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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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Research talks;Analysis and its Applications;Computer Science

The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, Poissonian, and multiplicative, and with several types of regularizers (TV, frame-based analysis, synthesis, or combinations thereof). I will then describe more recent work on using ADMM for other problems, namely blind deconvolution and image segmentation, as well as very recent work where ADMM is used with plug-in learned denoisers to achieve state-of-the-art results in class-specific image deconvolution. Finally, on the theoretical front, I will describe very recent work on tackling the infamous problem of how to adjust the penalty parameter of ADMM. The alternating direction method of multipliers (ADMM) is an optimization tool of choice for several imaging inverse problems, namely due its flexibility, modularity, and efficiency. In this talk, I will begin by reviewing our earlier work on using ADMM to deal with classical problems such as deconvolution, inpainting, compressive imaging, and how we have exploited its flexibility to deal with different noise models, including Gaussian, ...

65J22 ; 65K10 ; 65T60 ; 94A08

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- 394 p.
ISBN 978-3-7643-7983-4

Trends in mathematics

Localisation : Colloque 1er étage (XIAN)

analyse numérique # informatique # calcul formel # calcul algébrique # système polynômial # ensemble triangulaire # base de Gröbner # algorithme # optimisation

65-06 ; 68-06 ; 68W30 ; 65K10

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ISBN 978-3-7643-2887-0

I.S.N.M. , 0111

Localisation : Colloque 1er étage (FREI)

analyse et synthèse des systèmes non linéaires # calcul des variations # commande optimale # condition d'optimalité et algorithmes # controlabilité # méthode d'approximation successive # méthode de Rauleigh-Ritz et Galerkin # méthode numérique # observabilité # problème de minima # structure des systèmes # système mécaniques spatiaux

49Mxx ; 65K10 ; 65Kxx ; 65M60 ; 65N30

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ISBN 978-3-7643-0732-5

I.S.N.M. , 0023

Localisation : Colloque 1er étage (OBER)

champ de gravitation # controle de conduction de chaleur # cristallographie # erreur d'arrondi # fonction groupe-booléenne # minimalisation # méthode de métrique variable # méthode de pénalisation de courant # méthode des gradients conjugués # méthode nomographique # méthode numérique # optimisation dynamique # optimisation linéaire # planification de production # problème d'optimisation non linéaire

65-06 ; 65K10

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ISBN 978-3-7643-0668-7

I.S.N.M. , 0017

Localisation : Colloque 1er étage (OBER)

approximation non linéaire # conduite optimale # controle non linéaire # discrétisation d'optimisation convexe # dualité # formule de quadrature de type Gauss # méthode d'intégration cartésienne # méthode de décomposition de Bender # méthode numérique # polynome défini optimal # problème d'optimisation # programmation linéaire stochastique # slalom # système d'inéquation linéaire # équation aux dérivées partielles

65-06 ; 65K10

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- viii; 269 p.
ISBN 978-3-0348-0290-1

Advanced courses in mathematics - CRM Barcelona

Localisation : Colloque 1er étage (BARC)

optimisation mathématique # économie # programmation mathématique # problème de traffic

90-XX ; 91-XX ; 12Y05 ; 14P10 ; 65K10 ; 90B20 ; 90C22 ; 90C26 ; 90C30 ; 90C33 ; 90C40 ; 91A10 ; 91-06 ; 91B50 ; 90C90 ; 00B25

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- 355 p.
ISBN 978-0-444-86821-3

Localisation : Colloque 1er étage (RIO)

65-06 ; 65K10 ; 65Kxx ; 68Qxx ; 68Rxx

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- 271 p.
ISBN 978-0-8218-0471-1

DIMACS series in discrete mathematics and theoretical computer science , 0023

Localisation : Collection 1er étage

analyse numérique # biochimie # contrôle # mathématique discrète # modèle pliant # méthode d'optimisation globale # pliure # programmation # programmation convexe # programmation linéaire # programmation mathématique # programmation non linéaire # recherche opérationnel # structure protéine # système stochastique # technique variationnelle

65K10 ; 90C05 ; 90C27 ; 90C30 ; 92E10

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ISBN 978-3-540-51239-4

Lecture notes in control and information sciences , 0125

Localisation : Colloque 1er étage (CLER)

analyse de forme et optimisation # contrôlabilité pour de grands temps # contrôle des frontières # contrôle optimal d'inégalités hémivariationnelles # dissipation d'énergie interne # domaine perforé # géophysique du pétrole # homogénéisation itérée # modèle de mouillage distribué # méthode de Lagrange # méthode de Newton # petit ruban dans les flots laminaires # plaque thermoélastique # poly cristal # problème d'optimisation de domaine # propriété spectrale et asymptotique # science de l'information # stabilisation # système en chaîne vibrant # système élastique linéaire # équation d'onde # équation de la chaleur analyse de forme et optimisation # contrôlabilité pour de grands temps # contrôle des frontières # contrôle optimal d'inégalités hémivariationnelles # dissipation d'énergie interne # domaine perforé # géophysique du pétrole # homogénéisation itérée # modèle de mouillage distribué # méthode de Lagrange # méthode de Newton # petit ruban dans les flots laminaires # plaque thermoélastique # poly cristal # problème d'optimisation de domaine # ...

35K05 ; 65K10 ; 73Cxx ; 93-06 ; 93Bxx

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- 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

49-XX ; 35-XX ; 35J25 ; 35K20 ; 35R60 ; 49J20 ; 49K20 ; 49M05 ; 49M15 ; 49M25 ; 65K10 ; 65M60 ; 65N15 ; 65N30

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- 289 p.
ISBN 978-0-8218-2167-1

C.M.S. conference proceedings , 0027

Localisation : Collection 1er étage

théorie des nombres # analyse numérique # calcul de variation # optimisation # analyse non-linéaire # analyse constructive # fonction de Lyapunov # algorithme # calcul formel # calcul numérique # théorie du contrôle

11Y16 ; 11J70 ; 11Y65 ; 35J20 ; 41A65 ; 44A12 ; 46B20 ; 47A15 ; 49J52 ; 49K24 ; 58C20 ; 65K10 ; 90C25 ; 90C48

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- 175 p.
ISBN 978-0-8218-2710-9

DIMACS series in discrete mathematics and theoretical computer science , 0057

Localisation : Collection 1er étage

intelligence artificielle # analyse numérique # opitmisation # programmation par contrainte # résolution de problème # recherche opérationnelle # planification # théorie déterministe # théorie stochastique # optimisation combinatoriale

65K10 ; 68T20

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- 762 p.
ISBN 978-0-8218-1131-3

Lectures in applied mathematics , 0026

Localisation : Collection 1er étage

EDO # EDP non linéaire # algorithme PL # algorithme d'homotopie simplitielle # automatisation de génération de code adjoint # collision # continuation polynomial # contrôle stochastique singulier # méthode de type Newton # optimisation sans contrainte # problème de modèle non linéaire # résonance # scission de séparatrice et chaos # solution informatique d'équation de système non linéaire # spectrocopie diélectrique # système non linéaire d'équation # théorème de Sard # transport contaminant # équation d'onde non linéaire EDO # EDP non linéaire # algorithme PL # algorithme d'homotopie simplitielle # automatisation de génération de code adjoint # collision # continuation polynomial # contrôle stochastique singulier # méthode de type Newton # optimisation sans contrainte # problème de modèle non linéaire # résonance # scission de séparatrice et chaos # solution informatique d'équation de système non linéaire # spectrocopie diélectrique # système non linéaire ...

65H10 ; 65K05 ; 65K10 ; 65N10 ; 65N20

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ISBN 978-0-444-86203-7

Mathematical programming study , 0016

Localisation : Colloque 1er étage (MONT)

algorithme # algorithme convergent superlinéairement # amélioration de faisabilité # calcul de la direction recherche # contrainte non linéaire éparse # fonction non linéaire lisse # forcer la convergence # implémentation d'algorithme de Lagrange projeté # minimisation contrainte # méthode du gradient conjugué # méthode quasi-Newton réduite # optimisation contrainte non linéairement # programmation non linéaire # technique de chien de garde

49A36 ; 65K10

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Research schools

Can modern signal processing be used to overcome the diffraction limit? The classical diffraction limit states that the resolution of a linear imaging system is fundamentally limited by one half of the wavelength of light. This implies that conventional light microscopes cannot distinguish two objects placed within a distance closer than 0.5 × 400 = 200nm (blue) or 0.5 × 700 = 350nm (red). This significantly impedes biomedical discovery by restricting our ability to observe biological structure and processes smaller than 100nm. Recent progress in sparsity-driven signal processing has created a powerful paradigm for increasing both the resolution and overall quality of imaging by promoting model-based image acquisition and reconstruction. This has led to multiple influential results demonstrating super-resolution in practical imaging systems. To date, however, the vast majority of work in signal processing has neglected the fundamental nonlinearity of the object-light interaction and its potential to lead to resolution enhancement. As a result, modern theory heavily focuses on linear measurement models that are truly effective only when object-light interactions are weak. Without a solid signal processing foundation for understanding such nonlinear interactions, we undervalue their impact on information transfer in the image formation. This ultimately limits our capability to image a large class of objects, such as biological tissue, that generally are in large-volumes and interact strongly and nonlinearly with light.
The goal of this talk is to present the recent progress in model-based imaging under multiple scattering. We will discuss several key applications including optical diffraction tomography, Fourier Ptychography, and large-scale Holographic microscopy. We will show that all these application can benefit from models, such as the Rytov approximation and beam propagation method, that take light scattering into account. We will discuss the integration of such models into the state-of-the-art optimization algorithms such as FISTA and ADMM. Finally, we will describe the most recent work that uses learned-priors for improving the quality of image reconstruction under multiple scattering.
Can modern signal processing be used to overcome the diffraction limit? The classical diffraction limit states that the resolution of a linear imaging system is fundamentally limited by one half of the wavelength of light. This implies that conventional light microscopes cannot distinguish two objects placed within a distance closer than 0.5 × 400 = 200nm (blue) or 0.5 × 700 = 350nm (red). This significantly impedes biomedical discovery by ...

94A12 ; 94A08 ; 65T50 ; 65N21 ; 65K10 ; 62H35

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Research talks;Control Theory and Optimization;Partial Differential Equations;Mathematics in Science and Technology

Following the seminal work by Benamou and Brenier on the time continuous formulation of the optimal transport problem, we show how optimal transport techniques can be used in various areas, ranging from "the reconstruction problem" cosmology to a problem of volatility calibration in finance.

65K10 ; 85A30 ; 85A40 ; 35Q35

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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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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;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...

35K40 ; 35K55 ; 35K65 ; 35D30 ; 49N70 ; 49K20 ; 65K10 ; 65M06

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