Newton Methods for Mean Field Games: A Numerical Study
This work provides incremental improvements in computational methods for Mean Field Games, a domain-specific area in applied mathematics and game theory.
The paper tackles the numerical solution of second-order Mean Field Game problems by developing new discretization techniques, including finite difference and semi-Lagrangian schemes, to implement infinite-dimensional Newton iterations, with experiments showing robustness, accuracy, and efficiency on benchmark problems.
We address the numerical solution of second-order Mean Field Game problems through Newton iterations in infinite dimensions, introduced in [14], where quadratic convergence of the method was rigorously established. Building upon this theoretical framework, we develop new numerical discretization techniques, including both a finite difference and a semi-Lagrangian scheme, that enable an effective computational implementation of the infinite-dimensional iterations. The proposed methods are tested on several benchmark problems, and the resulting numerical experiments demonstrate their robustness, accuracy, and efficiency. A comparative analysis between the two schemes and existing approaches from the literature is also presented, highlighting the potential of Newton-based solvers for MFG systems.