| Abstract: |
| In this talk, we introduce a numerical method to approximate first-order time-dependent mean field game (MFG) systems posed in $R^d$ with non-separable, displacement monotone Hamiltonians and terminal costs, for arbitrary finite time-horizons and (possibly singular) initial player distributions with finite second moment. The numerical method is based on an implicit Euler discretization in time, together with sampling in space, of the characteristic Hamiltonian system associated with the continuous MFG system. We establish convergence of the scheme by proving an asymptotic error bound that implies optimal rates of convergence in the $L^{\infty}(L^2)$-norm for the approximations as the number of spatial samples tends to infinity jointly with the temporal time-step vanishing. We conclude the talk with numerical experiments that illustrate the performance of the scheme for a range of time horizons. |
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