Discrete potential mean field games: duality and numerical resolution

Bonnans, Frédéric; Lavigne, Pierre; Pfeiffer, Laurent

doi:10.1007/s10107-023-01934-8

Cited by 4 publications

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“…1.1), i.e., the system (MFG) can be interpreted as the first order optimality condition of an optimal control problem of the FP equation, see [22,24,25]. Then some optimization algorithms can be applied to solve this optimal control problem, such as the fictitious play [12], the generalized conditional gradient algorithm [25], ADMM and Chambolle-Pock's algorithm [2,10], etc. The last important feature of our theta-scheme is that it preserves the potential structure (when it exists), which allows the application of the previously mentioned methods directly on the discrete system.…”

Section: Introductionmentioning

confidence: 99%

Error estimates of a theta-scheme for second-order mean field games

2023

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We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of a coupled system of the Fokker-Planck and the Hamilton-Jacobi-Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our theta-scheme, under a CFL condition. Taking advantage of the regularity of the solution of the continuous problem, we estimate the consistency error of the theta-scheme. Our main result is a convergence rate of order $\mathcal{O}(h^r)$ for the theta-scheme, where $h$ is the step length of the space variable and $r \in (0,1)$ is related to the H\"older continuity of the solution of the continuous problem and some of its derivatives.

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