Multivariate Interpolation Methods for Parametric Model Reduction (MIM4PMOR)

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Dynamical systems are a principal tool in modeling and control of physical processes in engineering, economics, the natural and social sciences. In many areas, direct numerical simulation (DNS) has become essential for studying the rich complexity of these phenomena and for the design process. Due to the increasing complexity of the underlying mathematical models, unmatched by the increase in computing power, model reduction has become an indispensable tool in order to facilitate or even enable simulation (in particular parameter studies and uncertainty analysis), control, and optimization of dynamical systems. Here, we will focus on parametrized models where the preservation of parameters as symbolic quantities in the reduced-order model is essential. We will pursue two complementary approaches. The first one starts from empirical data and employs advanced interpolation techniques, overcoming limitations of standard projection methods. The empirical data may be provided by physical experimentation or by DNS. The second approach is model-based and employs a connection between bilinear and linear parameter-varying control systems to derive optimal interpolation conditions. Our common theme is to construct reduced-order models satisfying interpolation and certain optimality conditions. We are proposing to put these techniques on firm mathematical foundations in order to assure accuracy and to derive computationally efficient model reduction algorithms.