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#### Faculty / Organisational entity

Algebraic Systems Theory
(2004)

Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients).

Ein maßgeschneidertes Kommunikationssystem für eine mobile Applikation mit Dienstgüteanforderungen
(2004)

In diesem Beitrag wird die Maßschneiderung eines Ad-Hoc-Kommunikationssystems zur Fernsteuerung eines Luftschiffs über WLAN vorgestellt. Dabei steht die Dienstunterstützung bei der Übertragung mehrerer Datenströme im Vordergrund. Es werden verschiedene Dienstgütemechanismen erklärt und deren Entwicklung und Integration in ein Kommunikationsprotokoll mit Hilfe eines komponentenbasierten Ansatzes genauer erläutert.

We present a methodology to augment system safety step-by-step and illustrate the approach by the definition of reusable solutions for the detection of fail-silent nodes - a watchdog and a heartbeat. These solutions can be added to real-time system designs, to protect against certain types of system failures. We use SDL as a system design language for the development of distributed systems, including real-time systems.

IMRT planning on adaptive volume structures – a significant advance of computational complexity
(2004)

In intensity-modulated radiotherapy (IMRT) planning the oncologist faces the challenging task of finding a treatment plan that he considers to be an ideal compromise of the inherently contradictive goals of delivering a sufficiently high dose to the target while widely sparing critical structures. The search for this a priori unknown compromise typically requires the computation of several plans, i.e. the solution of several optimization problems. This accumulates to a high computational expense due to the large scale of these problems - a consequence of the discrete problem formulation. This paper presents the adaptive clustering method as a new algorithmic concept to overcome these difficulties. The computations are performed on an individually adapted structure of voxel clusters rather than on the original voxels leading to a decisively reduced computational complexity as numerical examples on real clinical data demonstrate. In contrast to many other similar concepts, the typical trade-off between a reduction in computational complexity and a loss in exactness can be avoided: the adaptive clustering method produces the optimum of the original problem. This flexible method can be applied to both single- and multi-criteria optimization methods based on most of the convex evaluation functions used in practice

We consider the contact of two elastic bodies with rough surfaces at the interface. The size of the micropeaks and valleys is very small compared with the macrosize of the bodies’ domains. This makes the direct application of the FEM for the calculation of the contact problem prohibitively costly. A method is developed that allows deriving a macrocontact condition on the interface. The method involves the twoscale asymptotic homogenization procedure that takes into account the microgeometry of the interface layer and the stiffnesses of materials of both domains. The macrocontact condition can then be used in a FEM model for the contact problem on the macrolevel. The averaged contact stiffness obtained allows the replacement of the interface layer in the macromodel by the macrocontact condition.

A spectral theory for constituents of macroscopically homogeneous random microstructures modeled as homogeneous random closed sets is developed and provided with a sound mathematical basis, where the spectrum obtained by Fourier methods corresponds to the angular intensity distribution of x-rays scattered by this constituent. It is shown that the fast Fourier transform applied to three-dimensional images of microstructures obtained by micro-tomography is a powerful tool of image processing. The applicability of this technique is is demonstrated in the analysis of images of porous media.

No doubt: Mathematics has become a technology in its own right, maybe even a key technology. Technology may be defined as the application of science to the problems of commerce and industry. And science? Science maybe defined as developing, testing and improving models for the prediction of system behavior; the language used to describe these models is mathematics and mathematics provides methods to evaluate these models. Here we are! Why has mathematics become a technology only recently? Since it got a tool, a tool to evaluate complex, "near to reality" models: Computer! The model may be quite old - Navier-Stokes equations describe flow behavior rather well, but to solve these equations for realistic geometry and higher Reynolds numbers with sufficient precision is even for powerful parallel computing a real challenge. Make the models as simple as possible, as complex as necessary - and then evaluate them with the help of efficient and reliable algorithms: These are genuine mathematical tasks.

The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbitheight is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analyzed numerically.

The article is concerned with the modelling of ionospheric current systems from induced magnetic fields measured by satellites in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered vectorial operator equation. Based on the knowledge of the singular system a regularization technique in terms of certain product kernels and corresponding convolutions can be formed. In order to reconstruct ionospheric current systems from satellite magnetic field data, an inversion of the Biot-Savart's law in terms of multiscale regularization is derived. The corresponding operator is formulated and the singular values are calculated. The method is tested on real magnetic field data of the satellite CHAMP and the proposed satellite mission SWARM.

A wavelet technique, the wavelet-Mie-representation, is introduced for the analysis and modelling of the Earth's magnetic field and corresponding electric current distributions from geomagnetic data obtained within the ionosphere. The considerations are essentially based on two well-known geomathematical keystones, (i) the Helmholtz-decomposition of spherical vector fields and (ii) the Mie-representation of solenoidal vector fields in terms of poloidal and toroidal parts. The wavelet-Mie-representation is shown to provide an adequate tool for geomagnetic modelling in the case of ionospheric magnetic contributions and currents which exhibit spatially localized features. An important example are ionospheric currents flowing radially onto or away from the Earth. To demonstrate the functionality of the approach, such radial currents are calculated from vectorial data of the MAGSAT and CHAMP satellite missions.