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In this thesis, we deal with the worst-case portfolio optimization problem occuring in discrete-time markets.
First, we consider the discrete-time market model in the presence of crash threats. We construct the discrete worst-case optimal portfolio strategy by the indifference principle in the case of the logarithmic utility. After that we extend this problem to general utility functions and derive the discrete worst-case optimal portfolio processes, which are characterized by a dynamic programming equation. Furthermore, the convergence of the discrete worst-case optimal portfolio processes are investigated when we deal with the explicit utility functions.
In order to further study the relation of the worst-case optimal value function in discrete-time models to continuous-time models we establish the finite-difference approach. By deriving the discrete HJB equation we verify the worst-case optimal value function in discrete-time models, which satisfies a system of dynamic programming inequalities. With increasing degree of fineness of the time discretization, the convergence of the worst-case value function in discrete-time models to that in continuous-time models are proved by using a viscosity solution method.

Magnetoelastic coupling describes the mutual dependence of the elastic and magnetic fields and can be observed in certain types of materials, among which are the so-called "magnetostrictive materials". They belong to the large class of "smart materials", which change their shape, dimensions or material properties under the influence of an external field. The mechanical strain or deformation a material experiences due to an externally applied magnetic field is referred to as magnetostriction; the reciprocal effect, i.e. the change of the magnetization of a body subjected to mechanical stress is called inverse magnetostriction. The coupling of mechanical and electromagnetic fields is particularly observed in "giant magnetostrictive materials", alloys of ferromagnetic materials that can exhibit several thousand times greater magnitudes of magnetostriction (measured as the ratio of the change in length of the material to its original length) than the common magnetostrictive materials. These materials have wide applications areas: They are used as variable-stiffness devices, as sensors and actuators in mechanical systems or as artificial muscles. Possible application fields also include robotics, vibration control, hydraulics and sonar systems.
Although the computational treatment of coupled problems has seen great advances over the last decade, the underlying problem structure is often not fully understood nor taken into account when using black box simulation codes. A thorough analysis of the properties of coupled systems is thus an important task.
The thesis focuses on the mathematical modeling and analysis of the coupling effects in magnetostrictive materials. Under the assumption of linear and reversible material behavior with no magnetic hysteresis effects, a coupled magnetoelastic problem is set up using two different approaches: the magnetic scalar potential and vector potential formulations. On the basis of a minimum energy principle, a system of partial differential equations is derived and analyzed for both approaches. While the scalar potential model involves only stationary elastic and magnetic fields, the model using the magnetic vector potential accounts for different settings such as the eddy current approximation or the full Maxwell system in the frequency domain.
The distinctive feature of this work is the analysis of the obtained coupled magnetoelastic problems with regard to their structure, strong and weak formulations, the corresponding function spaces and the existence and uniqueness of the solutions. We show that the model based on the magnetic scalar potential constitutes a coupled saddle point problem with a penalty term. The main focus in proving the unique solvability of this problem lies on the verification of an inf-sup condition in the continuous and discrete cases. Furthermore, we discuss the impact of the reformulation of the coupled constitutive equations on the structure of the coupled problem and show that in contrast to the scalar potential approach, the vector potential formulation yields a symmetric system of PDEs. The dependence of the problem structure on the chosen formulation of the constitutive equations arises from the distinction of the energy and coenergy terms in the Lagrangian of the system. While certain combinations of the elastic and magnetic variables lead to a coupled magnetoelastic energy function yielding a symmetric problem, the use of their dual variables results in a coupled coenergy function for which a mixed problem is obtained.
The presented models are supplemented with numerical simulations carried out with MATLAB for different examples including a 1D Euler-Bernoulli beam under magnetic influence and a 2D magnetostrictive plate in the state of plane stress. The simulations are based on material data of Terfenol-D, a giant magnetostrictive materials used in many industrial applications.

Destructive diseases of the lung like lung cancer or fibrosis are still often lethal. Also in case of fibrosis in the liver, the only possible cure is transplantation.
In this thesis, we investigate 3D micro computed synchrotron radiation (SR\( \mu \)CT) images of capillary blood vessels in mouse lungs and livers. The specimen show so-called compensatory lung growth as well as different states of pulmonary and hepatic fibrosis.
During compensatory lung growth, after resecting part of the lung, the remaining part compensates for this loss by extending into the empty space. This process is accompanied by an active vessel growing.
In general, the human lung can not compensate for such a loss. Thus, understanding this process in mice is important to improve treatment options in case of diseases like lung cancer.
In case of fibrosis, the formation of scars within the organ's tissue forces the capillary vessels to grow to ensure blood supply.
Thus, the process of fibrosis as well as compensatory lung growth can be accessed by considering the capillary architecture.
As preparation of 2D microscopic images is faster, easier, and cheaper compared to SR\( \mu \)CT images, they currently form the basis of medical investigation. Yet, characteristics like direction and shape of objects can only properly be analyzed using 3D imaging techniques. Hence, analyzing SR\( \mu \)CT data provides valuable additional information.
For the fibrotic specimen, we apply image analysis methods well-known from material science. We measure the vessel diameter using the granulometry distribution function and describe the inter-vessel distance by the spherical contact distribution. Moreover, we estimate the directional distribution of the capillary structure. All features turn out to be useful to characterize fibrosis based on the deformation of capillary vessels.
It is already known that the most efficient mechanism of vessel growing forms small torus-shaped holes within the capillary structure, so-called intussusceptive pillars. Analyzing their location and number strongly contributes to the characterization of vessel growing. Hence, for all three applications, this is of great interest. This thesis provides the first algorithm to detect intussusceptive pillars in SR\( \mu \)CT images. After segmentation of raw image data, our algorithm works automatically and allows for a quantitative evaluation of a large amount of data.
The analysis of SR\( \mu \)CT data using our pillar algorithm as well as the granulometry, spherical contact distribution, and directional analysis extends the current state-of-the-art in medical studies. Although it is not possible to replace certain 3D features by 2D features without losing information, our results could be used to examine 2D features approximating the 3D findings reasonably well.