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# Documents  Gobet, Emmanuel | enregistrements trouvés : 32

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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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## An introduction to BSDE Imkeller, Peter | CIRM H

Multi angle

Research schools;Mathematics in Science and Technology;Probability and Statistics

Backward stochastic differential equations have been a very successful and active tool for stochastic finance and insurance for some decades. More generally they serve as a central method in applications of control theory in many areas. We introduce BSDE by looking at a simple utility optimization problem in financial stochastics. We shall derive an important class of BSDE by applying the martingale optimality principle to solve an optimal investment problem for a financial agent whose income is partly affected by market external risk. We then present the basics of existence and uniqueness theory for solutions to BSDE the coefficients of which satisfy global Lipschitz conditions. Backward stochastic differential equations have been a very successful and active tool for stochastic finance and insurance for some decades. More generally they serve as a central method in applications of control theory in many areas. We introduce BSDE by looking at a simple utility optimization problem in financial stochastics. We shall derive an important class of BSDE by applying the martingale optimality principle to solve an optimal ...

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## Bandits in auctions (& more) Perchet, Vianney | CIRM H

Multi angle

Research schools;Computer Science;Probability and Statistics

In this talk, I will introduce the classical theory of multi-armed bandits, a field at the junction of statistics, optimization, game theory and machine learning, discuss the possible applications, and highlights the new perspectives and open questions that they propose We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is irreversible and modeled in terms of timing strategies characterized through threshold rules. Because the impact of changing costs on the producers is asymmetric, we are led to a nonzero-sum timing game describing the transitions among the discrete investment stages. Working in a continuous-time diffusion framework, we characterize and analyze the resulting Nash equilibrium and game values. Our analysis quantifies the dynamic competition effects and yields insight into dynamic preemption and over-investment in a general asymmetric setting. A case-study considering the impact of fluctuating emission costs on power producers investing in nuclear and coal-fired plants is also presented. In this talk, I will introduce the classical theory of multi-armed bandits, a field at the junction of statistics, optimization, game theory and machine learning, discuss the possible applications, and highlights the new perspectives and open questions that they propose We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate ...

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## Branching for PDEs Warin, Xavier | CIRM H

Multi angle

Research schools;Partial Differential Equations;Probability and Statistics

Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities polynomial in u and Du with convergence results. At last we give some numerical schemes to solve the semi-linear case and even the full non linear case but currently without convergence results. Branching methods have recently been developed to solve some PDEs. Starting from Mckean formulation, we give the initial branching method to solve the KPP equation. We then give a formulation to solve non linear equation with a non linearity polynomial in the value function u. The methodology is extended for general non linearities in the value function u. Then we develop the methodology to solve non linear equation with non linearities ...

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## Capacity expansion games with application to competition in power generation investments Aïd, René | CIRM H

Multi angle

Research schools;Control Theory and Optimization;Mathematics in Science and Technology

We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is irreversible and modeled in terms of timing strategies characterized through threshold rules. Because the impact of changing costs on the producers is asymmetric, we are led to a nonzero-sum timing game describing the transitions among the discrete investment stages. Working in a continuous-time diffusion framework, we characterize and analyze the resulting Nash equilibrium and game values. Our analysis quantifies the dynamic competition effects and yields insight into dynamic preemption and over-investment in a general asymmetric setting. A case-study considering the impact of fluctuating emission costs on power producers investing in nuclear and coal-fired plants is also presented. We consider competitive capacity investment for a duopoly of two distinct producers. The producers are exposed to stochastically fluctuating costs and interact through aggregate supply. Capacity expansion is irreversible and modeled in terms of timing strategies characterized through threshold rules. Because the impact of changing costs on the producers is asymmetric, we are led to a nonzero-sum timing game describing the transitions among the ...

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## Cubature methods and applications Crisan, Dan | CIRM H

Multi angle

Research schools;Dynamical Systems and Ordinary Differential Equations;Probability and Statistics

The talk will have two parts: In the first part, I will go over some of the basic feature of cubature methods for approximating solutions of classical SDEs and how they can be adapted to solve Backward SDEs. In the second part, I will introduce some recent results on the use of cubature method for approximating solutions of McKean-Vlasov SDEs.

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## Dynamic formulations of optimal transportation and variational MFGs Benamou, Jean-David | CIRM H

Multi angle

Research schools;Control Theory and Optimization;Partial Differential Equations

I will review several relaxations of the classical Monge Optimal Transport problem using a dynamic “time” extension and discuss the corresponding available numerical methods. They also apply to some instances of variational mean field games.

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## Forward and backward simulation of Euler scheme Gobet, Emmanuel | CIRM H

Multi angle

Research talks;Probability and Statistics

We analyse how reverting Random Number Generator can be efficiently used to save memory in solving dynamic programming equation. For SDEs, it takes the form of forward and backward Euler scheme. Surprisingly the error induced by time reversion is of order 1.

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## Global sensitivity analysis in stochastic systems Le Maître, Olivier | CIRM H

Multi angle

Research schools;Probability and Statistics

Stochastic models are used in many scientific fields, including mechanics, physics, life sciences, queues and social-network studies, chemistry. Stochastic modeling is necessary when deterministic ones cannot capture features of the dynamics, for instance, to represent effects of unresolved small-scale fluctuations, or when systems are subjected to important inherent noise. Often, stochastic models are not completely known and involve some calibrated parameters that should be considered as uncertain. In this case, it is critical to assess the impact of the uncertain model parameters on the stochastic model predictions. This is usually achieved by performing a sensitivity analysis (SA) which characterizes changes in a model output when the uncertain parameters are varied. In the case of a stochastic model, one classically applies the SA to statistical moments of the prediction, estimating, for instance, the derivatives with respect to the uncertain parameters of the output mean and variance. In this presentation, we introduce new approaches of SA in a stochastic system based on variance decomposition methods (ANOVA, Sobol). Compared to previous methods, our SA methods are global, with respect to both the parameters and stochasticity, and decompose the variance into stochastic, parametric and mixed contributions.
We consider first the case of uncertain Stochastic Differential Equations (SDE), that is systems with external noisy forcing and uncertain parameters. A polynomial chaos (PC) analysis with stochastic expansion coefficients is proposed to approximate the SDE solution. We first use a Galerkin formalism to determine the expansion coefficients, leading to a hierarchy of SDEs. Under the mild assumption that the noise and uncertain parameters are independent, the Galerkin formalism naturally separates parametric uncertainty and stochastic forcing dependencies, enabling an orthogonal decomposition of the variance, and consequently identify contributions arising
from the uncertainty in parameters, the stochastic forcing, and a coupled term. Non-intrusive approaches are subsequently considered for application to more complex systems hardly amenable to Galerkin projection. We also discuss parallel implementations and application to derived quantity of interest, in particular, a novel sampling strategy for non-smooth quantities of interest but smooth SDE solution. Numerical examples are provided to illustrate the output of the SA and the computational complexity of the method.
Second, we consider the case of stochastic simulators governed by a set of reaction channels with stochastic dynamics. Reformulating the system dynamics in terms of independent standardized Poisson processes permits the identification of individual realizations of each reaction channel dynamic and a quantitative characterization of the inherent stochasticity sources. By judiciously exploiting the inherent stochasticity of the system, we can then compute the global sensitivities associated with individual reaction channels, as well as the importance of channel interactions. This approach is subsequently extended to account for the effects of uncertain parameters and we propose dedicated algorithms to perform the Sobols decomposition of the variance into contributions from an arbitrary subset of uncertain parameters and stochastic reaction channels. The algorithms are illustrated in simplified systems, including the birth-death, Schlgl, and Michaelis-Menten models. The sensitivity analysis output is also contrasted with a local derivative-based sensitivity analysis method.
Stochastic models are used in many scientific fields, including mechanics, physics, life sciences, queues and social-network studies, chemistry. Stochastic modeling is necessary when deterministic ones cannot capture features of the dynamics, for instance, to represent effects of unresolved small-scale fluctuations, or when systems are subjected to important inherent noise. Often, stochastic models are not completely known and involve some ...

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## Least squares regression Monte Carlo for approximating BSDES and semilinear PDES Turkedjiev, Plamen | CIRM H

Multi angle

Research schools;Control Theory and Optimization;Probability and Statistics

In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the of the continuous time solution of the BSDE. The nodes of the discrete time processes can be expressed as conditional expectations. As we shall demonstrate, the choice of discrete time process is very important, as its properties will impact the performance of the overall numerical scheme. In the second stage, the conditional expectation is approximated in functional form using least squares regression on synthetically generated data - Monte Carlo simulations drawn from a suitable probability distribution. A key feature of the regression step is that the explanatory variables are built on a user chosen finite dimensional linear space of functions, which the user specifies by setting basis functions. The choice of basis functions is made on the hypothesis that it contains the solution, so regularity and boundedness assumptions are used in its construction. The impact of the choice of the basis functions is exposed in error estimates.
In addition to the choice of discrete time approximation and the basis functions, the Markovian structure of the problem gives significant additional freedom with regards to the Monte Carlo simulations. We demonstrate how to use this additional freedom to develop generic stratified sampling approaches that are independent of the underlying transition density function. Moreover, we demonstrate how to leverage the stratification method to develop a HPC algorithm for implementation on GPUs.
Thanks to the Feynmann-Kac relation between the the solution of a BSDE and its associated semilinear PDE, the approximation of the BSDE can be directly used to approximate the solution of the PDE. Moreover, the smoothness properties of the PDE play a crucial role in the selection of the hypothesis space of regressions functions, so this relationship is vitally important for the numerical scheme.
We conclude with some draw backs of the regression approach, notably the curse of dimensionality.
In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the ...

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## Lecture 1: Introduction to HPC, random generation, and OpenMP Lelong, Jérôme | CIRM H

Multi angle

Research schools

65Y05

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## Lecture 2: Introduction to HPC - MPI: design of parallel program and MPI Lelong, Jérôme | CIRM H

Multi angle

Research schools

65Y05

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## Mean field games with major and minor players Carmona, René | CIRM H

Multi angle

Research schools;Control Theory and Optimization;Probability and Statistics

We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and cyber-security. We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and ...

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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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## Metamodels for uncertainty quantification and reliability analysis Marelli, Stefano | CIRM H

Multi angle

Research schools;Probability and Statistics

Uncertainty quantification (UQ) in the context of engineering applications aims aims at quantifying the effects of uncertainty in the input parameters of complex models on their output responses. Due to the increased availability of computational power and advanced modelling techniques, current simulation tools can provide unprecedented insight in the behaviour of complex systems. However, the associated computational costs have also increased significantly, often hindering the applicability of standard UQ techniques based on Monte-Carlo sampling. To overcome this limitation, metamodels (also referred to as surrogate models) have become a staple tool in the Engineering UQ community. This lecture will introduce a general framework for dealing with uncertainty in the presence of expensive computational models, in particular for reliability analysis (also known as rare event estimation). Reliability analysis focuses on the tail behaviour of a stochastic model response, so as to compute the probability of exceedance of a given performance measure, that would result in a critical failure of the system under study. Classical approximation-based techniques, as well as their modern metamodel-based counter-parts will be introduced. Uncertainty quantification (UQ) in the context of engineering applications aims aims at quantifying the effects of uncertainty in the input parameters of complex models on their output responses. Due to the increased availability of computational power and advanced modelling techniques, current simulation tools can provide unprecedented insight in the behaviour of complex systems. However, the associated computational costs have also increased ...

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## Model-free control and deep learning Bellemare, Marc | CIRM H

Multi angle

Research schools;Computer Science

In this talk I will present some recent developments in model-free reinforcement learning applied to large state spaces, with an emphasis on deep learning and its role in estimating action-value functions. The talk will cover a variety of model-free algorithms, including variations on Q-Learning, and some of the main techniques that make the approach practical. I will illustrate the usefulness of these methods with examples drawn from the Arcade Learning Environment, the popular set of Atari 2600 benchmark domains. In this talk I will present some recent developments in model-free reinforcement learning applied to large state spaces, with an emphasis on deep learning and its role in estimating action-value functions. The talk will cover a variety of model-free algorithms, including variations on Q-Learning, and some of the main techniques that make the approach practical. I will illustrate the usefulness of these methods with examples drawn from the Arcade ...

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## Multilevel and multi-index sampling methods with applications - Lecture 1: Adaptive strategies for Multilevel Monte Carlo Tempone, Raul | CIRM H

Multi angle

Research schools;Partial Differential Equations;Probability and Statistics

We will first recall, for a general audience, the use of Monte Carlo and Multi-level Monte Carlo methods in the context of Uncertainty Quantification. Then we will discuss the recently developed Adaptive Multilevel Monte Carlo (MLMC) Methods for (i) It Stochastic Differential Equations, (ii) Stochastic Reaction Networks modeled by Pure Jump Markov Processes and (iii) Partial Differential Equations with random inputs. In this context, the notion of adaptivity includes several aspects such as mesh refinements based on either a priori or a posteriori error estimates, the local choice of different time stepping methods and the selection of the total number of levels and the number of samples at different levels. Our Adaptive MLMC estimator uses a hierarchy of adaptively refined, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform discretization MLMC method introduced independently by M. Giles and S. Heinrich. In particular, we show that our adaptive MLMC algorithms are asymptotically accurate and have the correct complexity with an improved control of the multiplicative constant factor in the asymptotic analysis. In this context, we developed novel techniques for estimation of parameters needed in our MLMC algorithms, such as the variance of the difference between consecutive approximations. These techniques take particular care of the deepest levels, where for efficiency reasons only few realizations are available to produce essential estimates. Moreover, we show the asymptotic normality of the statistical error in the MLMC estimator, justifying in this way our error estimate that allows prescribing both the required accuracy and confidence level in the final result. We present several examples to illustrate the above results and the corresponding computational savings. We will first recall, for a general audience, the use of Monte Carlo and Multi-level Monte Carlo methods in the context of Uncertainty Quantification. Then we will discuss the recently developed Adaptive Multilevel Monte Carlo (MLMC) Methods for (i) It Stochastic Differential Equations, (ii) Stochastic Reaction Networks modeled by Pure Jump Markov Processes and (iii) Partial Differential Equations with random inputs. In this context, the notion ...

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## Multilevel and multi-index sampling methods with applications - Lecture 2: Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation Tempone, Raul | CIRM H

Multi angle

Research schools;Partial Differential Equations;Probability and Statistics

We describe and analyze the Multi-Index Monte Carlo (MIMC) and the Multi-Index Stochastic Collocation (MISC) method for computing statistics of the solution of a PDE with random data. MIMC is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Instead of using first-order differences as in MLMC, MIMC uses mixed differences to reduce the variance of the hierarchical differences dramatically. These mixed differences yield new and improved complexity results, which are natural generalizations of Giles's MLMC analysis, and which increase the domain of problem parameters for which we achieve the optimal convergence. On the same vein, MISC is a deterministic combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. Provided enough mixed regularity, MISC can achieve better complexity than MIMC. Moreover, we show that, in the optimal case, the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one-dimensional spatial problem. We propose optimization procedures to select the most effective mixed differences to include in MIMC and MISC. Such optimization is a crucial step that allows us to make MIMC and MISC computationally efficient. We show the effectiveness of MIMC and MISC in some computational tests using the mimclib open source library, including PDEs with random coefficients and Stochastic Interacting Particle Systems. Finally, we will briefly discuss the use of Markovian projection for the approximation of prices in the context of American basket options. We describe and analyze the Multi-Index Monte Carlo (MIMC) and the Multi-Index Stochastic Collocation (MISC) method for computing statistics of the solution of a PDE with random data. MIMC is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Instead of using first-order differences as in MLMC, ...

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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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## 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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