A rational Lanczos algorithm for model reduction

Gallivan, Kyle A.; Grimme, E.; Dooren, Paul Van

doi:10.1007/bf02141740

Cited by 175 publications

(134 citation statements)

References 24 publications

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“…If the expansion is around zero the problem becomes Padé approximation. In general, for arbitrary of rational interpolation, which amounts to matching moments at the specified point For an overview as well as details on the material presented in this section we refer to [37,38,67,42,39,72,74,33,34,7].…”

Section: Krylov-based Approximation Methodsmentioning

confidence: 99%

Approximation of Large-Scale Dynamical Systems: An Overview

Antoulas

2004

IFAC Proceedings Volumes

146

179

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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.

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Section: Krylov-based Approximation Methodsmentioning

confidence: 99%

Approximation of Large-Scale Dynamical Systems: An Overview

Antoulas

2004

IFAC Proceedings Volumes

146

179

View full text Add to dashboard Cite

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“…After obtaining the best reduced order model from the iterative procedure, it might be possible to further compact the model with the information obtained from the singular values Σ of V comm (1). Figure 2 shows the flowchart for the truncation of the singular values.…”

Section: Model Compactingmentioning

confidence: 99%

“…Therefore, model order reduction (MOR) techniques are crucial to reduce the complexity of large scale models and the computational cost of the simulations, while retaining the important physical features of the original system [1][2][3]. Multipoint MOR methods have been developed over the years [1,4,5], which allows to generate accurate reduced models over the whole frequency range of interest. In this paper, the expansion points are selected adaptively using a reflective exploration technique.…”

Section: Introductionmentioning

confidence: 99%

Multipoint Model Order Reduction Using Reflective Exploration

Samuel

Knockaert

Dhaene

2016

Scientific Computing in Electrical Engineering

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Summary. Reduced order models obtained by model order reduction methods must be accurate over the whole frequency range of interest. Multipoint reduction algorithms allow to generate accurate reduced models. In this paper, we propose the use of reflective exploration technique for obtaining the expansion points adaptively for the reduction algorithm. At each expansion point the corresponding projection matrix is computed. Then, the projection matrices are merged and truncated based on their singular values to obtain a compact reduced order model.

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“…In [6], an approach based on the rational non symmetric Lanczos algorithm was proposed for computing a multi-point approximation of a rational function. Unfortunately, the non symmetric Lanczos algorithm can suffer a loss of orthogonality and may even break down.…”

Section: Relationship With Krylov Subspacesmentioning

confidence: 99%

“…It has been shown that such a rational function can be accurately and efficiently approximated by an appropriate Padé expansion [5,6,7,4,10,1]. The numerical algorithm proposed in this paper for identifying the eigenvalues missed in a given range of interest builds on the aforementioned observations.…”

Section: Introductionmentioning

confidence: 99%

A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver

Avery

Farhat

Hetmaniuk

2009

Numerical Meth Engineering

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SUMMARYWhen computing the solution of a generalized symmetric eigenvalue problem of the form Ku = λMu, the Sturm sequence check is the most popular method for reporting the number of missed eigenvalues within a range [σL, σR]. This method requires the factorization of the matrices K − σLM and K − σRM. When the size of the problem is reasonable and the matrices K and M are assembled, these factorizations are possible. When the eigensolver is equipped with an iterative solver, which is nowadays the preferred choice for large-scale problems, the factorization of K − σM is not desired or feasible and therefore the Sturm sequence check cannot be performed. To this effect, the purpose of this paper is to present a factorization-free algorithm for detecting and identifying the eigenvalues that were missed by an eigensolver equipped with an iterative linear equation solver within an interval of interest [σL, σR]. This algorithm constructs a scalar, rational, transfer function whose poles are exactly the eigenvalues of the symmetric pencil (K, M), approximates it by a Padé expansion, and computes the poles of this approximation to detect and identify the missed eigenvalues. The proposed algorithm is illustrated with an academic numerical example. Its potential for real engineering applications is also demonstrated with a large-scale structural vibrations problem.

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A rational Lanczos algorithm for model reduction

Cited by 175 publications

References 24 publications

Approximation of Large-Scale Dynamical Systems: An Overview

Approximation of Large-Scale Dynamical Systems: An Overview

Multipoint Model Order Reduction Using Reflective Exploration

A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver

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