SOBMOR: Structured Optimization-Based Model Order Reduction

Abstract

Model order reduction (MOR) methods that are designed to preserve structural features of a given full order model (FOM) often suffer from a lower accuracy when compared to their non structure preserving counterparts. In this paper, we present a framework for MOR based on direct parameter optimization. This means that the elements of the system matrices are iteratively varied to minimize an objective functional that measures the difference between the FOM and the reduced order model (ROM). Structural constraints are encoded in the parametrization of the ROM. The method only depends on frequency response data and can thus be applied to a wide range of dynamical systems.We illustrate the effectiveness of our method on a port-Hamiltonian and on a symmetric second order system in a comparison with other structure preserving MOR algorithms. * This work is supported by the German Research Foundation (DFG) within the project VO2243/2-1: "Interpolationsbasierte numerische Algorithmen in der robusten Regelung" and the DFG Cluster of Excellence MATH+ within the project AA4-5: "Energy-based modeling, simulation, and optimization of power systems under uncertainty".

Adaptive Sampling for Structure Preserving Model Order Reduction of Port-Hamiltonian Systems

“…In the following, we denote the Euclidian norm of the evaluated error transfer function by E(s) := H(s) − H(s) 2 . We focus our investigation on the H ∞ norm but the results for the H 2 norm are similar, which can also be observed in Schwerdtner and Voigt (2020).…”

“…The method in Gugercin et al (2012) adapts IRKA to also preserve the pH property of a dynamical system, while the method in Polyuga and van der Schaft (2008) (pH-BT) uses a balancing based approach. In our previous study (Schwerdtner and Voigt (2020)), we observe that for both methods the accuracy is drastically reduced compared to non-structure preserving IRKA or BT (this can also be observed in Fig. 4), and develop a new optimization based MOR framework that also preserves the pH structure but has a similar accuracy as the non-structure preserving MOR methods.…”

“…The purpose of this work is to overcome one of the major issues with our direct optimization approach as introduced in Schwerdtner and Voigt (2020). That approach is based on approximating the transfer function of the given FOM at a set of n S ∈ N sample points S := {s i } i=1...n S ⊂ iR, where iR denotes the imaginary axis.…”

“…Our article is organized as follows. In the next section, we formally introduce the terms needed to describe accuracy in the domain of MOR and summarize the main algorithm in Schwerdtner and Voigt (2020). In Section 3, we explain how adaptive sampling is integrated into our method, explain the adaptive sampling algorithm and reformulate the new overall method.…”

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We present an adaptive sampling strategy for the optimization-based structure preserving model order reduction (MOR) algorithm developed in [Schwerdtner, P. and Voigt, M. (2020). Structure preserving model order reduction by parameter optimization, Preprint arXiv:2011.07567].

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