Publications

Hybrid numerical-experimental testing is a standard approach for complex dynamical structures that are, on the one hand, not easy to model due to complexity and parameter uncertainty and, on the other hand, too expensive for full-scale experiments. The main idea is to subdivide the structure in a part that can be accurately simulated with numerical methods and an experimental component. The numerical simulation and the experiment are coupled in real-time by a so-called transfer system, which induces a time-delay into the system. In this paper, we study the solvability of the resulting hybrid numerical-experimental system, which is typically described by a set of nonlinear delay differential-algebraic equations, and extend existing results from the literature to this case.

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a-posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

Kolmogorov $n$-widths and Hankel singular values are two commonly used concepts in model reduction. Here we show that for the special case of linear time-invariant dynamical (LTI) systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov $n$-width of an LTI system equals its $(n+1)st$ Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov $n$-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov $n$-width.

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The propagation of primary discontinuities in initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Based on the (quasi-) Weierstraß form for regular matrix pencil, a complete characterization of the different propagation types is given and algebraic criteria in terms of the matrices are developed. The analysis, which is based on the method of steps, takes into account all possible inhomogeneities and history functions and thus serves as the worst-case scenario. Moreover, it reveals possible hidden delays in the DDAE. The new classification for DDAEs is compared to existing approaches in the literature and the impact of splicing conditions on the classification is studied.

We present a data-driven realization for systems with delay, which generalizes the Loewner framework. The realization is obtained with low computational cost directly from measured data of the transfer function. The internal delay is estimated by solving a least-square optimization over some sample data. Our approach is validated by several examples, which indicate the need for preserving the delay structure in the reduced model.