A Potpourri of Structured Krylov Subspace Methods: Approximated Exponential Integrators, Symplectic Model Order Reduction for Hamiltonian Systems, Krylov-Lagrangian Subspaces of Skew-Hamiltonian Matrices

This dissertation investigates structure-preserving Krylov subspace methods for the numerical integration and model reduction of large-scale Hamiltonian systems. Mathematical structures such as symplecticity, invariants, and conservation laws are fundamental for the accurate description of the qualitative behavior of dynamical systems. The first part of this work is devoted to the development and analysis of approximated exponential integrators that preserve the Hamiltonian structure. Various Krylov subspace methods for approximation are compared, and a short-recursion, structure-preserving variant is derived. These approximated integrators are subsequently applied within projection-based model reduction, where a new symplectic reduction method is proposed and successfully demonstrated on linear as well as nonlinear benchmark problems. Furthermore, challenges in the hyperreduction of nonlinear Hamiltonian systems are examined, revealing inherent structural limitations of existing approaches. The second part of this work deals with isotropic and Lagrangian subspaces that arise as Krylov subspaces of skew-Hamiltonian matrices. Within this framework, matrices with minimal Frobenius and 2-norms are characterized, and a constructive procedure for generating matrices with prescribed eigenvalues is presented.