A preconditioned block Arnoldi method for large scale Lyapunov and algebraic Riccati equations

Bouhamidi, A.; Hached, Mustapha; Jbilou, Khalide

doi:10.1007/s10898-015-0317-0

Cited by 4 publications

(4 citation statements)

References 25 publications

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“…where is any given matrix norm. We notice here that the computed approximation X k+1 should not be stored explicitly but could be given as a product of two low rank matrices X k+1 = ZZ T , so that only Z is used (see [21,27] for more details). The Lyapunov equations appearing in Algorithm 4 could be solved approximatively by using a Krylov based or ADI-based methods with a high accuracy to guarantee that the Newton iteration converges to the stabilizing solution [23].…”

Section: Krylov-based Methods Consider the Continuous-time Algebraic ...mentioning

confidence: 99%

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Computational methods for model reduction in large-scale MIMO dynamical systems

Jbilou

2023

Preprint

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In this work, we review different numerical methods for low order model reduction and linear control problems. These methods are essentially based on block, global, extended block or rational block Krylov subspace methods when we are dealing with large-scale dynamical systems. In this paper, the main schemes for model reduction are outlined, and personal contribution to some theoretical aspects is also given. Tools and approaches in model simplification, developed by some research groups are discussed, and the main results obtained during the last few years are summarized. AMS subject classifications. 65F10, 65F30

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Section: Krylov-based Methods Consider the Continuous-time Algebraic ...mentioning

confidence: 99%

“…Perturbation results and error bounds are given in [151]. Other Newton-based methods are presented in [22,23,23,27,37,38,78].…”

Section: Krylov-based Methods Consider the Continuous-time Algebraic ...mentioning

confidence: 99%

Computational methods for model reduction in large-scale MIMO dynamical systems

Jbilou

2023

Preprint

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“…When the dimension of the problem is large, this approach would be too demanding in terms of computation time and memory. In this case, iterative methods, as Krylov subspaces, Newton-type ( [4,6,12,22,23,17]) or ADI-type appear to be a standard choice, see ( [8,18,19,20]) for more details.…”

Section: The Bdf Methods For Solving Dresmentioning

confidence: 99%

Low-rank approximate solutions to large-scale differential matrix Riccati equations

Güldoğan

Hached²,

Jbilou

et al. 2018

Appl. Math. (Warsaw)

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In the present paper, we consider large-scale continuous-time differential matrix Riccati equations. To the authors' knowledge, the two main approaches proposed in the litterature are based on a splitting scheme or on a Rosenbrock / Backward Differentiation Formula (BDF) methods. The approach we propose is based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimensional differential matrix Riccati equation. The latter matrix differential problem is then solved by a Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. This process is repeated, increasing the dimension of the projection subspace until achieving a chosen accuracy. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments will be given.

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“…Additionally, there are many works focusing on the numerical algorithms for the CARE (1), such as the Schur method [18], the matrix sign function [19], the structure-preserving doubling algorithm [20], and Krylov subspace projection method [21]- [22]. Kleinman [23], Banks and Ito [24] applied the Newton method, due to its quadratic convergence, to solve this equation.…”

Section: Introductionmentioning

confidence: 99%

A general alternating-direction implicit Newton method for solving complex continuous-time algebraic Riccati matrix equation

Li¹,

Zhang²

2022

Preprint

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In this paper, applying the Newton method, we transform the complex continuoustime algebraic Riccati matrix equation into a Lyapunov equation. Then, we introduce an efficient general alternating-direction implicit (GADI) method to solve the Lyapunov equation. The inexact Newton-GADI method is presented to save computational amount effectively. Moreover, we analyze the convergence of the Newton-GADI method. The convergence rate of the Newton-GADI and Newton-ADI methods is compared by analyzing their spectral radii. Furthermore, we give a way to select the quasi-optimal parameter. Corresponding numerical tests are shown to illustrate the effectiveness of the proposed algorithms.

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A preconditioned block Arnoldi method for large scale Lyapunov and algebraic Riccati equations

Cited by 4 publications

References 25 publications

Computational methods for model reduction in large-scale MIMO dynamical systems

Computational methods for model reduction in large-scale MIMO dynamical systems

Low-rank approximate solutions to large-scale differential matrix Riccati equations

A general alternating-direction implicit Newton method for solving complex continuous-time algebraic Riccati matrix equation

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