A Robust Iterative Scheme for Symmetric Indefinite Systems

Manguoğlu, Murat; Mehrmann, Volker

doi:10.1137/18m1190860

Cited by 3 publications

(5 citation statements)

References 28 publications

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“…The matrix can be split into its symmetric and skew-symmetric part = + , where = 1 2 (( + ) + ( + )) ≥ 0 and = 1 2 (( + ) − ( + )). An analogous linear system structure occurs in discretized linear time-varying and nonlinear pHDAE systems, in the construction of reduced models , and by multiplying some equations by −1 in optimization methods; see also Güdücü et al (2022) and Manuoglu and Mehrmann (2019).…”

Section: Linear System Solversmentioning

confidence: 99%

Control of port-Hamiltonian differential-algebraic systems and applications

Mehrmann¹,

Unger²

2023

Acta Numerica

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We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling since it is invariant under power-conserving interconnection, congruence transformations and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity and numerical methods to check these properties.After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analysed, and time discretization and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed.The properties of port-Hamiltonian descriptor systems and their use in modelling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.

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Section: Linear System Solversmentioning

confidence: 99%

Control of port-Hamiltonian differential-algebraic systems and applications

Mehrmann¹,

Unger²

2023

Acta Numerica

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“…The matrix A can be split in its symmetric and skew-symmetric part A = H +S, where H = 1 2 ((E +τ R) T +(E +τ R)) ≥ 0 and S = 1 2 ((E +τ R) T −(E +τ R)). An analogous linear system structure occurs in discretized linear time-varying and nonlinear pHDAE systems, in the construction of reduced models [71], and by multiplying some equations with −1 in optimization methods,see also [92,153].…”

Section: Temporal Discretization and Linear System Solversmentioning

confidence: 99%

Control of port-Hamiltonian differential-algebraic systems and applications

Mehrmann¹,

Unger²

2022

Preprint

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The modeling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control are discussed. The structure is ideal for automated network-based modeling since it is invariant under power-conserving interconnection, congruence transformations, and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations present easy analysis tools for existence, uniqueness, regularity, and numerical methods to check these properties.After recalling the concepts for general linear and nonlinear descriptor systems, we demonstrate that many difficulties that arise in general descriptor systems can be easily overcome within the port-Hamiltonian framework. The properties of port-Hamiltonian descriptor systems are analyzed, time-discretization, and numerical linear algebra techniques are discussed. Structure-preserving regularization procedures for descriptor systems are presented to make them suitable for simulation and control. Model reduction techniques that preserve the structure and stabilization and optimal control techniques are discussed.The properties of port-Hamiltonian descriptor systems and their use in modeling simulation and control methods are illustrated with several examples from different physical domains. The survey concludes with open problems and research topics that deserve further attention.

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“…They are roughly invariant for incomplete factorization for all problems except rdbl1250l. For the cases when the rank is zero, we use the preconditioner in (22), otherwise we use the preconditioner in (21). We have also experimented with the preconditioner in (22) for problems where the rank is not zero but relatively small.…”

Section: Iterative Solution Of General Sparse Linear Systemsmentioning

confidence: 99%

“…For the cases when the rank is zero, we use the preconditioner in (22), otherwise we use the preconditioner in (21). We have also experimented with the preconditioner in (22) for problems where the rank is not zero but relatively small. However, the number of iterations has increased significantly even for the case where rank is equal to one (rdbl3200l with ildl(10 −2 )).…”

Section: Iterative Solution Of General Sparse Linear Systemsmentioning

confidence: 99%

“…where A ∈ R n×n is a general large sparse invertible matrix. If the coefficient matrix is symmetric and positive definite or symmetric and indefinite, one can use the Conjugate Gradient algorithm or the recently proposed two-level iterative scheme [22], respectively. In this paper, we propose a new robust two-level black-box scheme for solving general systems without any assumption on the symmetry or definiteness of the coefficient matrix.…”

Section: Introductionmentioning

confidence: 99%

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A two-level iterative scheme for general sparse linear systems based on approximate skew-symmetrizers

Manguoğlu¹,

Mehrmann²

2020

Preprint

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We propose a two-level iterative scheme for solving general sparse linear systems. The proposed scheme consists of a sparse preconditioner that increases the skew-symmetric part and makes the main diagonal of the coefficient matrix as close to the identity as possible. The preconditioed system is then solved via a particular Minimal Residual Method for Shifted Skew-Symmetric Systems (mrs). This leads to a two-level (inner and outer) iterative scheme where the mrs has short term recurrences and satisfies an optimally condition. A preconditioner for the inner system is designed via a skew-symmetry preserving deflation strategy based on the skew-Lanczos process. We demonstrate the robustness of the proposed scheme on sparse matrices from various applications.

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A Robust Iterative Scheme for Symmetric Indefinite Systems

Cited by 3 publications

References 28 publications

Control of port-Hamiltonian differential-algebraic systems and applications

Control of port-Hamiltonian differential-algebraic systems and applications

Control of port-Hamiltonian differential-algebraic systems and applications

A two-level iterative scheme for general sparse linear systems based on approximate skew-symmetrizers

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