

Localisation : Disparu

algèbre commutatif # algèbre de calcul # analyse numérique # calcul symbolique # corps # informatique théorique # intelligence artificielle # méthode de différence finie # polynomes # représentation de groupes # stabilité de Lyaponov # système expert # théorie de stabilité # équations différentielles avancées

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## Diffusion redistanciation schemes, Willmore problem and red blood cells Maitre, Emmanuel | CIRM H

Multi angle

Research talks

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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

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 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

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 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

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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## Discontinuous Galerkin solver design on hybrid computers Helluy, Philippe | CIRM H

Multi angle

Research talks;Mathematical Physics

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## The gradient discretisation method Droniou, Jérôme ; Eymard, Robert ; Gallouët, Thierry ; Guichard, Cindy ; Herbin, Raphaèle | Springer;Société de Mathématiques Appliquées et Industrielles 2018

Ouvrage

- xxiv; 497 p.
ISBN 978-3-319-79041-1

Mathématiques & applications , 0082

Localisation : Collection 1er étage

méthode de discrétisation du gradient # schéma de gradients # équation aux dérivées partielles elliptiques # équation aux dérivées partielles paraboliques # analyse de convergence uniforme # théorème d'Aubin-Simon discret # convergence par compacité # estimation des erreurs

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## Numerical solution of differential equations:introduction to finite difference and finite element methods Li, Zhilin ; Qiao, Zhonghua ; Tang, Tao | Cambridge University Press 2018

Ouvrage

- ix; 293 p.
ISBN 978-1-316-61510-2

Localisation : Ouvrage RdC (LI)

analyse numérique # équation différentielle # méthode des différences finies # méthode des éléments finis # Matlab

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## Hiérarchie de modèles en optique quantique:de Maxwell-Bloch à Schrödinger non-linéaire Bidégaray-Fesquet, Brigitte | Springer;Société de Mathématiques Appliquées et Industrielles 2006

Ouvrage

ISBN 978-3-540-27238-0

Mathématiques & applications , 0049

Localisation : Collection 1er étage

optique quantique # équation de Schrödinger # équation de Maxwell # équation de Bloch # équation de taux # équation enveloppante # problème de Cauchy # schéma de Yee # schéma de Crank-Nicholson # modèle de Debye # modèle de Lorentz

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## Optimal stochastic control, stochastic target problems, and backward SDE Touzi, Nizar ; Tourin, Agnès | Springer;The Fields Institute for Research in Mathematical Sciences 2013

Ouvrage

- x; 214 p.
ISBN 978-1-4614-4285-1

Fields institute monographs , 0029

Localisation : Collection 1er étage

contrôle stochastique optimal # problème de cible stochastique # équation différentielle stochastique rétrograde # mathématique financière # méthode des différences finies # équation de Hamilton-Jacobi-Bellman # solution de viscosité # finance quantitative # programmation dynamique # couverture en quantile # modèle de Black-Scholes # illiquidité # formule de Ito

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## Partial differential equations: modeling, analysis and numerical approximation Le Dret, Hervé ; Lucquin, Brigitte | Birkhäuser 2016

Ouvrage

- xi; 395 p.
ISBN 978-3-319-27065-4

International series of numerical mathematics , 0168

Localisation : Ouvrage RdC (LEDR)

équation différentielle partielle # équation différentielle elliptique # équation différentielle parabolique # équation différentielle hyperbolique # différence finie # méthode des volumes finis

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## Exercises in computational mathematics with MATLAB Lyche, Tom ; Merrien, Jean-Louis | Springer 2014

Ouvrage

- xii; 372 p.
ISBN 978-3-662-43510-6

Problem books in mathematics

Localisation : Ouvrage RdC (LYCH)

MATLAB # traitement automatique des données # calcul numérique # analyse numérique # algèbre linéaire # théorie de la matrice S # valeur propre # vecteur propre # méthode itérative # interpolation polynômiale # courbe de Bézier # polynôme de Bernstein # méthode des moindres carrés

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## Computing qualitatively correct approximations of balance laws:exponential-fit, well-balanced and asymptotic-preserving Gosse, Laurent | Springer 2013

Ouvrage

- xix; 340 p.
ISBN 978-88-470-2891-3

SIMAI Springer series , 0002

Localisation : Ouvrage RdC (GOSS)

équations aux dérivées partielles # mathématiques de l'ingénieur # asymptotic -preserving # équation de balance # schéma de Godunov # convergence faible # solutions visqueuses # modèles cinétiques # transfert radiatif # chimiotaxie # équations de semi-conducteurs # lissage exponentiel # équations cinétiques non linéaires # évaluations de l'erreur # modèle de Boltzmann-Poisson # équation de Klein-Kramers # modèle de Burgers/Fokker-Planck # stabilité équations aux dérivées partielles # mathématiques de l'ingénieur # asymptotic -preserving # équation de balance # schéma de Godunov # convergence faible # solutions visqueuses # modèles cinétiques # transfert radiatif # chimiotaxie # équations de semi-conducteurs # lissage exponentiel # équations cinétiques non linéaires # évaluations de l'erreur # modèle de Boltzmann-Poisson # équation de Klein-Kramers # modèle de Burgers/Fokker-Planck # ...

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