In this note, we develop Fourier approximation methods for the solutions of first-order nonlocal mean-field games (MFG) systems. Using Fourier expansion techniques, we approximate a given MFG system by a simpler one that is equivalent to a convex optimization problem over a finite-dimensional subspace of continuous curves. Furthermore, we perform a time-discretization for this optimization problem and arrive at a finite-dimensional saddle point problem. Finally, we solve this saddle-point problem by a variant of a primal dual hybrid gradient method.

中文翻译：

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一阶非局部平均场博弈的傅里叶近似方法

在本说明中，我们为一阶非局部平均场博弈 (MFG) 系统的解开发了傅立叶近似方法。使用傅立叶展开技术，我们通过一个更简单的系统来近似给定的 MFG 系统，该系统等效于连续曲线的有限维子空间上的凸优化问题。此外，我们对这个优化问题进行了时间离散化，得到了一个有限维鞍点问题。最后，我们通过原始对偶混合梯度方法的变体解决了这个鞍点问题。