Error Control Based Model Reduction for Parameter Optimization of Elliptic Homogenization Problems

In this work we are considered with parameter optimization of elliptic multiscale problems with macroscopic optimization functionals and microscopic material design parameters. An efficient approximation is obtained by the reduced basis approach. A posteriori error estimates for the reduced forward problem are obtained in the periodic homogenization setting, using the so called two scale weak formulation of the multiscale problem. The resulting error indicators allow for an efficient offline/online decomposition and are used for an efficient reduced basis construction, both for the homogenization limit, as well as for the approximation of the corresponding cell problems.