## Fachbereich Mathematik

### Refine

#### Faculty / Organisational entity

- Fachbereich Mathematik (808)
- Fraunhofer (ITWM) (2)

#### Year of publication

#### Document Type

- Preprint (542)
- Doctoral Thesis (175)
- Report (35)
- Article (24)
- Diploma Thesis (20)
- Lecture (6)
- Study Thesis (2)
- Working Paper (2)
- Bachelor Thesis (1)
- Periodical (1)

#### Language

- English (808) (remove)

#### Keywords

- Wavelet (12)
- Inverses Problem (10)
- Mehrskalenanalyse (8)
- Boltzmann Equation (7)
- Location Theory (7)
- Approximation (6)
- Navier-Stokes-Gleichung (6)
- Elastoplastizität (5)
- Numerical Simulation (5)
- Algebraische Geometrie (4)

- Maximum entropy for reduced moment problems (1999)
- The existence of maximum entropy solutions for a wide class of reduced moment problems on arbitrary open subsets of Rd is considered. In particular, new results for the case of unbounded domains are obtained. A precise condition is presented under which solvability of the moment problem implies existence of a maximum entropy solution.

- The Quantum Zero Space Charge Model for Semiconductors (1999)
- The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain \(\Omega\subset I\!\!R^d,d=1,2\) or \( d=3\) is the minimizer of the total energy \({\mathcal E}_{\epsilon\lambda}\); \({\mathcal E}_{\epsilon\lambda}\) involves the squares of the scaled Planck's constant \(\epsilon\) and the scaled minimal Debye length \(\lambda\). In applications one frequently has \(\lambda^2\ll 1\). In these cases the zero-space-charge approximation is rigorously justified. As \(\lambda \to 0 \), the particle densities converge to the minimizer of a limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal energies one gets an differential-algebraic system for the limiting \((\lambda=0)\) particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit \(\epsilon \to 0\) exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small \(\epsilon\), a reasonable approximation of the quantum model if and only if the quantum gap vanishes. The simultaneous limit \(\epsilon =\lambda \to 0\) is analyzed.

- A General Hilbert Space Approach to Wavelets and Its Application in Geopotential Determination (1999)
- A general approach to wavelets is presented within a framework of a separable functional Hilbert space H. Basic tool is the construction of H-product kernels by use of Fourier analysis with respect to an orthonormal basis in H. Scaling function and wavelet are defined in terms of H-product kernels. Wavelets are shown to be 'building blocks' that decorrelate the data. A pyramid scheme provides fast computation. Finally, the determination of the earth's gravitational potential from single and multipole expressions is organized as an example of wavelet approximation in Hilbert space structure.

- Constructive Approximation and Numerical Methods in Geodetic; Research Today - An Attempt of a Categorization Based on anUncertainty Principle (1999)
- This review article reports current activities and recent progress on constructive approximation and numerical analysis in physical geodesy. The paper focuses on two major topics of interest, namely trial systems for purposes of global and local approximation and methods for adequate geodetic application. A fundamental tool is an uncertainty principle, which gives appropriate bounds for the quantification of space and momentum localization of trial functions. The essential outcome is a better understanding of constructive approximation in terms of radial basis functions such as splines and wavelets.

- A Singular-Perturbed Two-Phase Stefan Problem Due to Slow Diffusion (1999)
- The asymptotic behaviour of a singular-perturbed two-phase Stefan problem due to slow diffusion in one of the two phases is investigated. In the limit the model equations reduce to a one-phase Stefan problem. A boundary layer at the moving interface makes it necessary to use a corrected interface condition obtained from matched asymptotic expansions. The approach is validated by numerical experiments using a front-tracking method.

- The Stationary Current-Voltage Characteristics of the Quantum Drift Diffusion Model (1999)
- This paper is concerned with numerical algorithms for the bipolar quantum drift diffusion model. For the thermal equilibrium case a quasi-gradient method minimizing the energy functional is introduced and strong convergence is proven. The computation of current - voltage characteristics is performed by means of an extended emph{Gummel - iteration}. It is shown that the involved fixed point mapping is a contraction for small applied voltages. In this case the model equations are uniquely solvable and convergence of the proposed iteration scheme follows. Numerical simulations of a one dimensional resonant tunneling diode are presented. The computed current - voltage characteristics are in good qualitative agreement with experimental measurements. The appearance of negative differential resistances is verified for the first time in a Quantum Drift Diffusion model.

- A Finite Difference Interpretation of the Lattice Boltzmann Method (1999)
- Compared to conventional techniques in computational fluid dynamics, the lattice Boltzmann method (LBM) seems to be a completely different approach to solve the incompressible Navier-Stokes equations. The aim of this article is to correct this impression by showing the close relation of LBM to two standard methods: relaxation schemes and explicit finite difference discretizations. As a side effect, new starting points for a discretization of the incompressible Navier-Stokes equations are obtained.

- Remarks on Orthogonal Polynomials and Balanced Realizations (1992)
- Given a proper antistable rational transfer function g, a balanced realization of g is contructed as a matrix representation of the abstract shift realization introduced in Fuhrmann [1976]. The required basis is constructed as a union of sets of polynomials orthogonal with respect to weights given by the square of the absolute values of minimal degree Schmidt vectors of the corresponding Hankel operators. This extends results of Fuhrmann [1991], obtained in the generic case.