Adaptive Reduced Basis Methods for Multiscale Problems and Large-scale PDE-constrained Optimization

Doctoral examination procedure finished at: Doctoral examination procedure at University of Münster

Start date of doctoral examination procedure: 01/03/2018

End date of doctoral examination procedure: 22/06/2022

Name of the doctoral candidate: Keil, Tim

Doctoral subject: Mathematik

Doctoral degree: Dr. rer. nat.

Awarded by: Department 10 - Mathematics and Computer Science

List of all reviewers: Ohlberger, Mario; Volkwein, Stefan

Model order reduction is an enormously growing field that is particularly suitable for numerical simulations in real-life applications such as engineering and various natural science disciplines. Here, partial differential equations are often parameterized towards, e.g., a physical parameter. Furthermore, it is likely to happen that the repeated utilization of standard numerical methods like the finite element method (FEM) is considered too costly or even inaccessible. 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 trustregion 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-optimizethen- 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. Throughout this thesis, numerical experiments of well-defined benchmark problems are used to analyze the proposed methods thoroughly and to show their respective strength compared to approaches from the literature.