Team Leader NLMA

Prof. Dr. Peter Benner
Prof. Dr. Peter Benner
Phone: +49 391 6110 450
Fax: +49 391 6110 453

Numerical Linear and Multilinear Algebra

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. 

 

Current Projets

Projects Scientists Funded by Partners
Numerical methods for nonlinear eigenvalue problems Peter Benner, Xin Liang DAAD PPP (Croatia) Ninoslav Truhar, Suzana Miodragović (J.J. Strossmayer Univerity Osijek)

Structured (Hamiltonian, even) eigenvalue problems

Peter Benner MPI Zvonimir Bujanović (U Zagreb), Heike Faßbender (TU Braunschweig), Volker Mehrmann, Matthias Voigt (TU Berlin), Hongguo Xu (U Kansas, Lawrence, KS), Chao Yang (LLBL Berkeley, CA)
Rank-structured eigenproblems (in electronic structure calculation) Peter Benner, Sergey Dolgov, Venera Khoromskaia (co-financed by MPI MIS, Leipzig) MPI, MPI MIS Steffen Börm (CAU Kiel), Boris Khoromskii (MPI MIS, Leipzig), Thomas Mach (KU Leuven), Chao Yang (LLBL Berkeley, CA)
Recycling techniques for Krylov subspace solvers Peter Benner, Lihong Feng MPI Kapil Ahuja (IIT Indore), Eric de Sturler (Virginia Tech)
Preconditioners for saddle point problems in PDE-constrained optimization and control Peter Benner, Sergey Dolgov, Akwum Onwunta, Jens Saak, Heiko Weichelt DFG SPP1253, IMPRS, MPI Martin Stoll (MPI/NDS) 
Tensor techniques for high-dimensional linear systems Peter Benner, Sergey Dolgov, Akwum Onwunta CDS, IMPRS Martin Stoll (MPI/NDS) 
 
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