Approximation of Large-Scale Dynamical Systems: An Overview

Abstract

In this paper we review the state of affairs in the area of approximation of large-scale systems. We distinguish among three basic categories, namely the SVD-based, the Krylov-based and the SVD-Krylov-based approximation methods. The first two were developed independently of each other and have distinct sets of attributes and drawbacks. The third approach seeks to combine the best attributes of the first two.

Projection-Based Model Reduction for Finite Element Approximation of Shallow Water Flows

“…It has been applied to a host of general, large-scale parameterized dynamical systems. Fundamentally, it is based on approximating the high-dimensional solution to the dynamical system in a lowdimensional subspace that is chosen by considering modes that capture a targeted portion of the system's energy [4,10]. To obtain a dynamical system of low dimension, POD proceeds by projecting the original high-dimensional equations on the solution subspace.…”

A model reduction strategy for computing the forced response of elastic waveguides using the wave finite element method

“…Model order reduction (MOR) techniques have been widely treated in the literature within the frameworks of CMS approaches [4], SVD-based and Krylov-based methods [5]. Within the CMS framework, an optimal modal reduction technique based on the study of an error norm for coupling interface forces has been proposed in refs.…”

“…the first m wave modes as ranked in this preliminary step) for 1 ≤ m ≤ n, where n is the total number of incident / reflected wave modes contained in the full wave basis. This procedure yields the minimization of E s to be an easy task -indeed, this requires us to plot the function m → E s and to identify its minimum value -which is weakly expensive from the computational point of view 5 . The fact that such a minimum value is likely to occur follows from the comments in Section 3.3.3.…”

Balanced truncation model reduction for descriptor systems

“…There exist various model reduction approaches for standard state space systems (E = I) such as balanced truncation, Hankel norm approximation and moment matching approximation, e.g., [1,2]. In this paper, developing the ideas from [7,8], we generalize a balanced truncation model reduction method for descriptor systems.…”