Rough polyharmonic splines (RPS) is a variational method which has recently been developed for linear divergence-form operators with arbitrary rough coefficients. RPS method does not rely on concepts of ergodicity or scale-separation, but on compactness properties of the solution space. In this paper, we extend RPS approach method for the optimal control problem governed by parabolic systems with rough $L^{\infty }$ coefficients. RPS method is used for the spatial discretization, while the temporal discretization is performed by the finite difference method. As the iterative solution of the optimal control problem requires solving the state and co-state equations many times with different right hand sides, RPS method only requires one-time pre-computation on the fine scale and the following iterations can be done to coarse degrees of freedom. First of all, we extend an approximation method for the multiscale optimal control problem. Secondly, we obtain the error estimates of the multiscale optimal control problem. Finally, numerical experiments are presented to validate the theoretical analysis.

粗糙多调和样条（RPS）是最近针对具有任意粗糙系数的线性发散型算子开发的一种变分方法。RPS方法不依赖于遍历性或比例分离的概念，而是依赖于解空间的紧凑性。在本文中，我们将RPS方法扩展到具有抛物线型系统的最优控制问题。${\mathrm{大号}}^{\infty }$系数。RPS方法用于空间离散化，而时间离散化则通过有限差分法执行。由于最优控制问题的迭代解决方案需要用不同的右侧多次求解状态方程和共状态方程，因此RPS方法仅需要在精细尺度上进行一次预计算，并且可以粗略地进行以下迭代自由。首先，我们针对多尺度最优控制问题扩展了一种近似方法。其次，我们获得了多尺度最优控制问题的误差估计。最后，通过数值实验验证了理论分析的正确性。