# Research

### Variational Characterization for Nonlinear Eigenproblems

For a wide class of linear selfadjoint operators *A* on a Hilbert space the eigenvalues of the
linear eigenvalue problem can be characterized by three
fundamental variational principles, namely by Rayleigh’s principle, by Poincaré’s minmax characterization,
and by the maxmin principle of Courant, Fischer and Weyl.

In this project we study corresponding characterizations for nonlinear egenproblems.
They are applied in vibrations of fluid–solid structures, a priori bounds for automated multi–level structuring,
locking strategies for iterative projection methods, and electronic structure calculation.

### Fluid-Structure Interaction

Free vibrations of fluid-solid structures are governed by linear eigenvalue problems

which are nonsymmetric, but are known to have real eigenvalues.

They are often tackled in the following way: one first solves the symmetric
eigenproblems governing free vibrations of the
fluid and of the structure
independently by an efficient method like Lanczos or AMLS, and then the
original problem is projected to the space spanned by the obtained eigenmodes.
Hence, the coupling is not considered when constructing the search
space, but only in the projected problem. In this project we study variational
characterizations of the original problem, and resulting structure preserving
iterative projection methods and variants of AMLS which incorporate the
coupling already into the reduction process.

### Iterative Projection Methods for Nonlinear Eigenproblems

For linear sparse eigenproblems very efficient methods
are iterative projection methods, where approximations to the wanted eigenvalues and eigenvectors are obtained from projections
of the eigenproblem to subspaces of small dimension which are expanded in the course of the algorithm.

In this project we develop methods of this type for nonlinear eigenvalue problems of Arnoldi and Jacobi–Davidson type.
Since no normal forms are known for this type of problems, a crucial point when approximating more than one eigenvalue is to
inhibit the method to converge to the same eigenvalue repeatedly.

### Automated Multi-Level Substructuring

Over the last few years, a new method for performing frequency response and eigenvalue analysis of complex
finite element (FE)
structures known as Automated Multi-Level Substructuring (AMLS for short) has been developed by Bennighof an Co-workers.

In AMLS the large finite element model is recursively divided into many substructures on several levels based on the sparsity
structure of the system matrices. Assuming that the interior degrees of freedom of substructures depend quasistatically on the
interface degrees of freedom, and modeling the deviation from quasistatic dependence in terms of a small number of selected
substructure eigenmodes, the size of the finite element model is reduced substantially yet yielding satisfactory accuracy over
a wide frequency range of interest. Recent studies in vibro-acoustic analysis of passenger car bodies where very large FE models
with more than one million degrees of freedom appear and several hundreds of eigenfrequencies and eigenmodes are needed have shown
that AMLS is considerably faster than Lanczos type approaches for this type of problems.

Taking advantage of a variational characterization of eigenvalues
for nonlinear eigenproblems we proved an a priori bound for AMLS. Our plan is to extend it to an posteriori bound and to develop
efficient cut-off strategies for AMLS. Moreover, we consider generalizations of AMLS to nonlinear eigenproblems

The efficiency of this approach was demonstrated by a gyroscopic eigenproblem modeling a rotating tire, and rational eigenproblems modeling vibrations of a fluid–solid structure and damped vibrations of a structure.

### Electron energy level calculations for semiconductor nanostructures

Semiconductor nanostructures have attracted tremendous
interest in the past few years because of their special physical properties and their potential for applications in micro–
and optoelectronic devices. In such nanostructures, the free carriers are confined to a small region of space by potential barriers,
and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels.
The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus
reducing the degrees of freedom to zero.

We consider the problem to compute relevant energy states and corresponding wave functions of (three dimensional)
semiconductor nanostructures. The governing equation characterizing the energy states *E* and corresponding wave
functions is the Schrödinger equation

where the confinement potential *V* and the electron effective mass *m* are
discontinous across the interface between the dot and the matrix. Assuming non-parabolicity for the electron effective mass,
*m* depends nonlinearly on the energy state *E*, and the Schrödinger equation becomes a rational eigenvalue
problem. Discretizing by FE or FV methods it results in a nonlinear matrix eigenvalue problem which is typically large and sparse.
Iterative projection methods of Arnoldi and Jacobi-Davidson type are very efficient. One of the ongoing projects is to apply the
automated multi-level substructuring to this type of problems.

### Model reduction based on hierarchical substructuring

Modelling time-invariant linear systems often yields state space models of
very large dimension, e.g. if they are obtained by discretizing a partial differential
equation. Model reduction is an ubiquitous tool in analysis and
simulation of such problems. Here the original system is approximated by
one of significantly smaller order which preserves the system characteristics (e.g. stability, minimality).

In this project we study order reduction methods which are based on hierarchical
substructuring as used in AMLS for huge eigenvalue problems. The
original problem is recursively divided into smaller substructures on several
levels, and known order reduction methods are applied to LTI-systems corresponding
to these substructures. Thus, a basis of an appropriate subspace is
constructed to which the original problem is projected.

### Methods for regularized total least squares problems based on eigensolvers

The total least squares method is a successful approach for linear problems
if both the system matrix A and the right hand side b are contaminated
by some noise. Quite often these problems are ill-posed. Then least
squares or total least squares methods often yield physically meaningless solutions,
and regularization is necessary to stabilize the solution.

In this project we study and analyze methods which are based on the solution
of a sequence of linear or quadratic eigenproblems. We pay particular
attention to the reuse of information from previous iteration steps which may
accelerate the methods considerably.