Existence of positive solutions for an approximation of stationary mean-field games

Almayouf, Nojood; Bachini, Elena; Chapouto, Andreia; Ferreira, Rita; Gomes, Diogo A.; Jordão, Daniela; Evangelista, David; Karagulyan, Avetik; Monasterio, Juan de; Nurbekyan, Levon; Pagliar, Giorgia; Piccirilli, Marco; Pratapsi, Sagar; Prazeres, Mariana; Junior, João dos Reis; Rodrigues, André; Romero, Orlando; Sargsyan, Maria; Seneci, Tommaso; Song, Chuliang; Terai, Kengo; Tomisaki, Ryota; Velasco-Perez, Hector; Voskanyan, Vardan; Yang, Xianjin

doi:10.2140/involve.2017.10.473

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…When ν > 0, well-posedness of system (1.1) has been studied in several articles, starting with the works by J.-M. Lasry and P.-L. Lions [51,53], followed by [45,46,33,43,7,61] in the case of smooth solutions and [33,11,39,34] in the case of weak solutions.…”

Section: Introductionmentioning

confidence: 99%

Proximal Methods for Stationary Mean Field Games with Local Couplings

Briceño-Arias¹,

Kalise²,

Silva³

2018

SIAM J. Control Optim.

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We address the numerical approximation of Mean Field Games with local couplings. For powerlike Hamiltonians, we consider both the stationary system introduced in [51,53] and also a similar system involving density constraints in order to model hard congestion effects [65,57]. For finite difference discretizations of the Mean Field Game system as in [3], 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, the monotonicity of the coupling term is not used, which allow us to recover general existence results proved in [3]. Next, assuming next that the coupling term is monotone, 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.

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Section: Introductionmentioning

confidence: 99%

Proximal Methods for Stationary Mean Field Games with Local Couplings

Briceño-Arias¹,

Kalise²,

Silva³

2018

SIAM J. Control Optim.

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“…The theory for first-order MFGs is less developed. The existence of solutions for first or second-order stationary MFGs was examined in [16] (also see [2]) using monotone operators and, using a variational approach, certain first-order MFGs with congestion were examined in [12].…”

Section: Introductionmentioning

confidence: 99%

Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions

Ferreira

Gomes

Tada

2019

Proc. Amer. Math. Soc.

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In this paper, we study first-order stationary monotone mean-field games (MFGs) with Dirichlet boundary conditions. While for Hamilton-Jacobi equations Dirichlet conditions may not be satisfied, here, we establish the existence of solutions of MFGs that satisfy those conditions. To construct these solutions, we introduce a monotone regularized problem. Applying Schaefer's fixed-point theorem and using the monotonicity of the MFG, we verify that there exists a unique weak solution to the regularized problem. Finally, we take the limit of the solutions of the regularized problem and using Minty's method, we show the existence of weak solutions to the original MFG. u ′ (x) = 0

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Existence of Weak Solutions to Stationary Mean-Field Games through Variational Inequalities

Ferreira¹,

Gomes²

2018

SIAM J. Math. Anal.

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Here, we consider stationary monotone mean-field games (MFGs) and study the existence of weak solutions. First, we introduce a regularized problem that preserves the monotonicity. Next, using variational inequality techniques, we prove the existence of solutions to the regularized problem. Then, using Minty's method, we establish the existence of solutions for the original MFG. Finally, we examine the properties of these weak solutions in several examples. Our methods provide a general framework to construct weak solutions to stationary MFGs with local, nonlocal, or congestion terms.

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Existence of positive solutions for an approximation of stationary mean-field games

Cited by 4 publications

References 21 publications

Proximal Methods for Stationary Mean Field Games with Local Couplings

Proximal Methods for Stationary Mean Field Games with Local Couplings

Existence of weak solutions to first-order stationary mean-field games with Dirichlet conditions

Existence of Weak Solutions to Stationary Mean-Field Games through Variational Inequalities

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