Max Planck Institute for Dynamics of Complex Technical Systems
Large Scale and Nonlinear Eigenvalue Problems
Large-Scale linear and nonlinear eigenvalue problems belong to the main fields of research in modern numerical mathematics. Approximations of a number of eigenvalues and the associated eigenvectors are usually obtained using iterative methods. In this project we focus on iterative methods which are based on the treatment of eigenvalue problems via a Newton scheme. These are, e.g., inverse- and Rayleigh quotient iteration, as well as the closely related Jacobi-Davidson methods.
Spectral norm of a transfer functions.
Spectral norm of a transfer functions.
As a special application we investigate the application of these eigensolvers in the context of model order reduction, which amounts to finding dominant poles of the transfer function of a linear, time-invariant control system. These dominant poles yield a significant contribution to the input-output dynamics of the systems and produce the peaks in the frequency response plot of the transfer function as it can be seen in the pictures below.
In particular, the following issues are investigated:
Extending and adapting existing methods to nonlinear eigenvalue problems, where the emphasis is drawn to methods based on the two-sided and generalized Jacobi-Davidson and related methods.
Exploiting the structure of certain classes of nonlinear eigenproblems, e.g. polynomial, rational, delay eigenproblems, as those occur frequently in model order reduction.
Improving the efficiency in both the linear and nonlinear case by solving the occurring linear systems of equations inexactly using Krylov subspace methods. This includes the determination and application of adequate preconditioners.
Application of these methods to nonlinear eigenproblems occurring in natural sciences outside the model order reduction context, for instance, mechanics, chemistry, etc.
Three- and two-dimensional view of the surface of the transfer functions spectral norm, with highlighted dominant eigenvalues.
Three- and two-dimensional view of the surface of the transfer functions spectral norm, with highlighted dominant eigenvalues.
