论文标题
轻松的局部信任区域减少了优化多尺度问题的基础方法
A Relaxed Localized Trust-Region Reduced Basis Approach for Optimization of Multiscale Problems
论文作者
论文摘要
在此贡献中，我们关注的是参数优化问题，这些问题受到多尺度PDE状态方程的约束。作为此类问题的有效数值解决方案方法，我们介绍和分析了一种新的放松和局部信任区域减少的基础方法。根据Petrov-Galerkin局部正交分解方法获得定位及其最近引入的两尺度降低基础近似值。我们为最佳系统以及两尺度降低的目标函数得出有效的可定位误差估计值。尽管外部信任区域优化循环的放松仍然可以产生严格的汇合结果，但由于迭代算法的初始阶段较大的步骤，结果方法会收敛得更快。生成的算法是并行的，以利用本地化。为多尺度热块基准问题提供了数值实验。该实验证明了该方法的效率，特别是对于大规模问题，基于传统有限元近似方案的方法是过度或完全失败的。
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.
