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Proximal methods for stationary mean field games with local couplings

Briceño-Arias, L. M.; Kalise, D.; Silva, F. J.

Authors

L. M. Briceño-Arias

F. J. Silva



Abstract

© 2018 Society for Industrial and Applied Mathematics. We address the numerical approximation of mean field games with local couplings. For power-like Hamiltonians, we consider a stationary system and also a system involving density constraints modeling hard congestion effects. For finite difference discretization of the mean field game system developed in [Y. Achdou and I. Capuzzo-Dolcetta, SIAM J. Numer. Anal., 48 (2010), pp. 1136-1162], we follow a variational approach. We prove that the aforementioned schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, monotonicity assumptions on the coupling term are not needed, which allows us to recover general existence results proved by Achdou and Capuzzo-Dolcetta. Next, assuming that the coupling term is nondecreasing, the variational problem is cast as a convex optimization problem, for which we study and compare several proximal-type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter, which can eventually be zero. We assess the performance of the methods via numerical experiments.

Journal Article Type Article
Publication Date Jan 1, 2018
Journal SIAM Journal on Control and Optimization
Print ISSN 0363-0129
Electronic ISSN 1095-7138
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 56
Issue 2
Pages 801-836
APA6 Citation BriceƱo-Arias, L. M., Kalise, D., & Silva, F. J. (2018). Proximal methods for stationary mean field games with local couplings. SIAM Journal on Control and Optimization, 56(2), 801-836. https://doi.org/10.1137/16M1095615
DOI https://doi.org/10.1137/16M1095615

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