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A survey of model reduction methods for large-scale systems. (English) Zbl 1048.93014

Olshevsky, Vadim (ed.), Structured matrices in mathematics, computer science, and engineering I. Proceedings of an AMS-IMS-SIAM joint summer research conference, University of Colorado, Boulder, CO, USA, June 27–July 1, 1999. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-1921-6/pbk). Contemp. Math. 280, 193-219 (2001).
In this excellent paper the authors present an overview of model reduction methods for large-scale systems. They also give a comparative study of seven algorithms for model reduction applied to six dynamical systems of low to moderate complexity.
The model reduction problem has been very popular over the last two decades because of its ability to overcome the difficulties in the study of complex physical phenomena using direct numerical simulations of dynamical systems. It is known from the literature that when the level of detail is increased, these numerical simulations produce large amounts of information, and as a consequence huge amounts of storage and computational time are required. The goal is to produce a low-dimensional system \((\widehat{\Sigma})\) that has the same response characteristics as the original system \(\left(\Sigma\right)\) with far smaller storage requirements and much lower evaluation time. This approximation of the original system must be such that the following properties are satisfied: 1. The approximation error is small and there exists a global error bound. 2. System properties like stability and passivity are preserved. 3. The procedure is computationally stable and efficient.
In the first part of the paper the authors present approximation methods which are related to singular value decomposition. Based on several theorems, e.g. Schmidt-Mirsky, Eckart-Young, and Adamyan-Arov-Krein, to name a few, four algorithms which make use of the Hankel singular values are presented. These algorithms are: balanced model reduction, the approximate balanced reduction, the singular perturbation method and Hankel norm approximation.
In the second part, methods for matching moments (i.e. the coefficients of the Laurent expansion of the transfer function around some point of the complex plane) are reviewed. Three algorithms are presented, namely: the Arnoldi procedure, the Lanczos procedure and the rational Krylov method. The authors also present in detail a new approach named cross Grammian which is related to the implicitly restarted dual Arnoldi approach; although it is not a moment matching method it belongs to the general set of Krylov projection methods.
The third part of the paper is devoted to a comparison of the resulting seven algorithms applied on six dynamical systems of low to moderate complexity. Several figures are given and the following quantities are plotted in each case: (i) the largest singular value of the frequency response of the full and the reduced order models, (ii) the same quantity for the corresponding error systems, (iii) the Nyquist plots of the full and reduced order systems.
The paper concludes with a number of remarks on unifying features of the model reduction methods presented, and complexity considerations.
For the entire collection see [Zbl 0972.00034].

MSC:

93B11 System structure simplification
93B15 Realizations from input-output data
93B40 Computational methods in systems theory (MSC2010)
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