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: Thomas Richter (Otto von Guericke University Magdeburg)
Funded by: Deutsche Forschungsgemeinschaft (DFG), DFG-GRK 2297
Funding period: 2017-2021
Contact: Peter Benner
Project description: MathCoRe stands for Mathematical Complexity Reduction - a Research Training Group (RTG) located at Otto-von-Guericke-Universität Magdeburg (OvGU). The RTG is a Graduiertenkolleg (DFG-GRK 2297) funded by Deutsche Forschungsgemeinschaft (DFG). Headed by the Faculty of Mathematics (FMA) it is run as a cooperation with the Faculty of Electrical Engineering and Information Technology (FEIT) and the Max Planck Institute for the Dynamics of Complex Technical Systems(MPI)

The combination of expertise from different mathematical areas under the theme of Complexity Reduction provides the RTG with a unique profile that specifically shapes the scientific understanding of the young researchers graduating within the RTG. A fundamental goal of our Philosophy is to make the PhD students work on projects that connect several mathematical areas and to let them profit from supervision by two principal investigators with different mathematical backgrounds. more

Tensor Methods for Machine Learning

Partners: Martin Stoll (TU Chemnitz), Sergey Dolgov (U Bath)
Funded by: IMPRS-ProEng
Contact: Peter Benner, Kirandeep Kour
Project description: Low-rank tensor decomposition methods are the crucial algorithms to reduce computational cost while preserving the important information and structure of multidimensional data. Therefore, in this project we develop new tensor methods to build efficient and stable machine learning models (tensorized kernel methods). Mainly, we focus on "generalization" and "stability" issues in SciML.

Structured (Hamiltonian, even) Eigenvalue Problems

Partners: Heike Faßbender, Philip Saltenberger (TU Braunschweig), Federico Poloni (U Pisa), Yuji Nakatsuksa (U Oxford), Vasile Sima (ICI Bucarest), Carolin Penke
Funded by: MPI DyktS
Contact: Peter Benner
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.

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), Carolin Penke
Funded by: MPI DyktS, MPI MIS and MPCDF
Funding period: MPI DyktS, MPI MIS (2018-2020) and MPCDF (2017-2020)
Contact: Peter Benner
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.

Customized Numerical Linear Algebra and Optimization for Vibrating Systems

Partners: Zoran Tomljanović (University of Osijek), Davide Palitta (University of Bologna)
Funded by: DAAD
Contact:
Jennifer Przybilla, Tim Mitchell, Jens Saak, Peter Benner 
Project description: 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.

Communication-Avoiding Low-Rank Tensor Computations

Partners: Grey Ballard (Wake Forest University), Hussam Al Daas (U Oxford)
Funded by: MPI DyktS
Contact: Peter Benner
Project description: In this project, we are interested in developing and providing highly scalable and robust parallel algorithms and solvers for low-rank tensor computations.
Partners: Peter Benner (MPI Magdeburg, OVGU), Sebastian Sager (OVGU), Kai Sundermacher (MPI Magdeburg, OVGU), Martin Stoll (TU Chemnitz)
Industrial partners:  AVACON und BASF
Funded by: BMBF
Contact: Peter Benner, Shaimaa Monem

 

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Large Dense Eigenvalue Problems

Funded by: MPI DyktS
Contact: Martin Köhler, Peter Benner
Project description: Many simulation and modelling tasks result in large-scale eigenvalue problems. Although most of them are sparse and can be handled with appropriate iterative solvers, some of them are described via dense matrices. Due to the runtime complexity of standard dense eigenvalues solvers, these problems are often considered infeasible to solve for large dimensions. In this project, we aim at the development of matrix function based divide and conquer algorithms for the solution of large and dense eigenvalues problems. We employ scalable level-3 BLAS matrix operations as main building blocks, such that an acceleration using GPUs is easily possible.

Concluded Projects

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 DyktS
Contact: Davide Palitta (2018-2021, current: University of Bologna, Italy)
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.

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

Partners: Boris Khoromskij (MPI, Leipzig), Venera Khoromskaia (MPI, Leipzig)
Funded by: MPI DyktS
Contact: Peter Benner, Cleophas Kweyu (2014-2019, current: Moi University, Kenya)

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 DyktS, MPI MIS
Contact: Peter Benner, Venera Khoromskaia

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 (2014-2016, current: Tsinghua University, Beijing, P.R.China), Cleophas Kweyu (2014-2019, current: Moi University, Kenya)

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