On non-Hermitian positive (semi)definite linear algebraic systems arising from dissipative Hamiltonian DAEs

  title={On non-Hermitian positive (semi)definite linear algebraic systems arising from dissipative Hamiltonian DAEs},
  author={Candan G{\"u}d{\"u}c{\"u} and J{\"o}rg Liesen and Volker Mehrmann and Daniel B. Szyld},
. We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An important feature of the linear algebraic systems is that the (non-Hermitian) system matrix has a positive definite or semidefinite Hermitian part. In the positive definite case we can solve the linear algebraic systems iteratively by Krylov subspace methods based… 

A flexible short recurrence Krylov subspace method for matrices arising in the time integration of port Hamiltonian systems and ODEs/DAEs with a dissipative Hamiltonian

New, right preconditioned variants of this approach which as their crucial new feature allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences.

A Rosenbrock framework for tangential interpolation of port-Hamiltonian descriptor systems

A new structure-preserving model order reduction (MOR) framework for large-scale port-Hamiltonian descriptor systems (pH-DAEs) is presented, which produces reduced-order models (ROMs) of minimal dimension, which tangentially interpolate the original model’s transfer function and are guaranteed to be again in pH-DAE form.

Structure-Preserving Model Order Reduction for Index Two Port-Hamiltonian Descriptor Systems

This work develops optimization-based structure-preserving model order reduction (MOR) methods for port-Hamiltonian descriptor systems of differentiation index one that include a simplified treatment of algebraic constraints and often a higher accuracy of the resulting reduced-order model.



Linear Algebra Properties of Dissipative Hamiltonian Descriptor Systems

A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left

Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems

A class of preconditioned Hermitian/skew-Hermitian splitting iteration methods is established, showing that the new method converges unconditionally to the unique solution of the linear system.

Modified HSS iteration methods for a class of complex symmetric linear systems

The modified Hermitian and skew-Hermitian splitting (MHSS) iteration method is unconditionally convergent and each iteration of this method requires the solution of two linear systems with real symmetric positive definite coefficient matrices.

A generalized conjugate gradient method for non-symmetric systems of linear equations

A generalized conjugate gradient method for solving systems of linear equations having nonsymmetric coefficient matrices with positive-definite symmetric part based on splitting the matrix into its symmetric and skew-symmetric parts, which simplifies in this case, as only one of the two usual parameters is required.

Iterative Solution of Skew-Symmetric Linear Systems

The preconditioned iterations the authors develop are based on preserving the skew-symmetry, and an incomplete $2\times2$ block $LDL^T$ decomposition is introduced, which illustrates the convergence properties of the algorithms and the effectiveness of the preconditionsing approach.

Hypocoercivity and controllability in linear semi-dissipative ODEs and DAEs

A detailed analysis of the stability of dynamical systems of evolution equations (finite or infinite-dimensional) is still very problem dependent and computationally challenging, see [6, 18, 19, 27].

A Generalization of the Hermitian and Skew-Hermitian Splitting Iteration

  • M. Benzi
  • Mathematics
    SIAM J. Matrix Anal. Appl.
  • 2009
It is shown that the new scheme can outperform the standard HSS method in some situations and can be used as an effective preconditioner for certain linear systems in saddle point form.

Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations

It is shown that under structure-preserving perturbations the asymptotical stability of a DH system is much more robust than under general perturbation, since the distance to instability can be much larger when struc...