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.
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.
|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)|