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Mean field games of controls: Finite difference approximations

Université de Paris and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions, (LJLL), F-75006 Paris, France

This contribution is part of the Special Issue: Critical values in nonlinear pdes – Special Issue dedicated to Italo Capuzzo Dolcetta
Guest Editor: Fabiana Leoni

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We consider a class of mean field games in which the agents interact through both their states and controls, and we focus on situations in which a generic agent tries to adjust her speed (control) to an average speed (the average is made in a neighborhood in the state space). In such cases, the monotonicity assumptions that are frequently made in the theory of mean field games do not hold, and uniqueness cannot be expected in general. Such model lead to systems of forward-backward nonlinear nonlocal parabolic equations; the latter are supplemented with various kinds of boundary conditions, in particular Neumann-like boundary conditions stemming from reflection conditions on the underlying controled stochastic processes. The present work deals with numerical approximations of the above megntioned systems. After describing the finite difference scheme, we propose an iterative method for solving the systems of nonlinear equations that arise in the discrete setting; it combines a continuation method, Newton iterations and inner loops of a bigradient like solver. The numerical method is used for simulating two examples. We also make experiments on the behaviour of the iterative algorithm when the parameters of the model vary.
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Citation: Yves Achdou, Ziad Kobeissi. Mean field games of controls: Finite difference approximations. Mathematics in Engineering, 2021, 3(3): 1-35. doi: 10.3934/mine.2021024

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