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For the accurate modeling of real‐world phenomena, high‐dimensional nonlinear dynamical systems are indispensable. Quadratic‐bilinear nonlinearities are of particular importance since many systems with general nonlinearities can be remodeled into this form. Such nonlinear systems typically encode physical properties via their internal differential structure. This leads to a high demand for structure‐preserving model order reduction techniques for quadratic‐bilinear systems that allow for the accurate representation of their dynamics by a small number of differential equations while retaining the internal system structure. In this paper, we provide formulae for structured regular subsystem transfer functions of quadratic‐bilinear systems with internal differential structures and provide a corresponding interpolation theory for the efficient construction of the resulting structured reduced‐order models. The presented theoretical results are illustrated in numerical experiments including a time‐delayed nonlinear reaction–diffusion equation and a molecular simulation. [ABSTRACT FROM AUTHOR]

Fluid‐structure interaction models are used to study how a material interacts with different fluids at different Reynolds numbers. Examining the same model not only for different fluids but also for different solids allows to optimize the choice of materials for construction even better. A possible answer to this demand is parameter‐dependent discretization. Furthermore, low‐rank techniques can reduce the complexity needed to compute approximations to parameter‐dependent fluid‐structure interaction discretizations. Low‐rank methods have been applied to parameter‐dependent linear fluid‐structure interaction discretizations. The linearity of the operators involved allows to translate the resulting equations to a single matrix equation. The solution is approximated by a low‐rank method. In this paper, we propose a new method that extends this framework to nonlinear parameter‐dependent fluid‐structure interaction problems by means of the Newton iteration. The parameter set is split into disjoint subsets. On each subset, the Newton approximation of the problem related to the median parameter is computed and serves as initial guess for one Newton step on the whole subset. This Newton step yields a matrix equation whose solution can be approximated by a low‐rank method. The resulting method requires a smaller number of Newton steps if compared with a direct approach that applies the Newton iteration to the separate problems consecutively. In the experiments considered, the proposed method allows to compute a low‐rank approximation up to twenty times faster than by the direct approach. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we discuss a novel model reduction framework for linear structured dynamical systems. The transfer functions of these systems are assumed to have a special structure, for example, coming from second‐order linear systems or time‐delay systems, and they may also have parameter dependencies. Firstly, we investigate the connection between classic interpolation‐based model reduction methods with the reachability and observability subspaces of linear structured parametric systems. We show that if enough interpolation points are taken, the projection matrices of interpolation‐based model reduction encode these subspaces. Consequently, we are able to identify the dominant reachable and observable subspaces of the underlying system. Based on this, we propose a new model reduction algorithm combining these features and leading to reduced‐order systems. Furthermore, we discuss computational aspects of the approach and its applicability to a large‐scale setting. We illustrate the efficiency of the proposed approach with several numerical large‐scale benchmark examples. [ABSTRACT FROM AUTHOR]

While extracting information from data with machine learning plays an increasingly important role, physical laws and other first principles continue to provide critical insights about systems and processes of interest in science and engineering. This work introduces a method that infers models from data with physical insights encoded in the form of structure and that minimizes the model order so that the training data are fitted well while redundant degrees of freedom without conditions and sufficient data to fix them are automatically eliminated. The models are formulated via solution matrices of specific instances of generalized Sylvester equations that enforce interpolation of the training data and relate the model order to the rank of the solution matrices. The proposed method numerically solves the Sylvester equations for minimal-rank solutions and so obtains models of low order. Numerical experiments demonstrate that the combination of structure preservation and rank minimization leads to accurate models with orders of magnitude fewer degrees of freedom than models of comparable prediction quality that are learned with structure preservation alone. [ABSTRACT FROM AUTHOR]

We study the tangential interpolation problem for a passive transfer function in standard state-space form. We derive new interpolation conditions based on the computation of a deflating subspace associated with a selection of spectral zeros of a parameterized para-Hermitian transfer function. We show that this technique improves the robustness of the low order model and that it can also be applied to nonpassive systems, provided they have sufficiently many spectral zeros in the open right half-plane. We analyze the accuracy needed for the computation of the deflating subspace, in order to still have a passive lower order model and we derive a novel selection procedure of spectral zeros in order to obtain low order models with a small approximation error. [ABSTRACT FROM AUTHOR]