Rough polyharmonic splines method for optimal control problem governed by parabolic systems with rough coefficient

Abstract

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∞ 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.