Over the past three decades, model order reduction (MOR) has become an established tool to reduce the computational cost for obtaining high fidelity solutions of (partial) differential-algebraic equations required in parameter studies, controller design, and optimization. The key observation used in MOR is that the solution evolves in a low-dimensional manifold in many applications, which can be embedded approximately in a low-dimensional subspace. The best linear subspace of a given dimension is characterized by the Kolmogorov n-widths, which coincide with the Hankel singular values for linear time-invariant systems. From a MOR perspective, we address questions related to
- MOR method for systems with slowly decaying Kolmogorov n-widths, for instance prevalent in transport or convection-dominate dphenomena,
- Structure-preserving MOR methods for port-Hamiltonian systems,
- MOR for switched systems
The previous methods all require a state-space description of the dynamics. If such a description is not available, then non-intrusive methods are the method of choice. Even though a description in terms of differential equations may not be accessible, expert knowledge may be available. Exploiting this knowledge in constructing the surrogate model from data ensures the physical behavior or may allow for an accurate description with only a few unknowns. In terms of system identification techniques we work on (amongst other things)
- the Loewner framework,
- dynamic mode decomposition, and
- physics-informed machine learning techniques.
The port-Hamiltonian (pH) framework constitutes an innovative energy-based model paradigm that offers a systematic approach for the interactions of (physical) systems with each other and the environment via interconnection structures. It establishes a shift from discipline-oriented research to problem-oriented interdisciplinary research. It is a fundamental building block for future high-tech initiatives in advanced system engineering, such as digital twins. The inherent structure of pH systems encodes control theoretical concepts such as passivity and stability, facilitates structure-preserving approximation schemes, enables short-term recursions in linear systems solves, and enlarges the distance to instability. In our research, we focus on
- model applications such as poroelastic network models,
- structure-preserving model order reduction schemes,
- learning port-Hamiltonian systems from data, and
- infinite-dimensional port-Hamiltonian systems
For an overview of port-Hamiltonian descriptor systems, we refer to our recent survey article.
Delay differential-algebraic equations (DDAEs) emerge in many applications such as time-delayed feedback control, blood flow models, and wide-area power networks. A particularly interesting DDAE appears in the context of hybrid numerical-experimental models that arise in earthquake engineering, where a first-principle simulation model is coupled with an actual experiment. Due to the inevitable time delay that results from the transfer system between the numerical and experimental system, this becomes an infinite-dimensional system. Despite the different application areas, the solution theory for general DDAEs is far from complete. One of the major challenges in the analytic and numerical treatment of (operator) differential-algebraic equations (DAEs) is the so-called index, which, roughly speaking, measures how often parts of the system must be differentiated to construct a solution explicitly. If time delays are involved in these system parts, further difficulties arise since so-called neutral or advanced equations may be implicitly encoded in the DDAE. Even in the linear time-invariant case, a complete characterization of solutions is difficult and requires a distributional solution concept. Besides theoretical and numerical advances in the study of DDAEs, we also use them as a powerful tool to analyze the convergence of semi-explicit time-integrators for linear poroelasticity. There, we developed a novel proof technique, where we interpret the semi-explicit scheme as implicit scheme for a related delay equation.