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In this contribution, we are concerned with parameter optimization problemsthat are constrained by multiscale PDE state equations. As an efficientnumerical solution approach for such problems, we introduce and analyze a newrelaxed and localized trust-region reduced basis method. Localization isobtained based on a Petrov-Galerkin localized orthogonal decomposition methodand its recently introduced two-scale reduced basis approximation. We deriveefficient localizable a posteriori error estimates for the optimality system,as well as for the two-scale reduced objective functional. While the relaxationof the outer trust-region optimization loop still allows for a rigorousconvergence result, the resulting method converges much faster due to largerstep sizes in the initial phase of the iterative algorithms. The resultingalgorithm is parallelized in order to take advantage of the localization.Numerical experiments are given for a multiscale thermal block benchmarkproblem. The experiments demonstrate the efficiency of the approach,particularly for large scale problems, where methods based on traditionalfinite element approximation schemes are prohibitive or fail entirely.

A Relaxed Localized Trust-Region Reduced Basis Approach for Optimization of Multiscale Problems