In this contribution, we are concerned with parameter optimization problems that are constrained by multiscale PDE state equations. As an efficient numerical solution approach for such problems, we introduce and analyze a new relaxed and localized trust-region reduced basis method. Localization is obtained based on a Petrov-Galerkin localized orthogonal decomposition method and its recently introduced two-scale reduced basis approximation. We derive efficient localizable a posteriori error estimates for the optimality system, as well as for the two-scale reduced objective functional. While the relaxation of the outer trust-region optimization loop still allows for a rigorous convergence result, the resulting method converges much faster due to larger step sizes in the initial phase of the iterative algorithms. The resulting algorithm is parallelized in order to take advantage of the localization. Numerical experiments are given for a multiscale thermal block benchmark problem. The experiments demonstrate the efficiency of the approach, particularly for large scale problems, where methods based on traditional finite element approximation schemes are prohibitive or fail entirely.

翻译：本文关注受多尺度偏微分方程状态方程约束的参数优化问题。作为此类问题的高效数值求解方法，我们提出并分析了一种新的松弛且局部的信任域缩减基方法。基于Petrov-Galerkin局部正交分解方法及其最近引入的双尺度缩减基近似，实现了局部化处理。我们推导了最优性系统以及双尺度缩减目标泛函的高效可局部化后验误差估计。外环信任域优化过程的松弛仍能保证严格的收敛性结果，但由于迭代算法初始阶段步长更大，该方法收敛速度显著加快。所提算法通过并行化充分利用局部化特性。针对多尺度热块基准问题进行了数值实验，结果表明该方法具有高效性，尤其适用于传统有限元近似方案不可行或完全失效的大规模问题。