On Numerical Approximations of Fractional and Nonlocal Mean Field Games

Abstract

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.

A Hessian-Dependent Functional With Free Boundaries and Applications to Mean-Field Games

“…For further references on the topics, we mention the monographs [1,7,13,22]. Fully nonlinear mean-field games are the subject of [3,16].…”

A high-order Lagrange-Galerkin scheme for a class of Fokker-Planck equations and applications to mean field games

“…In the framework of MFGs, in [1] and [13] the authors propose a semi-implicit finite difference scheme and a Semi-Lagrangian (SL) type scheme, respectively, to approximate the solutions to FP equations. The scheme proposed in [13], which does not impose a CFL condition and hence allows for large time steps compared to space steps, has been extended in [14] to deal with nonlinear FP equations and in [16] to approximate FP equations with non-local diffusions terms.…”

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

“…But in the case of powers of the discrete Laplacian, only very simple non-degenerate constant coefficient problems were considered, and no numerical experiments were performed. This paper gives extensions of the schemes and convergence results in [18] to a very large class of fully nonlinear equations, including non-convex, strongly degenerate, and variable coefficients problems. We show that the resulting schemes are consistent, monotone, stable, and convergent.…”

“…These weights satisfy a discrete version of the Lévy integrability condition. Previously powers of discrete Laplacians have been used to discretize linear equations [25], porous medium equations [28], and very recently also certain HJB equations [18]. In [18], (optimal fractional) error bounds for numerical schemes for convex fractional equations are studied.…”

“…Previously powers of discrete Laplacians have been used to discretize linear equations [25], porous medium equations [28], and very recently also certain HJB equations [18]. In [18], (optimal fractional) error bounds for numerical schemes for convex fractional equations are studied. But in the case of powers of the discrete Laplacian, only very simple non-degenerate constant coefficient problems were considered, and no numerical experiments were performed.…”