Structure-preserving model reduction on subspaces and manifolds

Abstract

Mathematical models are a key enabler to understand complex processes across all branches of research and development since such models allow us to simulate the behavior of the process without physically realizing it. However, detailed models are computationally demanding and, thus, are frequently prohibited from being evaluated (a) multiple times for different parameters, (b) in real time or (c) on hardware with low computational power. The field of model (order) reduction (MOR) aims to approximate such detailed models with more efficient surrogate models that are suitable for the tasks (a-c). In classical MOR, the solutions of the detailed model are approximated in a problem-specific, low-dimensional subspace, which is why we refer to it as MOR on subspaces. The subspace is characterized by a reduced basis that can be computed from given data with a so-called basis generation technique. The two key aspects in this thesis are: (i) structure-preserving MOR techniques and (ii) MOR on manifolds. Preserving given structures throughout the reduction is important to obtain physically consistent reduced models. We demonstrate this for Lagrangian and Hamiltonian systems, which are dynamical systems that guarantee preservation of energy over time. MOR on manifolds, on the other hand, broadens the applicability of MOR to problems that cannot be treated efficiently with MOR on subspaces.