On the stability of projection-based linear reduced-order models: Descriptor vs non-descriptor forms

David Amsallem; Charbel Farhat

Home/Archive/arXiv

arXiv · 1209.5694

On the stability of projection-based linear reduced-order models: Descriptor vs non-descriptor forms

David Amsallem·Charbel Farhat

View original source ↗Read source document ↗Export metadata ↓

Abstract

Two comprehensive approaches are considered for constructing projection-based reduced-order computational models for linear dynamical systems. The first one reduces the governing equations written in the descriptor form, using a Galerkin or Petrov-Galerkin projection onto a reduced-order basis or pair of them, respectively. These bases can be constructed by any preferred method such as the Proper Orthogonal Decomposition, Balanced Proper Orthogonal Decomposition, or Moment Matching method. The second approach transforms first the governing equations into their non-descriptor form before applying the same projection-based model order reduction method. For several structural and coupled fluid-structure dynamical systems, it is observed that the first reduction approach leads to reduced-order models that exhibit significantly better numerical stability and accuracy properties than those resulting from the second one. These observations are reinforced by theoretical results that anticipate and confirm the better stability properties obtained in general when reducing the descriptor rather than non-descriptor form of the equations governing a linear dynamical system.

Explore related subjects

math.DS

Keep this discovery

BibTeX RIS

David Amsallem, Charbel Farhat. 2013-01-07. On the stability of projection-based linear reduced-order models: Descriptor vs non-descriptor forms. https://arxiv.org/abs/1209.5694

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗

arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗

arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗