Iterative strategies for solving linearized discrete mean field games systems
Iterative strategies for solving linearized discrete mean field games systems
复制标题
求解线性化离散平均场博弈系统的迭代策略
DOI：
10.3934/nhm.2012.7.197
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
V. Pérez
中科院分区：
文献类型：
--
作者：
Y. Achdou;V. Pérez
平均场博弈(MFG)描述了随机微分博弈的渐近行为
玩家的数量趋向于$+\infty$。在适当的假设下，
它们导致了一类新的两个偏微分方程组：前向Bellman方程和后向Fokker-Planck方程耦合。
在以前的文章中，已经提出并研究了保持系统结构的有限差分格式。
它们导致了有限维上的大型非线性方程组。
数值求解后者的一种可能方法是使用不精确的牛顿方法：牛顿步长由求解一个线性化的离散MFG系统组成。
由于MFG系统的前进性和后进性，不可能使用时间推进法。在目前的工作中，我们提出了三类迭代策略
对于线性化离散MFG系统的求解，
其中大多数涉及到合适的多重网格解算器或预条件。
Mean fields games (MFG) describe the asymptotic behavior of stochastic differential games in which the
number of players tends to $+\infty$. Under suitable assumptions,
they lead to a new kind of system of two partial differential equations: a forward Bellman equation coupled with a backward Fokker-Planck equation.
In earlier articles, finite difference schemes preserving the structure of the system have been proposed and studied.
They lead to large systems of nonlinear equations in finite dimension.
A possible way of numerically solving the latter is to use inexact Newton methods: a Newton step consists of solving a linearized discrete MFG system.
The forward-backward character of the MFG system makes it impossible to use time marching methods. In the present work, we propose three families of iterative strategies
for solving the linearized discrete MFG systems,
most of which involve suitable multigrid solvers or preconditioners.