# Non-Funded Research Activity

## Large Scale and Nonlinear Eigenvalue Problems

**Project director:**

- Prof. Dr. Peter Benner

Max Planck Institut for Dynamics Complex Technical Systems Magdeburg,

Computational Methods in Systems and Control Theory,

Sandtorstr. 1, 39106 Magdeburg, Germany

Tel: +49 (0)391-6110-450

E-mail: benner@mpi-magdeburg.mpg.de

**Researcher:**

- Patrick Kürschner

Max Planck Institut for Dynamics Complex Technical Systems Magdeburg,

Computational Methods in Systems and Control Theory,

Sandtorstr. 1, 39106 Magdeburg, Germany

Tel: +49 (0)391-6110-424

E-mail: kuerschner@mpi-magdeburg.mpg.de

**Duration:**since 2000

**Project description:**-
Large-Scale linear and nonlinear eigenvalue problems belong to the main fields of research in modern numerical mathematics.
Approximations of a number of eigenvalues and the associated eigenvectors are usually obtained using iterative methods.

In this project we focus on iterative methods which are based on the treatment of eigenvalue problems via a Newton scheme. These are, e.g., inverse- and Rayleigh quotient iteration, as well as the closely related Jacobi-Davidson methods.

As special application we investigate the application of these eigensolvers in the context of model order reduction, which amounts to finding dominant poles of a transfer function of a linear, time-invariant control systems. These dominant poles yield a significant contribution to the input-output dynamics of the systems and produce the peaks in the frequency response plot of the transfer function as it can be seen in the pictures below.

Three- and two-dimensional view of the surface of the transfer functions spectral norm, with highlighted dominant eigenvalues.

In particular, the following issues are investigated:- Extending and adapting existing methods to nonlinear eigenvalue problems, where the emphasis is drawn methods based on the two-sided and generalized Jacobi-Davidson and related methods.
- Exploiting the structure of certain classes of nonlinear eigenproblems, e.g. polynomial, rational, delay eigenproblems, as those occur frequently in model order reduction.
- Improving the efficiency in both the linear and nonlinear case by solving the occuring linear systems of equations inexactly using Krylov subspace methods. This includes the determination and application of adequate preconditioners.
- Application of these methods to nonlinear eigenproblems occurring in natural sciences outside the model order reduction context, for instance, mechanics, chemistry, etc.

**Related publications:**- @article{FreKP18,

author = {Freitag,

M. and K{\"u}rschner,

P. and Pestana,

J.},

title = {GMRES convergence bounds for eigenvalue problems},

journal = CMAM,

year = 2018,

pages ="203-222" doi={10.1515/cmam-2017-0017},

volume=18,

number=2 }

GMRES convergence bounds for eigenvalue problems ;

Freitag, Melina; Kürschner, Patrick; Pestana, Jennifer;*Comput. Meth. Appl. Mat.*: 18(2), pp. 203-222;

2018.@article {FreK14,

author = {Freitag,

Melina A. and K{\"u}rschner,

Patrick},

title = {Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration},

journal = {Numerical Linear Algebra with Applications},

issn = {1099-1506},

url = {http://onlinelibrary.wiley.com/doi/10.1002/nla.1945/},

doi = {10.1002/nla.1945},

pages = {175--196},

volume={22},

number={1},

year = {2014},

}

Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration;

Freitag, Melina; Kürschner, Patrick;*Numerical Linear Algebra with Applications*: 22(1), pp. 175--196;

2015.@inproceedings{FreK13,

Title = {Inner-outer methods for large-scale two-sided eigenvalue problems},

Author = {Melina Freitag and Patrick K{\"u}rschner},

Booktitle = {Oberwolfach Report - Numerical Solution of PDE Eigenvalue Problems},

Pages = {86--89},

Year = {2013},

Volume = {56},

doi = {10.4171/OWR/2013/56} }

Inner-outer methods for large-scale two-sided eigenvalue problems;

Freitag, Melina; Kürschner, Patrick;*Numerical Solution of PDE Eigenvalue Problems*: Oberwolfach Report 56/2013;

2013.

doi:10.4171/OWR/2013/56.@INPROCEEDINGS{BenHK11,

author={Benner,

P. and Hochstenbach,

M.E. and K\"u rschner,

P.},

booktitle={Communications,

Computing and Control Applications (CCCA),

2011 International Conference on},

title={Model order reduction of large-scale dynamical systems with {J}acobi-{D}avidson style eigensolvers},

year={2011},

month={march},

volume={},

number={},

pages={1 -6},

doi={10.1109/CCCA.2011.6031208},

url={http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6031208} }

Model order reduction of large-scale dynamical systems with Jacobi-Davidson style eigensolvers;

Benner, Peter; Hochstenbach, Michiel; Kürschner, Patrick;*International Conference on Communications, Computing and Control Applications, 2011 (CCCA11)*:

IEEE; 2011. ISBN/ISSN: 978-1-4244-9796-6

IEEE Catalog Number: CFP1154M-ART

DOI: 10.1109/CCCA.2011.6031208 .@article{BenHK11,

author = {Benner,

P. and K\"urschner,

P. and Hochstenbach,

M.},

title = {{T}wo-sided harmonic subspace extractions for the generalized eigenvalue problem},

journal = {PAMM},

volume = {11},

number = {1},

publisher = {WILEY-VCH Verlag},

issn = {1617-7061},

url = {http://dx.doi.org/10.1002/pamm.201110359},

doi = {10.1002/pamm.201110359},

pages = {739--740},

year = {2011},

}

Two-sided harmonic subspace extractions for the generalized eigenvalue problem;

Benner, Peter; Hochstenbach, Michiel; Kürschner, Patrick;*Proceedings in Applied Mathematics and Mechanics*: Volume 11, Issue 1, pages 741–742, December 2011;

Wiley InterScience; 2011.

doi: 10.1002/pamm.201110359.@MASTERSTHESIS{Kuer10,

author = {P. K\"urschner},

title = {Two-Sided Eigenvalue Methods for Modal Approximation},

school = {Chemnitz University of Technology,

Department of Mathematics,

Germany},

year = {2010},

owner = {kupa},

timestamp = {2010.07.16} }

Two-Sided Eigenvalue Methods for Modal Approximation;

Patrick Kürschner;

Chemnitz University of Technology; 2010. **Related talks:**P. Kürschner

Inner-outer methods for large-scale two-sided eigenvalue problems,

*Oberwolfach Workshop on Numerical Solution of PDE Eigenvalue Problems*, Mathematisches Forschungsinstitut Oberwolfach, November 17-23, 2013P. Benner

The Symplectic Lanczos Process for Hamiltonian-Positive Matrices,

*GAMM 84th Annual Scientific Conference*, Novi Sad, Serbia, March 18-22, 2013P. Kürschner

Solving inhomogeneous eigenvalue problems,

*GAMM 84th Annual Scientific Conference*, Novi Sad, Serbia, March 18-22, 2013P. Kürschner

On inexact dominant pole based model truncation,

*Bath Numerical Analysis Seminar*, Bath, UK, Septermber 4, 2012

and*GMA / GAMM Workshops*, Anif, Austria, September, 17-21, 2012P. Kürschner

Tuned preconditioners for inexact two-sided inverse and Rayleigh quotient iteration,

*2012 SIAM Confernce on Applied Linear Algebra*, Valencia, Spain, June 18-22, 2012

and*3rd IMA Conference on Numerical Linear Algebra and Optimisation*, University of Birmingham, UK, September 10-12, 2012P. Kürschner

Dominant pole computation of MIMO second order systems,

*17th ILAS Conference*,Braunschweig, August 22-26, 2011P. Kürschner

Two-sided harmonic subspace extractions for the generalized eigenvalue problem,*GAMM 82nd Annual Scientific Conference*, Graz, Austria, April 18-21, 2011P. Kürschner

Model order reduction of large-scale dynamical systems with Jacobi-Davidson style eigensolvers,

*2011 International Conference on Communications, Computing and Control Applications (CCCA’11),*, Hammamet, Tunisia, March 3-5, 2011P. Kürschner

Modern Numerical Methods for Large-Scale Eigenvalue Problems,

*Welcome colloquium dedicated to Prof. Benner*, Max Planck Institute Magdeburg, November 5, 2010