Numerical Linear and Multilinear Algebra

We study numerical methods for linear and nonlinear eigenvalue problems. This includes the development and analysis of new algorithms, (backward) error analysis, and the derivation of the associated (relative) perturbation theory. Special attention is given to linear, generalized, and polynomial eigenproblems with spectral symmetries. Special cases include:
  • linear eigenproblems for Hamiltonian and symplectic matrices,
  • generalized eigenproblems for skew-Hamiltonian/Hamiltonian, even, and positive definite matrix pencils,
  • as well as even, gyroscopic, and hyperbolic polynomial eigenvalue problems.
Such problems often arise in systems, control and stability theory, FE analysis of corner singularities, discrete approximations to the Schrödinger equation such as the Hartree-Fock and Bethe-Salpeter equations, and many other areas. Another important class of structured eigenproblems investigated by the NLMA team is related to rank-structured matrices and matrix pairs. This includes H- and H2-matrices resulting from FEM and BEM discretizations of PDE eigenvalue problems, but also matricizations of tensor equations in electronic structure calculation.    

Moreover, we investigate the solution of special linear systems of equations arising in PDE control and model reduction algorithms. This includes in particular
  • recycling techniques for Krylov subspace solvers for systems with multiple-right hand sides and constant (or slowly varying) coefficient matrices,
  • preconditioning techniqes for saddle point problems, and
  • using tensor techniques to solve high-dimensional problems like stochastic Galerkin systems.

Projects

Partners: Prof. Dr. Thomas Richter (Otto von Guericke University Magdeburg)
Funded by: Deutsche Forschungsgemeinschaft (DFG), DFG-GRK 2297
Contact: Roman Weinhandl, Peter Benner more
<p>Structured (Hamiltonian, even) eigenvalue problems</p>

Structured (Hamiltonian, even) eigenvalue problems

Partners:  Heike Faßbender, Philip Saltenberger (TU Braunschweig), Federico Poloni (U Pisa), Yuji Nakatsuksa (U Oxford), Vasile Sima (ICI Bucarest), Matthias Voigt (TU Berlin), Hongguo Xu, (U Kansas, Lawrence, KS)  
Funded by: MPI
Contact: Peter Benner, Carolin Penke
Project descriptionStructured eigenvalue problems are at the heart of various applications in science and engineering including model order reduction, control engineering and electronic structure calculations of physical materials. We aim to exploit available structure leading to more efficient and more accurate algorithms.

<p>Efficient solvers for the Bethe-Salpeter equations</p>

Efficient solvers for the Bethe-Salpeter equations

PartnersAndreas Marek, Markus Rampp (MPCDF Garching), Claudia Draxl (HU Berlin), Chao Yang (Berkeley Labs, CA), Heike Faßbender (TU Braunschweig)
Funded by: MPI, MPI MIS and MPCDF
Funding period: MPI, MPI MIS (2018-2020) and MPCDF (2017-2020)
Contact: Peter Benner, Carolin Penke
Project description: Ab initio spectroscopy aims to calculate optical properties of novel materials without the need for empirical measurements. A state-of-the-art approach employs the Bethe-Salpeter equation as derived in many-body perturbation theory, describing the electron-hole propagation of a system. We develop solution strategies to deal with the resulting large-scale eigenvalue problems suitable for high performance computing environments.
<span>Advanced Krylov Subspace Methods </span>

Advanced Krylov Subspace Methods 

Partners:  Valeria Simoncini (Università di Bologna), Stefano Massei (Eindhoven University of Technology), Daniel Kressner (EPFL), Kathryn Lund (Charles University)
Funded by: MPI
Contact: Davide Palitta
Project description:  

Krylov subspace methods are one of the most classic yet powerful algorithms in numerical linear and multi-linear algebra. Their range of applications spans from the solution of eigenvalue problems and algebraic equations like linear systems, matrix, and tensor equations, to the numerical evaluation of matrix functions. Here we mainly focus on the new generation of Krylov subspace methods, namely extended and rational (block) Krylov routines, which include standard (block) polynomial Krylov methods as a specific case. We apply these methods to a variety of algebraic problems. A particular emphasis is given to the solution of large-scale matrix equations which arise, e.g., from the discretization of partial differential equations and as intermediate steps within certain model order reduction techniques.

<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.
<p><span>Communication-avoiding low rank tensor computations<br /><br /></span></p>

Communication-avoiding low rank tensor computations

Partners: Grey Ballard (Wake Forest University)
Funded by: MPI Magdeburg
Contact: Peter Benner, Hussam Al Daas
Project descriptionIn this project, we are interested in developing and providing highly scalable and robust parallel algorithms and solvers for low rank tensor computations.
P2Chem: New mixed-integer optimization methods for efficient synthesis and flexible management from power-to-chemicals processes
Partners: Prof. Dr. Peter Benner (MPI Magdeburg, OVGU), Prof. Dr. Sebastian Sager (OVGU), Prof. Dr. Kai Sundermacher MPI Magdeburg, OVGU)Prof. Dr. Martin Stoll (TU Chemnitz)
Industrial partners:  AVACON und BASF
Funded by: BMBF
Contact: Peter Benner, Shaimaa Monem

 

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Concluded Projects

Regularization of the Poisson-Boltzmann Equation by the Range-Separated Tensor Format

Regularization of the Poisson-Boltzmann Equation by the Range-Separated Tensor Format

Partners: Boris Khoromskij (MPI, Leipzig), Venera Khoromskaia (MPI, Leipzig)
Funded by: MPI Magdeburg
Contact: Peter Benner, Cleophas Kweyu (former member)
Eigenvalue problems with rank structure

Eigenvalue problems with rank structure

Partners: Steffen Börm (CAU Kiel), Boris Khoromskii (MPI MIS), Thomas Mach (KU Leuven), Chao Yang (Berkeley Lab, CA), Sergey Dolgov (U Bath, UK)
Funded by: MPI, MPI MIS
Contact: Peter Benner, Venera Khoromskaia
<p>Numerical methods of nonlinear eigenvalue problems</p>

Numerical methods of nonlinear eigenvalue problems

Partners: Ninoslav Truhar, Suzana Miodragović (J.J. Strossmayer Univerity Osijek)
Funded by: DAAD PPP (Croatia)
Contact: Peter Benner, Xin Liang

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