This paper proposes a multiscale method for solving the numerical solution of mean field games which accelerates the convergence and addresses the problem of determining the initial guess. Starting from an approximate solution at the coarsest level, the method constructs approximations on successively finer grids via alternating sweeping, which not only allows for the use of classical time marching numerical schemes but also enables applications to both local and nonlocal problems. At each level, numerical relaxation is used to stabilize the iterative process. A second-order discretization scheme is derived for higher-order convergence. Numerical examples are provided to demonstrate the efficiency of the proposed method in both local and nonlocal, 1-dimensional and 2-dimensional cases.
翻译:本文提出一种多尺度的方法,用以解决平均场游戏的数字解决方案,加速趋同,并解决确定初步猜想的问题。从粗粗层次的近似解决方案开始,该方法通过交替扫描构建对连续细网格的近似值,不仅允许使用经典时间推移数字方法,而且还允许对本地和非本地问题进行应用。在每个级别,都使用数字放松来稳定迭接进程。为较高级趋同制定了二级分解方案。提供了数字示例,以证明拟议方法在地方和非本地、一维和二维案例中的效率。