Matrix Equations

Matrix Equations

Nearly all other research field of the CSC group involve matrix equation computations at a certain point. The work of this team is dedicated to the efficient numerical solution of the variety of those equations in all kinds of working environments.

Matrix equations of the one kind or the other are a central tool in all kinds of applications. In optimal control the linear quadratic control problem features a feedback solution that is determined via the solution of an algebraic Riccati equation. The system Gramians of a linear time invariant dynamical system are the solutions of two adjoint Lyapunov equations. Riccati, Lyapunov and Sylvester equations play an important role in different model order reduction techniques for continuous time linear dynamical systems. They all have discrete time counterparts as e.g. the well known Stein equations. Krylov subspace and eigenvalue methods can be related to certain Sylvester equations, as well.

Internal Projects

Improved and Structure-Aware Solvers for Large-Scale Matrix Equations
Partner: Uni Augsburg, UJI Castellón, Uni Zagreb, FAU Erlangen, Uni Bath
Funded by: MPI, DFG SPP1253 (until 10/2013), EU-MORNET (2/2018 - 3/2018)
Funding Period: since 10/2010
Contact: Peter Benner, Jens Saak
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Projection-based solvers for differential matrix equations

Projection-based solvers for differential matrix equations

Funded by: MPI
Funding Period: since 01/2016
Contact: Jan Heiland, Maximilian Behr
<span>Numerical Solution of Nonsymmetric Quadratic Matrix Equations</span>

Numerical Solution of Nonsymmetric Quadratic Matrix Equations

Partner: Victoria University (Melbourne), Università di Pisa, Università degli Studi di Perugia
Funded by: Australian Research Council - ARC (2018-2021), MPI
Funding Period: since 05/2018
Contact: Peter Benner, Martin Köhler, Davide Palitta
Partner: Uni Augsburg, Yachay Tech
Funded by: MPI
Funding Period: since 10/2010
Contact: Peter Benner, Jens Saak, Björn Baran

Solving Parametric Sylvester Matrix Integral Equations for Model Order Reduction

Funded by: MPI
Funding Period: since 06/2014
Contact: Jens Saak, Manuela Hund

High Performance Solvers for Dense Differential and Algebraic Matrix Equations

Partner: TU Dresden, UJI Castellón, UdelaR Montevideo
Funded by: MPI
Funding Period: since 01/2011
Contact: Jens Saak, Martin Köhler
<span>Customized numerical linear algebra and optimization for vibrating systems</span>

Customized numerical linear algebra and optimization for vibrating systems

Partners: Zoran Tomljanović (University of Osijek)
Contact: Davide Palitta, Tim Mitchell, Jens Saak, Peter Benner 
Project desciption: The selection of dampers locations and related viscosities is a very challenging task in the optimization of vibrating systems.  One of the main difficulties is dealing with the expensive eigenvalue problems and matrix equations that arise in this setting.  We aim to design novel and efficient algorithms by leveraging specialized techniques in numerical linear algebra.  For example, by transforming the problems into offline and online phases, we have been able to develop new algorithms with total costs that are significantly smaller than other techniques.
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