  This paper concerns the convex optimal control problem governed by multiscale
elliptic equations with arbitrarily rough $L^\infty$ coefficients, which has
important applications in composite materials and geophysics. We use one of the
recently developed numerical homogenization techniques, the so-called Rough
Polyharmonic Splines (RPS) and its generalization (GRPS) for the efficient
resolution of the elliptic operator on the coarse scale. Those methods have
optimal convergence rate which do not rely on the regularity of the
coefficients nor the concepts of scale-separation or ergodicity. As the
iterative solution of the OCP-OPT formulation of the optimal control problem
requires solving the corresponding (state and co-state) multiscale elliptic
equations many times with different right hand sides, numerical homgogenization
approach only requires one-time pre-computation on the fine scale and the
following iterations can be done with computational cost proportional to coarse
degrees of freedom. Numerical experiments are presented to validate the
theoretical analysis.
