A remark on Uzawa’s algorithm and an application to mean field games systems

Bertucci, Charles

doi:10.1051/m2an/2019084

Cited by 6 publications

(4 citation statements)

References 24 publications

(27 reference statements)

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“…• Mean field games related to impulse control and optimal exit time have been studied by C. Bertucci, both from a theoretical and a numerical viewpoint, see [21]. In particular for MFGs related to impulse control problems, there remains a lot of difficult open issues.…”

Section: Discussionmentioning

confidence: 99%

Mean Field Games and Applications: Numerical Aspects

Achdou

Laurière

2020

Lecture Notes in Mathematics

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The theory of mean field games aims at studying deterministic or stochastic differential games (Nash equilibria) as the number of agents tends to infinity. Since very few mean field games have explicit or semi-explicit solutions, numerical simulations play a crucial role in obtaining quantitative information from this class of models. They may lead to systems of evolutive partial differential equations coupling a backward Bellman equation and a forward Fokker-Planck equation. In the present survey, we focus on such systems. The forward-backward structure is an important feature of this system, which makes it necessary to design unusual strategies for mathematical analysis and numerical approximation. 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. Finally, we discuss in details two applications of mean field games to the study of crowd motion and to macroeconomics, a comparison with mean field type control, and present numerical simulations.

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Section: Discussionmentioning

confidence: 99%

Mean Field Games and Applications: Numerical Aspects

Achdou

Laurière

2020

Lecture Notes in Mathematics

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“…Very few papers present numerical algorithms for Mean-field games of optimal stopping and discuss their convergence. [12] prove the convergence of fictitious play for potential games and [9] studies the Uzawa algorithm for the (possibly non-potential) MFG system introduced in [8] in the stationary case, under the assumption of strict monotonicity of the reward map. This LPFP algorithm has already been used for applications to water management and electricity markets in [13] and [3], respectively, but no theoretical convergence results have been provided.…”

Section: Introductionmentioning

confidence: 99%

Linear programming fictitious play algorithm for mean field games with optimal stopping and absorption

Dumitrescu¹,

Leutscher²,

Tankov³

2023

ESAIM: M2AN

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We develop the fictitious play algorithm in the context of the linear programming approach for mean field games of optimal stopping and mean field games with regular control and absorption. This algorithm allows to approximate the mean field game population dynamics without computing the value function by solving linear programming problems associated with the distributions of the players still in the game and their stopping times/controls. We show the convergence of the algorithm using the topology of convergence in measure in the space of subprobability measures, which is needed to deal with the lack of continuity of the flows of measures. Numerical examples are provided to illustrate the convergence of the algorithm.

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“…Very few papers present numerical algorithms for Mean-field games of optimal stopping and discuss their convergence. [BDT20] prove the convergence of fictitious play for potential games and [Ber20] studies the Uzawa algorithm for the (possibly non-potential) MFG system introduced in [Ber17] in the stationary case, under the assumption of strict monotonicity of the reward map. This LPFP algorithm has already been used for applications to water management and electricity markets in [BDT22] and [ADT21], respectively, but no theoretical convergence results have been provided.…”

Section: Introductionmentioning

confidence: 99%