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. Moreover, we investigate the solution of special linear systems of equations arising in PDE control and model reduction algorithms.
In order to get a deep insight into the underlying process, dynamics, structure or devices, modeling and simulation are unavoidable in many research and application fields. The resulting mathematical models are usually in the form of partial differential equations. To simulate such models, spatial (-time) discretization is necessary, which results in large-scale, complex systems with enormous number of equations. The simulation becomes time-consuming because of the large scale and complexity of the systems.
Developed from well-established mathematical theories and robust numerical algorithms, model order reduction (MOR) has been recognized as an efficient tool for fast simulation. Using model order reduction, small systems of far less number of equations (reduced models) are derived, and can substitute the original large-scale systems for simulation. As a result, the simulation can be sped up by several orders of magnitude.
One of our core areas is the computed aided control of system governed by differential equations. We in particular focus on the numerical aspects of optimal control problems with PDE constraints. We concentrate on the efficient solution of the underlying matrix equations in the form of linear saddle point systems or matrix Riccati equations. Additionally, we develop numerical algorithms for robust control and stabilization of descriptor systems that are obtained from the discretization of differential equations with additional constraints. We focus on the construction of (suoptimal H-infinity controllers or the computation of system norms by using spectral information of certain structured matrix pencils. All of the above mentioned techniques are accompanied by the incorporation of our algorithms into efficient and robust software environments.
Modeling and simulation of fluids in large networks is a challenging problem, especially if the simulation is used to provide solutions to optimal control problems or other optimization questions. Our first goal is to be able to simulate this systems in a stable and efficient manner as truely transient systems, considering in particular the underlying network stucture.
Matrix equations of the one kind or the other are a central tool in all kinds of applications. In optimal control the linear quadratic control problem features a feedback solution that is determined via the solution of an algebraic Ricatti equation. The system Gramians of a linear time invariant dynamical system are te solutions of two adjoint Lyapunov equations. Ricatti, Lyapunov and Sylvester equations play an important role in different model order reduction techniques for continuous time linear dynamical systems. They all have discrete time counterparts as e.g. the well known Stein equations. Krylov subspace and eigenvalue methods can be related to certain Sylvester equations as well. Nearly all other research fields of the CSC group involve matrix equation computations at a certain point. The work of this team is dedicated to the efficient numerical solution of the variety of those equations in all kinds of working environments.
Scientific Computing is an interdisciplinary field of research including
Computer Science, and
where researchers of all fields jointly work on the computer based solution of active applications. Besides the cassical questions of parallel implementation, energy-aware formulations of all kinds of computational methods and exploitation of special purpose low power hardware are becoming more and more important.