OPTIMAL CONTROL FOR MULTISCALE ELLIPTIC EQUATIONS WITH ROUGH COEFFICIENTS

This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough L infinity coefficients, which has not only complex coupling be-tween nonseparable scales and nonlinearity, but also 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 (GRP-S) 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 periodicity. As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corre-sponding (state and co-state) multiscale elliptic equations many times with different right hand sides, numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost propor-tional to coarse degrees of freedom. Numerical experiments are presented to validate the theoretical analysis.