Interpolatory model reduction of parameterized bilinear dynamical systems
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Interpolatory model reduction of parameterized bilinear dynamical systems Andrea Carracedo Rodriguez1 · Serkan Gugercin1 · Jeff Borggaard1
Received: 17 July 2017 / Accepted: 19 April 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Interpolatory projection methods for model reduction of nonparametric linear dynamical systems have been successfully extended to nonparametric bilinear dynamical systems. However, this has not yet occurred for parametric bilinear systems. In this work, we aim to close this gap by providing a natural extension of interpolatory projections to model reduction of parametric bilinear dynamical systems. We introduce necessary conditions that the projection subspaces must satisfy to obtain parametric tangential interpolation of each subsystem transfer function. These conditions also guarantee that the parameter sensitivities (Jacobian) of each subsystem transfer function are matched tangentially by those of the corresponding reduced-order model transfer function. Similarly, we obtain conditions for interpolating the parameter Hessian of the transfer function by including additional vectors in the projection subspaces. As in the parametric linear case, the basis construction for two-sided projections does not require computing the Jacobian or the Hessian. Keywords Model reduction · Parametric · Bilinear · Interpolation Mathematics Subject Classification (2010) 41A05 · 93A15 · 93C10 · 35B30 · 93C15 · 93B40 Communicated by: Peter Benner Andrea Carracedo Rodriguez
[email protected] Serkan Gugercin [email protected] Jeff Borggaard [email protected] 1
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA
A. Carracedo Rodriguez et al.
1 Introduction Simulation of dynamical systems has become an essential part in the development of science to study complex physical phenomena. However, as the ever increasing need for accuracy has lead to ever larger dimensional dynamical systems, this increased dimension often makes the desired numerical simulations prohibitively expensive to perform. Model order reduction (MOR) is one remedy for this predicament. MOR tackles this issue by constructing a substantially lower dimensional representation of the full-order dynamical system, which is cheap to simulate, yet provides high-fidelity, i.e., it provides an accurate approximation of the original quantities of interest. In many applications such as optimization, design, control, uncertainty quantification, and inverse problems, the dynamics of the system are defined by a set of parameters that describe initial conditions, material properties, etc. Since carrying out model reduction for every parameter value is not computationally feasible, the goal in the parameterized setting is to construct a parametric reduced model that can approximate one or more quantities of interest well for the whole parameter range of interest. This lead to the parametric model reduction framework. For more specific details on both parametric and nonparametric model reduction, we re
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