Multilevel minimization in trust-region framework: algorithmic and software developments

The field of scientific computing is associated with the modeling of complex physical phenomena. The resulting numerical models are often described by differential equations, which, in many cases, can be related to non-convex minimization problems. Thus, after discretization, the solution of large-scale non-convex optimization problem is required. Various iterative solution strategies can be used to solve such optimization problems. However, the convergence speed of the majority of them deteriorates rapidly with increasing problem size. Multilevel methods are known to overcome this difficulty, and therefore we focus on a class of globally convergent multilevel solution strategies called the recursive multilevel trust-region (RMTR) method. The RMTR method combines globalization properties of trust-region method and the efficiency of multilevel methods. Despite its robustness and efficiency, the practical implementation of the RMTR method is a technically demanding task, which relies upon a suitable multilevel framework. This framework requires the careful design of two main components: i) multilevel hierarchy and transfer operators, ii) coarse-level models. To maximize the efficiency of the RMTR method, these components must be created with knowledge of the particular optimization problem in mind. In this thesis, we propose three novel variants of the RMTR method. Our first variant of the RMTR method is tailored for solving phase-field fracture problems. It employs novel coarse-level models, that allow the representation of fine-level fractures on the coarser levels. Our second RMTR variant is developed for thin-shell cloth simulations. Here, we employ a subdivision-based multilevel hierarchy and transfer operators. Our third variant of the RMTR method is designed for the training of the deep residual networks (ResNets). We construct the multilevel hierarchy and transfer operators by leveraging a dynamical system's view-point, which casts ResNet as the discretization of an initial value problem. We analyze the convergence properties of all three novel variants of the RMTR method. To this aim, we consider numerical examples from respective scientific fields. A comparison with a single-level trust-region method is made and demonstrates the efficiency of the proposed RMTR variants. Furthermore, we introduce our open-source library UTOPIA, which incorporates the parallel implementation of the multilevel methods presented in this work. Weak and strong scaling properties of our implementation are investigated up to 12,000 processors and a billion degrees of freedom