A tensorial-parallel Chebyshev method for a differential game theory problem

TL;DR

This paper introduces a tensorial-parallel Chebyshev interpolation method for multidimensional differential game problems, significantly reducing computational costs through parallelization and efficient evaluation, demonstrated on a pollution game example.

Contribution

It presents a novel tensorial-parallel Chebyshev approach that enhances computational efficiency for high-dimensional differential games, addressing the curse of dimensionality.

Findings

Reduced computational time compared to spline-parallelized methods

Effective handling of high-dimensional problems with Chebyshev interpolation

Demonstrated accuracy and efficiency in pollution differential game

Abstract

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.