This thesis presents recent advances in model order reduction methods with the primary aim to construct online-efficient reduced surrogate models for parameterized multiscale phenomena and accelerate large-scale PDE-constrained parameter optimization methods. In particular, we present several different adaptive RB approaches that can be used in an error-aware trust-region framework for progressive construction of a surrogate model used during a certified outer optimization loop. In addition, we elaborate on several different enhancements for the trust-region reduced basis (TR-RB) algorithm and generalize it for parameter constraints. Thanks to the a posteriori error estimation of the reduced model, the resulting algorithm can be considered certified with respect to the high-fidelity model. Moreover, we use the first-optimize-then-discretize approach in order to take maximum advantage of the underlying optimality system of the problem. In the first part of this thesis, the theory is based on global RB techniques that use an accurate FEM discretization as the high-fidelity model. In the second part, we focus on localized model order reduction methods and develop a novel online efficient reduced model for the localized orthogonal decomposition (LOD) multiscale method. The reduced model is internally based on a two-scale formulation of the LOD and, in particular, is independent of the coarse and fine discretization of the LOD. The last part of this thesis is devoted to combining both results on TR-RB methods and localized RB approaches for the LOD. To this end, we present an algorithm that uses adaptive localized reduced basis methods in the framework of a trust-region localized reduced basis (TR-LRB) algorithm. The basic ideas from the TR-RB are followed, but FEM evaluations of the involved systems are entirely avoided.
翻译:本文介绍了在模型排序减少方法方面的最新进展, 其主要目的是为参数化多尺度现象建立在线高效、 降低的替代模型, 并加快大规模 PDE 限制的参数优化方法。 特别是, 我们介绍了几种不同的适应性 RB 方法, 可用于错误意识信任区域框架, 以逐步构建在认证外部优化循环中使用的替代模型。 此外, 我们详细介绍了对信任区域降低基础( TR- RB) 进行的若干不同改进, 并概括了参数限制。 由于对减少的模型进行事后误差估算, 由此产生的算法可以被视为与高纤维化模型有关的认证。 此外, 我们采用了多种适应性RB- dislor 方法, 使用第一个优化- 最佳信任区域框架( TR- Restric) 逐步构建替代模型, 将当前成本- TR- 降低成本- 成本- 成本- 成本- 成本- 成本- 成本- 成本- 成本- 成本- 成本- 成本- 系统使用精确- 系统, 以精确化- 常规- 常规- 常规- 常规- 常规- 运行- 常规- 常规- 常规- 和内部- 常规- 常规- 常规- 和内部- 常规- 常规- 常规- 和内部- 常规- 常规- 常规- 常规- 常规- 常规- 常规- 常规- 常规- 常规- 和内部- 常规- 和内部- 常规- 常规- 常规- 常规- 常规- 常规- 常规- 和内部- 常规- 常规- 常规- 常规- 和内部- 常规- 常规- 常规- 常规- 常规- 常规- 常规- 常规- 方法- 常规- 常规- 常规- 格式- 方法- 常规- 方法- 常规- 常规-