Fast frequency response computation for Rayleigh damping

Meerbergen, Karl

doi:10.1002/nme.2058

Cited by 26 publications

(30 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If there is at least ( − k)/2 diagonal blocks of size 2×2 corresponding to distinct nonreal complex conjugate eigenvalues of algebraic multiplicity smaller than n (i.e., if q ≥ ( − k)/2), then we can move ( − k)/2 diagonal blocks of size 2×2 to appear after T (k) 11 and the obtained leading × submatrix would be nonderogatory. If, on the contrary, q < ( − k)/2, then we will have to move all diagonal blocks of nonreal complex conjugate eigenvalues of multiplicity less than n and − k − 2q distinct real eigenvalues to appear after T (k) 11 . The question is: Do we have − k − 2q distinct real eigenvalues?…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…Choose k 1 2 × 2 blocks corresponding to distinct nonreal complex conjugate eigenvalues of algebraic multiplicity less than n and move them so that they appear directly after T (k) 11 . Denote the new submatrix T (k+2k1) 11…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…Published by SIAM under the terms of the Creative Commons 4.0 license eigenvalues of algebraic multiplicity less than n so that they appear after T (k+2k1) 11 . The leading × submatrix is nonderogatory.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…Since s > 0, one real eigenvalue of algebraic multiplicity less than n can be placed after T (k+2k1) 11 so that the × principal submatrix is nonderogatory.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…− k − 2k 1 distinct real eigenvalues of algebraic multiplicity less than n so that they appear after T (k+2k1) 11 and the resulting × principal submatrix is nonderogatory. Apply the induction hypothesis as above.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

See 4 more Smart Citations

Reduction of Matrix Polynomials to Simpler Forms

Nakatsukasa¹,

Taslaman²,

Tisseur³

et al. 2018

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. A square matrix can be reduced to simpler form via similarity transformations. Here "simpler form" may refer to diagonal (when possible), triangular (Schur), or Hessenberg form. Similar reductions exist for matrix pencils if we consider general equivalence transformations instead of similarity transformations. For both matrices and matrix pencils, well-established algorithms are available for each reduction, which are useful in various applications. For matrix polynomials, unimodular transformations can be used to achieve the reduced forms but we do not have a practical way to compute them. In this work we introduce a practical means to reduce a matrix polynomial with nonsingular leading coefficient to a simpler (diagonal, triangular, Hessenberg) form while preserving the degree and the eigenstructure. The key to our approach is to work with structure preserving similarity transformations applied to a linearization of the matrix polynomial instead of unimodular transformations applied directly to the matrix polynomial. As an application, we illustrate how to use these reduced forms to solve parameterized linear systems.

show abstract

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Auxiliary Resultsmentioning

confidence: 99%

“…Since s > 0, one real eigenvalue of algebraic multiplicity less than n can be placed after T (k+2k1) 11 so that the × principal submatrix is nonderogatory.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Auxiliary Resultsmentioning

confidence: 99%

See 3 more Smart Citations

Reduction of Matrix Polynomials to Simpler Forms

Nakatsukasa¹,

Taslaman²,

Tisseur³

et al. 2018

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

Model reduction for linear simulated moving bed chromatography systems using Krylov‐subspace methods

Feng

et al. 2014

AIChE Journal

View full text Add to dashboard Cite

Simulated moving bed (SMB) chromatography is a well-established technology for separating chemical compounds. To describe an SMB process, a finite-dimensional multistage model arising from the discretization of partial differential equations is typically employed. However, its relatively high dimension poses severe computational challenges to various model-based analysis. To overcome this challenge, two Krylov-type model order reduction (MOR) methods are proposed to accelerate the computation of the cyclic steady states (CSSs) of SMB processes with linear isotherms. A "straightforward method" that carefully deals with the switching behavior in MOR is first proposed. Its improvement, a "subspace-exploiting method," thoroughly exploits each reduced model to achieve further acceleration. Simulation studies show that both methods achieve high accuracy and significant speedups. The subspace-exploiting method turns out to be computationally much more efficient. Two challenging analyses of SMB processes, namely uncertainty quantification and CSS optimization, further demonstrate the accuracy, efficiency, and applicability of the proposed methods.

show abstract

Using Krylov‐Padé model order reduction for accelerating design optimization of structures and vibrations in the frequency domain

Meerbergen

2012

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARY In many engineering problems, the behavior of dynamical systems depends on physical parameters. In design optimization, these parameters are determined so that an objective function is minimized. For applications in vibrations and structures, the objective function depends on the frequency response function over a given frequency range, and we optimize it in the parameter space. Because of the large size of the system, numerical optimization is expensive. In this paper, we propose the combination of Quasi‐Newton type line search optimization methods and Krylov‐Padé type algebraic model order reduction techniques to speed up numerical optimization of dynamical systems. We prove that Krylov‐Padé type model order reduction allows for fast evaluation of the objective function and its gradient, thanks to the moment matching property for both the objective function and the derivatives towards the parameters. We show that reduced models for the frequency alone lead to significant speed ups. In addition, we show that reduced models valid for both the frequency range and a line in the parameter space can further reduce the optimization time. Copyright © 2012 John Wiley & Sons, Ltd.

show abstract

Fast frequency response computation for Rayleigh damping

Cited by 26 publications

References 18 publications

Reduction of Matrix Polynomials to Simpler Forms

Reduction of Matrix Polynomials to Simpler Forms

Model reduction for linear simulated moving bed chromatography systems using Krylov‐subspace methods

Using Krylov‐Padé model order reduction for accelerating design optimization of structures and vibrations in the frequency domain

Contact Info

Product

Resources

About