Optimal Control for Multiscale Elliptic Equations with Rough Coefficients (Journal of Computational Mathematics, 2023)
This paper concerns the convex optimal control problem governed by multiscale ellipticequations with arbitrarily rough $L^∞$ coefficients, which has not only complex coupling between nonseparable scales and nonlinearity, but also important applications in compositematerials and geophysics. We use one of the recently developed numerical homogenizationtechniques, 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 haveoptimal 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 coupledOCP-OPT formulation for the optimal control problem requires solving the corresponding (state and co-state) multiscale elliptic equations many times with different right handsides, numerical homogenization approach only requires one-time pre-computation on thefine scale and the following iterations can be done with computational cost proportional tocoarse degrees of freedom. Numerical experiments are presented to validate the theoreticalanalysis.
Keywords: Optimal control; Rough coefficients; Multiscale elliptic equations; Numerical homogenization; Rough polyharmonic splines; Iterative algorithm.