This paper concerns the design of a multidimensional Chebyshev interpolation based method for a differential game theory problem. In continuous game theory problems, it might be difficult to find analytical solutions, so numerical methods have to be applied. As the number of players grows, this may increase computational costs due to the curse of dimensionality. To handle this, several techniques may be applied and paralellization can be employed to reduce the computational time cost. Chebyshev multidimensional interpolation allows efficient multiple evaluations simultaneously along several dimensions, so this can be employed to design a tensorial method which performs many computations at the same time. This method can also be adapted to handle parallel computation and, the combination of these techniques, greatly reduces the total computational time cost. We show how this technique can be applied in a pollution differential game. Numerical results, including error behaviour and computational time cost, comparing this technique with a spline-parallelized method are also included.

翻译：本文针对微分博弈问题，设计了一种基于多维切比雪夫插值的数值方法。在连续博弈问题中，由于解析解难以获得，需采用数值方法求解。随着博弈参与方数量的增加，维度灾难可能导致计算成本急剧上升。为解决该问题，可采用多种技术手段，其中并行化能有效降低计算时间成本。切比雪夫多维插值方法支持沿多个维度同时进行高效多值计算，据此可设计张量化方法实现同步大规模运算。该方法还可适配并行计算架构，通过技术组合显著降低总体计算时间成本。本文以污染微分博弈为典型案例展示该技术的应用效果，同时给出包含误差特性与计算时间成本的数值结果，并与基于样条插值的并行化方法进行对比分析。