Journal ArticleDOI
A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov Equations
TLDR
The cyclic low-rank Smith method is presented, which is an iterative method for the computation of low- rank approximations to the solution of large, sparse, stable Lyapunov equations, and a heuristic for determining a set of suboptimal alternating direction implicit (ADI) shift parameters is proposed.Abstract:
In this paper we present the cyclic low-rank Smith method, which is an iterative method for the computation of low-rank approximations to the solution of large, sparse, stable Lyapunov equations. It is based on a generalization of the classical Smith method and profits by the usual low-rank property of the right-hand side matrix. The requirements of the method are moderate with respect to both computational cost and memory. Furthermore, we propose a heuristic for determining a set of suboptimal alternating direction implicit (ADI) shift parameters. This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation. Numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the ADI parameters.read more
Citations
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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
TL;DR: Model reduction aims to reduce the computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior as mentioned in this paper. But model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books.
Journal ArticleDOI
On Dynamic Mode Decomposition: Theory and Applications
TL;DR: A theoretical framework in which dynamic mode decomposition is defined as the eigendecomposition of an approximating linear operator, which generalizes DMD to a larger class of datasets, including nonsequential time series, and shows that under certain conditions, DMD is equivalent to LIM.
Journal ArticleDOI
A survey of model reduction by balanced truncation and some new results
TL;DR: In this article, a survey of balancing related model reduction methods and their corresponding error norms is presented, and also some new results are introduced, including a modified positive real balancing scheme with an absolute error bound.
Journal ArticleDOI
Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems
TL;DR: This paper gives an overview of the recent progress in other Krylov subspace techniques for a variety of dynamical systems, including second-order and nonlinear systems, and case studies arising from circuit simulation, structural dynamics and microelectromechanical systems are presented.
Journal ArticleDOI
Computational Methods for Linear Matrix Equations
TL;DR: The aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.
References
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Book
Iterative Methods for Sparse Linear Systems
TL;DR: This chapter discusses methods related to the normal equations of linear algebra, and some of the techniques used in this chapter were derived from previous chapters of this book.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.