# On Numerical approximations of fractional and nonlocal Mean Field Games

@article{Chowdhury2021OnNA, title={On Numerical approximations of fractional and nonlocal Mean Field Games}, author={Indranil Chowdhury and Olav Ersland and Espen Robstad Jakobsen}, journal={ArXiv}, year={2021}, volume={abs/2105.00073} }

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full… Expand

#### References

SHOWING 1-10 OF 56 REFERENCES

A Fully Discrete Semi-Lagrangian Scheme for a First Order Mean Field Game Problem

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2014

It is proved that the resulting discretization admits at least one solution and, in the scalar case, a convergence result for the scheme is proved. Expand

A Semi-Lagrangian scheme for a degenerate second order Mean Field Game system

- Mathematics
- 2014

In this paper we study a fully discrete Semi-Lagrangian approximation of a second order Mean Field Game system, which can be degenerate. We prove that the resulting scheme is well posed and, if the… Expand

Convergence of approximation schemes for fully nonlinear second order equations

- Mathematics
- 29th IEEE Conference on Decision and Control
- 1990

The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is… Expand

On fractional and nonlocal parabolic mean field games in the whole space

- Journal of Differential Equations
- 2021

A Finite Element Like Scheme for Integro-Partial Differential Hamilton-Jacobi-Bellman Equations

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2009

A finite element like scheme for fully nonlinear integro-partial differential equations arising in optimal control of jump-processes is constructed and it is proved that they converge in very general situations, including degenerate equations, multiple dimensions, relatively low regularity of the data, and for most (if not all) types ofJump-models used in finance. Expand

On fully nonlinear parabolic mean field games with examples of nonlocal and local diffusions

- Mathematics
- 2021

We prove existence and uniqueness of solutions of a class of abstract fully nonlinear mean field game systems. We justify that such problems are related to controlled local or nonlocal diffusions, or… Expand

A machine learning framework for solving high-dimensional mean field game and mean field control problems

- Computer Science, Medicine
- Proceedings of the National Academy of Sciences
- 2020

This paper provides a flexible machine learning framework for the numerical solution of potential MFG and MFC models by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. Expand

Mean Field Games and Applications: Numerical Aspects

- Computer Science, Mathematics
- Lecture Notes in Mathematics
- 2020

In this survey, several aspects of a finite difference method used to approximate the previously mentioned system of PDEs are discussed, including convergence, variational aspects and algorithms for solving the resulting systems of nonlinear equations. Expand

Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II - The Finite Horizon Case

- Mathematics, Computer Science
- ArXiv
- 2019

Two algorithms are proposed for the solution of the optimal control of ergodic McKean-Vlasov dynamics based on the approximation of the theoretical solutions by neural networks, which allows the use of modern machine learning tools, and efficient implementations of stochastic gradient descent. Expand

Linear-Quadratic Mean-Field Reinforcement Learning: Convergence of Policy Gradient Methods

- Computer Science, Mathematics
- ArXiv
- 2019

This work proves rigorously the convergence of exact and model-free policy gradient methods in a mean-field linear-quadratic setting and provides graphical evidence of the convergence based on implementations of these algorithms. Expand