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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 ...

49K20 ; 49N70 ; 35F21 ; 35K40 ; 35K55 ; 35Q84 ; 65K10 ; 65M06 ; 65M12 ; 91A23 ; 91A15

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- 386 p.
ISBN 978-0-8218-3662-0

Contemporary mathematics , 0383

Localisation : Collection 1er étage

analyse numérique # méthode des éléments finis # EDP # convergence des méthodes numériques # génération de maillage # erreur # méthode multigrille # mécanique des fluides # équation de Navier-Stokes

65M12 ; 65M50 ; 65N12 ; 65N15 ; 65N30 ; 65N50 ; 65N55 ; 76-02 ; 76D05

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- 323 p.
ISBN 978-0-8218-3533-3

Contemporary mathematics , 0371

Localisation : Collection 1er étage

opérateur aux dérivées partielles # EDP # théorème d'existence # comportement des solutions d'EDP # analyse numérique # EDP non-linéaire # équation elliptique # équation parabolique # équation de Navier-Stokes # loi de conservation hyperbolique # mécanique des fluides # fiabilité statistique # simulation # traitement d'image # onde de choc # limite libre # stabilité # limite singulière

35-06 ; 35-02 ; 35A05 ; 35B05 ; 65M12 ; 65N12 ; 76N10 ; 76P05 ; 76D03 ; 76B03

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- viii; 263 p.
ISBN 978-3-642-16228-2

Lecture notes in computational science and engineering , 0079

Localisation : Colloque 1er étage (BONN)

analyse numérique # méthodes sans maillage # équations différentielles #- Solutions numériques des équations différentielles # équations aux dérivées partielles # solutions numériques des EDP

65N99 ; 65M12 ; 65Y99 ; 65-06 ; 65Mxx ; 65Nxx ; 65Z05 ; 00B25

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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

65M06 ; 65M12

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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

65M06 ; 65M12

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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

65M06 ; 65M12

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- 433 p.
ISBN 978-0-8218-3177-9

Contemporary mathematics , 0323

Localisation : Collection 1er étage

matrice # transformation de Fourier # algorithme # interpolation rationnelle # décodage d'algorithme # algèbre de Lie # optimisation # résolution d'équation matricielle

68Q25 ; 65Y20 ; 65F05 ; 65F10 ; 65G50 ; 65M12 ; 15A57 ; 15A18 ; 47N70 ; 47N40

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- 120 p.
ISBN 978-90-6196-527-5

CWI tract , 0135

Localisation : Collection 1er étage

femmes et mathématiques # nombre normal # structure de groupe modulaire # fraction continue # théorie métrique des fractions continues # entropie # dynamique symbolique # transformation de Fourier # ondelettes # stabilité des méthodes numériques # convergence des méthodes numériques # convergence

11K16 ; 11F06 ; 11A55 ; 11K50 ; 28D20 ; 37B10 ; 42A38 ; 42C40 ; 65M12 ; 93B05

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Research talks

We present a Godunov type numerical scheme for a class of scalar conservation laws with nonlocal flux arising for example in traffic flow modeling. The scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme and also allows to show well-posedness of the model. In a second step, we consider the extension of the non-local traffic flow model to road networks by defining appropriate conditions at junctions. Based on the proposed numerical scheme we show some properties of the approximate solution and provide several numerical examples. We present a Godunov type numerical scheme for a class of scalar conservation laws with nonlocal flux arising for example in traffic flow modeling. The scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme and also allows to show well-posedness of the model. In a second step, we consider the extension of the non-local traffic flow model to road networks by defining appropriate conditions at junctions. Based on ...

35L65 ; 65M12 ; 90B20

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Research talks

We review Optimized Schwarz waveform relaxation methods which are space-time domain decomposition methods. The main ideas are explained on the heat equation, and extension to advection-diffusion equations are illustrated by numerical results. We present the Schwarz for TrioCFD project, which aims at using this kind of methods for the Stokes equations.

65M55 ; 65M60 ; 65M12 ; 65Y20

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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 ...

49K20 ; 49N70 ; 35F21 ; 35K40 ; 35K55 ; 35Q84 ; 65K10 ; 65M06 ; 65M12 ; 91A23 ; 91A15

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Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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 ...

49K20 ; 49N70 ; 35F21 ; 35K40 ; 35K55 ; 35Q84 ; 65K10 ; 65M06 ; 65M12 ; 91A23 ; 91A15

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Research talks

The aim of the talk is to introduce a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift-diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is rewritten in a nonlinear form before being discretized. This is a joint work with Clément Cancès (Lille) and Stella Krell (Nice). The aim of the talk is to introduce a nonlinear Discrete Duality Finite Volume scheme to approximate the solutions of drift-diffusion equations. The scheme is built to preserve at the discrete level even on severely distorted meshes the energy / energy dissipation relation. This relation is of paramount importance to capture the long-time behavior of the problem in an accurate way. To enforce it, the linear convection diffusion equation is ...

65M08 ; 65M12

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Research talks;Control Theory and Optimization;Partial Differential Equations

We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We show the strong convergence of the approximation and then discussed several examples for $N = 1$ and $N = 2$. The reconstruction of both the state and the source term is also discussed, as well as the boundary case. The parabolic case - more delicate as it requires the use of appropriate weights - will be also addressed. Joint works with Nicolae Cîndea and Diego Araujo de Souza. We explore a direct method allowing to solve numerically inverse type problems for hyperbolic type equations. We first consider the reconstruction of the full solution of the equation posed in $\Omega \times (0, T )$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technic and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic ...

35L10 ; 65M12 ; 93B40

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Research talks;Partial Differential Equations

The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard interpolatory Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at specified nodes. Due to the fact that previously proposed commutator-free Magnus integrators of order five or higher involve negative coefficients in the linear combinations, severe instabilities are observed for spatially semi-discretised partial differential equations of parabolic type or for master equations describing dissipative quantum systems, respectively. In order to remedy this issue, two different approaches for the design of efficient Magnus integrators of orders four, five, and six are pursued: (i) the study of commutator-free Magnus integrators involving complex coefficients with positive real part, and (ii) the study of unconventional Magnus integrators that comprise in addition a single exponential involving a commutator. Numerical experiments for test equations of Schrödinger and parabolic type confirm that the identified novel Magnus integrators are superior to Magnus integrators previously proposed in the literature. The class of commutator-free Magnus integrators is known to provide a favourable alternative to standard interpolatory Magnus integrators, in particular for large-scale applications arising in the time integration of non-autonomous linear evolution equations. A high-order commutator-free Magnus integrator is given by a composition of several exponentials that comprise certain linear combinations of the values of the defining operator at ...

35Q41 ; 65M12

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- 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

65M06 ; 65M08 ; 65M12 ; 65M15 ; 65M60 ; 65N06 ; 65N08 ; 65N12 ; 65N15 ; 65N30

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- xvi; 104 p.
ISBN 978-1-4614-5810-4

SpringerBriefs in mathematics

Localisation : Ouvrage RdC (MARI)

approximation linéaire # propagation non uniforme # relation de dispersion # grille uniforme # équation de vague # approximation discontinue de Galerkin # différence finie # algorithme de filtrage

65M60 ; 65-01 ; 65T50 ; 93B07 ; 93C20 ; 35L05 ; 65M12 ; 65F15 ; 70H05 ; 78A05 ; 93B05 ; 93C05 ; 93C15

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- viii; 226 p.
ISBN 978-3-03719-078-4

Series of lectures in mathematics

Localisation : Ouvrage RdC (SPLI)

équations aux dérivées partielles # équations différentielles non linéaires # lois de conservation # équation de Euler # équation de cnvection-diffusion # médias poreux # Matlab

35L65 ; 35K65 ; 65M99 ; 65M15 ; 65M12 ; 35L67 ; 47N40 ; 35A35

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- xvi; 470 p.
ISBN 978-3-540-71040-0

Springer series in computational mathematics , 0041

Localisation : Ouvrage RdC (SHEN)

théorie spectrale # méthode spectrale # équation aux dérivées partielles # convergence # analyse d'erreur # équation différentielle d'ordre supérieur # domaine illimité # codes de Matlab

65M70 ; 65M12 ; 65N15 ; 65N22 ; 65F05 ; 35J25 ; 35J40 ; 35K15 ; 42C05 ; 65N35 ; 65-02 ; 65M15 ; 65N12

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