Geodesic
On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local couplings
L. Briceño-Arias1 ; D. Kalise1 ; Z. Kobeissi2 ; M. Laurière3 ; Á. Mateos González4 ; F. J. Silva5
1 Universidad Técnica Federico Santa María, Departamento de Matemática, Av. Vicuña Mackenna 3939, San Joaquín, Santiago, Chile
2 Laboratoire Jacques-Louis Lions, Univ. Paris Diderot, Sorbonne Paris Cité, UMR 7598, UPMC, CNRS, 75205, Paris, France
3 ORFE, Princeton University, Princeton, NJ 08540, USA
4 Institut Montpellliérain Alexander Grothendieck (IMAG), UMR CNRS 5149, Université de Montpellier, 34090 Montpellier, France, and Institut des Sciences de l'Évolution de Montpellier (ISEM), UMR CNRS 5554, Université de Montpellier, 34095 Montpellier, France
5 Toulouse School of Economics, Université de Toulouse I Capitole, 31015 Toulouse, France and Institut de recherche XLIMDMI, UMR-CNRS 7252 Faculté des sciences et techniques Université de Limoges, 87060 Limoges, France
ESAIM. Proceedings, Tome 65 (2019), pp. 330-348.

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We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [14] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [3]. In order to solve the finite dimensional variational problems, in [14] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [20], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [14] by suitable preconditioned iterative algorithms.
DOI : 10.1051/proc/201965330

L. Briceño-Arias 1 ; D. Kalise 1 ; Z. Kobeissi 2 ; M. Laurière 3 ; Á. Mateos González 4 ; F. J. Silva 5

1 Universidad Técnica Federico Santa María, Departamento de Matemática, Av. Vicuña Mackenna 3939, San Joaquín, Santiago, Chile
2 Laboratoire Jacques-Louis Lions, Univ. Paris Diderot, Sorbonne Paris Cité, UMR 7598, UPMC, CNRS, 75205, Paris, France
3 ORFE, Princeton University, Princeton, NJ 08540, USA
4 Institut Montpellliérain Alexander Grothendieck (IMAG), UMR CNRS 5149, Université de Montpellier, 34090 Montpellier, France, and Institut des Sciences de l'Évolution de Montpellier (ISEM), UMR CNRS 5554, Université de Montpellier, 34095 Montpellier, France
5 Toulouse School of Economics, Université de Toulouse I Capitole, 31015 Toulouse, France and Institut de recherche XLIMDMI, UMR-CNRS 7252 Faculté des sciences et techniques Université de Limoges, 87060 Limoges, France 
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     author = {L. Brice\~no-Arias and D. Kalise and Z. Kobeissi and M. Lauri\`ere and \'A. Mateos Gonz\'alez and F. J. Silva},
     title = {On the implementation of a primal-dual algorithm for second order time-dependent {Mean} {Field} {Games} with local couplings},
     journal = {ESAIM. Proceedings},
     pages = {330--348},
     publisher = {mathdoc},
     volume = {65},
     year = {2019},
     doi = {10.1051/proc/201965330},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201965330/}
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L. Briceño-Arias; D. Kalise; Z. Kobeissi; M. Laurière; Á. Mateos González; F. J. Silva. On the implementation of a primal-dual algorithm for second order time-dependent Mean Field Games with local couplings. ESAIM. Proceedings, Tome 65 (2019), pp. 330-348. doi : 10.1051/proc/201965330. http://geodesic.mathdoc.fr/articles/10.1051/proc/201965330/

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