图上含个体噪声的平均场博弈交错格式的收敛性与变分结构
Convergence and variational structure of a staggered scheme for mean field games with individual noise on graphs
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中文总结 AI 辅助
本文针对有限图上含个体噪声的平均场博弈系统提出时间交错数值格式，证明其一阶收敛性，建立势MFG的变分刻画，通过可行原始-对偶牛顿法求解优化问题，数值实验验证了收敛速率及拓扑相关特性。
中文摘要 AI 辅助
我们针对有限图上含个体噪声的平均场博弈（MFG）系统，提出并分析了一种时间交错数值格式。数值求解这类耦合的正倒向系统十分棘手，因为密度在开概率单纯形上演化，且系数可能在其边界处退化。该格式保持质量守恒，且满足与Lasry-Lions单调性论证相容的离散基本恒等式，这使得数值解具有唯一性。通过建立密度的步长一致正下界及值变量的一致界，我们证明了每个内部离散解的一阶收敛性。对于势MFG，我们通过将该格式与凸离散作用量的KKT系统等同，建立了变分刻画，从而得到离散解的存在性及基于优化的实现方式。所得到的优化问题采用质量守恒坐标下的可行原始-对偶牛顿法求解。数值实验证实了预测的收敛速率，并说明了依赖拓扑的迁移及拥堵驱动的路径选择。
英文摘要
We propose and analyze a time-staggered numerical scheme for mean field game (MFG) systems with individual noise on finite graphs. Numerically solving such coupled forward--backward systems is delicate because the density evolves in the open probability simplex and the coefficients may degenerate at its boundary. The scheme preserves mass and satisfies a discrete fundamental identity compatible with the Lasry--Lions monotonicity argument, leading to uniqueness of the numerical solution. By establishing a timestep-uniform positive lower bound for the density and uniform bounds for the value variable, we prove first-order convergence for every interior discrete solution. For potential MFGs, we establish a variational characterization by identifying the scheme with the KKT system of a convex discrete action, yielding existence of the discrete solution and an optimization-based realization. The resulting optimization problem is solved by a feasible primal--dual Newton method in mass-preserving coordinates. Numerical experiments confirm the predicted convergence rate and illustrate topology-dependent transport and congestion-driven route choice.
