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The application behind the subject of this thesis are multiscale simulations on highly heterogeneous particle-reinforced composites with large jumps in their material coefficients. Such simulations are used, e.g., for the prediction of elastic properties. As the underlying microstructures have very complex geometries, a discretization by means of finite elements typically involves very fine resolved meshes. The latter results in discretized linear systems of more than \(10^8\) unknowns which need to be solved efficiently. However, the variation of the material coefficients even on very small scales reveals the failure of most available methods when solving the arising linear systems. While for scalar elliptic problems of multiscale character, robust domain decomposition methods are developed, their extension and application to 3D elasticity problems needs to be further established.
The focus of the thesis lies in the development and analysis of robust overlapping domain decomposition methods for multiscale problems in linear elasticity. The method combines corrections on local subdomains with a global correction on a coarser grid. As the robustness of the overall method is mainly determined by how well small scale features of the solution can be captured on the coarser grid levels, robust multiscale coarsening strategies need to be developed which properly transfer information between fine and coarse grids.
We carry out a detailed and novel analysis of two-level overlapping domain decomposition methods for the elasticity problems. The study also provides a concept for the construction of multiscale coarsening strategies to robustly solve the discretized linear systems, i.e. with iteration numbers independent of variations in the Young's modulus and the Poisson ratio of the underlying composite. The theory also captures anisotropic elasticity problems and allows applications to multi-phase elastic materials with non-isotropic constituents in two and three spatial dimensions.
Moreover, we develop and construct new multiscale coarsening strategies and show why they should be preferred over standard ones on several model problems. In a parallel implementation (MPI) of the developed methods, we present applications to real composites and robustly solve discretized systems of more than \(200\) million unknowns.

Due to tremendous improvements of high-performance computing resources as well
as numerical advances computational simulations became a common tool for modern
engineers. Nowadays, simulation of complex physics is more and more substituting a
large amount of physical experiments. While the vast compute power of large-scale
high-performance systems enabled for simulating more complex numerical equations,
handling the ever increasing amount of data with spatial and temporal resolution
burdens new challenges to scientists. Huge hardware and energy costs desire for
ecient utilization of high-performance systems. However, increasing complexity of
simulations raises the risk of failing simulations resulting in a single simulation to be
restarted multiple times. Computational Steering is a promising approach to interact
with running simulations which could prevent simulation crashes. The large amount
of data expands gaps in the amount of data that can be calculated and the amount of
data that can be processed. Extreme-scale simulations produce more data that can
even be stored. In this thesis, I propose several methods that enhance the process
of steering, exploring, visualizing, and analyzing ongoing numerical simulations.

Many real life problems have multiple spatial scales. In addition to the multiscale nature one has to take uncertainty into account. In this work we consider multiscale problems with stochastic coefficients.
We combine multiscale methods, e.g., mixed multiscale finite elements or homogenization, which are used for deterministic problems with stochastic methods, such as multi-level Monte Carlo or polynomial chaos methods.
The work is divided into three parts.
In the first two parts we study homogenization with different stochastic methods. Therefore we consider elliptic stationary diffusion equations with stochastic coefficients.
The last part is devoted to the study of mixed multiscale finite elements in combination with multi-level Monte Carlo methods. In the third part we consider multi-phase flow and transport equations.

This thesis is concerned with a phase field model for brittle fracture.
The high potential of phase field modeling in computational fracture mechanics lies in the generality of the approach and the straightforward numerical implementation, combined with a good accuracy of the results in the sense of continuum fracture mechanics.
However, despite the convenient numerical application of phase field fracture models, a detailed understanding of the physical properties is crucial for a correct interpretation of the numerical results. Therefore, the driving mechanisms of crack propagation and nucleation in the proposed phase field fracture model are explored by a thorough numerical and analytical investigation in this work.

This thesis deals with generalized inverses, multivariate polynomial interpolation and approximation of scattered data. Moreover, it covers the lifting scheme, which basically links the aforementioned topics. For instance, determining filters for the lifting scheme is connected to multivariate polynomial interpolation. More precisely, sets of interpolation sites are required that can be interpolated by a unique polynomial of a certain degree. In this thesis a new class of such sets is introduced and elements from this class are used to construct new and computationally more efficient filters for the lifting scheme.
Furthermore, a method to approximate multidimensional scattered data is introduced which is based on the lifting scheme. A major task in this method is to solve an ordinary linear least squares problem which possesses a special structure. Exploiting this structure yields better approximations and therefore this particular least squares problem is analyzed in detail. This leads to a characterization of special generalized inverses with partially prescribed image spaces.

This thesis is divided into two parts. Both cope with multi-class image segmentation and utilize
non-smooth optimization algorithms.
The topic of the first part, namely unsupervised segmentation, is the application of clustering
to image pixels. Therefore, we start with an introduction of the biconvex center-based clustering
algorithms c-means and fuzzy c-means, where c denotes the number of classes. We show that
fuzzy c-means can be seen as an approximation of c-means in terms of power means.
Since noise is omnipresent in our image data, these simple clustering models are not suitable
for its segmentation. To this end, we introduce a general and finite dimensional segmentation
model that consists of a data term stemming from the aforementioned clustering models plus a
continuous regularization term. We tackle this optimization model via an alternating minimiza-
tion approach called regularized c-centers (RcC). Thereby, we fix the centers and optimize the
segment membership of the pixels and vice versa. In this general setting, we prove convergence
in the sense of set-valued algorithms using Zangwill’s Theory [172].
Further, we present a segmentation model with a total variation regularizer. While updating
the cluster centers is straightforward for fixed segment memberships of the pixels, updating the
segment membership can be solved iteratively via non-smooth, convex optimization. Thereby,
we do not iterate a convex optimization algorithm until convergence. Instead, we stop as soon as
we have a certain amount of decrease in the objective functional to increase the efficiency. This
algorithm is a particular implementation of RcC providing also the corresponding convergence
theory. Moreover, we show the good performance of our method in various examples such as
simulated 2d images of brain tissue and 3d volumes of two materials, namely a multi-filament
composite superconductor and a carbon fiber reinforced silicon carbide ceramics. Thereby, we
exploit the property of the latter material that two components have no common boundary in
our adapted model.
The second part of the thesis is concerned with supervised segmentation. We leave the area
of center based models and investigate convex approaches related to graph p-Laplacians and
reproducing kernel Hilbert spaces (RKHSs). We study the effect of different weights used to
construct the graph. In practical experiments we show on the one hand image types that
are better segmented by the p-Laplacian model and on the other hand images that are better
segmented by the RKHS-based approach. This is due to the fact that the p-Laplacian approach
provides smoother results, while the RKHS approach provides often more accurate and detailed
segmentations. Finally, we propose a novel combination of both approaches to benefit from the
advantages of both models and study the performance on challenging medical image data.

The research presented in this PhD thesis is a contribution to the field of anion recognition in competitive aqueous solvent mixtures. Neutral anion receptors having a cage-type architecture have been developed on the basis of triply-linked bis(cyclopeptides) and their binding properties toward various inorganic anions have been studied.
The synthetic approaches chosen to assemble the targeted container molecules rely on dynamic chemistry under the template effects of anions such as sulfate and halides. As reversible reactions metal-ligand exchange and thiol-disulfide exchange were used. Disulfide exchange has previously provided singly- and doubly-linked bis(cyclopeptide) receptors whose anion affinities in 2:1 acetonitrile/water mixtures approached the nanomolar range. Metal-ligand interactions have so far not been used to assemble bis(cyclopeptides) in our group. The cyclopeptide building blocks required for both approaches, namely cyclic hexapeptides containing alternating 6-aminopicolinic acid and either (2S,4S)-4-cyanoproline or (2S,4S)-4-thioproline subunits could be synthesized successfully.
Self-assembly of the bis(cyclopeptide) held together by coordinative interactions has been attempted by treating the cyclopeptide trinitrile with square-planar palladium (II) complexes. The reaction was followed with different NMR spectroscopic techniques. Unfortunately, none of the experiments provides conclusive evidence that the targeted triply-linked cage was indeed formed.
Bis(cyclopeptides) containing three dithiol derived linkers between the cyclopeptide rings could be synthesizes successfully. Two complexes were isolated, albeit in small amounts, one containing linkers derived from 1,2-ethanedithiol and the other one from 1,3-benzenedithiol that contain a sulfate anion incorporated in the cavity between the cyclopeptide rings. Formation of triply-linked bis(cyclopeptides) containing different types of linkers could be achieved by performing the synthesis in the presence of different dithiols. Unfortunately, the two C3 symmetrical bis(cyclopeptides) containing a single linker type could not be isolated in analytically pure form so that only qualitative binding studies could be performed. Investigations in this context indicate extraordinary sulfate affinity for these bis(cyclopeptides). In particular, affinity of the receptor containing the 1,2-ethanedithiol linkers for sulfate anions is so high that is even able to dissolve barium sulfate under appropriate conditions and presumably exceeds the sulfate affinity of the doubly-linked bis(cyclopeptides). The sulfate anion present in the cavity of this bis(cyclopeptide) can be replaced by a large number of other anions, i.e. by selenate, perrhenate, nitrate, tetrafluoroborate, hexafluorophosphate and halides. None of these complexes proved to be as stable as the corresponding sulfate complex. In addition, 1H-NMR spectroscopic investigations provided information about the solution structure of the bis(cyclopeptide) anion complexes. Sulfate release from the cavity of the receptor is a slow process while exchange of other anions is significantly faster. Another interesting feature that has been observed for sulfate and selenate complexes of the 1,2-ethanedithiol-containing bis(cyclopeptide) is the very slow H/D rate with which protons on amide groups located inside the cavity of the cage are replaced by deuterium atoms in protic deuterated solvents. This effect in combination with the observation that the different deuterated bis(cyclopeptide) species exhibit individual amide NH signals in the 1H-NMR spectrum are indicative for well defined complex geometries with strong hydrogen-bonding interactions between the anion and the amide NH groups of the receptor. Following the H/D exchange rate in the presence of various salts indicated that anion exchange proceeds via the dissociated complex and not by direct replacement of one anion by another one.

This thesis is concerned with tropical moduli spaces, which are an important tool in tropical enumerative geometry. The main result is a construction of tropical moduli spaces of rational tropical covers of smooth tropical curves and of tropical lines in smooth tropical surfaces. The construction of a moduli space of tropical curves in a smooth tropical variety is reduced to the case of smooth fans. Furthermore, we point out relations to intersection theory on suitable moduli spaces on algebraic curves.

This work shall provide a foundation for the cross-design of wireless networked control systems with limited resources. A cross-design methodology is devised, which includes principles for the modeling, analysis, design, and realization of low cost but high performance and intelligent wireless networked control systems. To this end, a framework is developed in which control algorithms and communication protocols are jointly designed, implemented, and optimized taking into consideration the limited communication, computing, memory, and energy resources of the low performance, low power, and low cost wireless nodes used. A special focus of the proposed methodology is on the prediction and minimization of the total energy consumption of the wireless network (i.e. maximization of the lifetime of wireless nodes) under control performance constraints (e.g. stability and robustness) in dynamic environments with uncertainty in resource availability, through the joint (offline/online) adaptation of communication protocol parameters and control algorithm parameters according to the traffic and channel conditions. Appropriate optimization approaches that exploit the structure of the optimization problems to be solved (e.g. linearity, affinity, convexity) and which are based on Linear Matrix Inequalities (LMIs), Dynamic Programming (DP), and Genetic Algorithms (GAs) are investigated. The proposed cross-design approach is evaluated on a testbed consisting of a real lab plant equipped with wireless nodes. Obtained results show the advantages of the proposed cross-design approach compared to standard approaches which are less flexible.

This thesis combined gas phase mass spectrometric investigations of ionic transition metal clusters that are either homogeneous \((Nb_n^{+/-}, Co_n^{+/-})\) or heterogeneous \(([Co_nPt_m]^{+/-})\), of their organo metallic reaction products, and of organic molecules (aspartame and Asp-Phe) and their alkali metal ion adducts.At the Paris FEL facility CLIO a newly installed FT-ICR mass spectrometer has been modified by inclusion of an ion bender that allows for the usage of additional ion sources beyond the installed ESI source. The installation of an LVAP metal cluster source served to produce metal cluster adsorbate complex ions of the type \([Nb_n(C_6H_6)]^{+/-}\). IR-MPD of the complexes \([Nb_n(C_6H_6)]^{+/-} (n = 18, 19)\) resulted in \([Nb_n(C_6)]^{+/-} (n = 18, 19)\) fragments. Spectra are broad, possibly because of vibronic / electronic transitions. In Kaiserslautern the capabilities of the LVAP source were extended by adding a gas pick up unit. Complex gases containing C-H bonds otherwise break within the cluster forming plasma. More stable gases like CO seem to attach at least partially intact. Metal cluster production with argon tagged onto the cluster failed when introducing argon through the pick up source, but succeeded when using argon as expansion gas. A new mass spectrometer concept of an additional multipole collision cell for metal cluster adsorbate formation is currently under construction. Subsequent cooling shall achieve high resolution IR-MPD spectra of transition metal cluster adsorbate complexes.Prior work on reaction of transition metal clusters with benzene was extended by investigating the reaction with benzene and benzene-d6 of size selected cationic cobalt clusters \(Co_n^+\) and of anionic cobalt clusters \(Co_n^-\) in the size range \(n = 3 - 28\) and of bimetallic cobalt platinum clusters \([Co_nPt_m]^{+/-}\) in the size range \(n + m \le 8\). Dehydrogenation by cationic cobalt clusters \(Co_n^+\) is sparse, it is effective in small bimetallic clusters \([Co_nPt_m]^+ (n + m \le 3)\). Thus single platinum atoms promote benzene dehydrogenation while further cobalt atoms quench it. Dehydrogenation is ubiquitous in reactions of anionic cobalt clusters. Mixed triatomic clusters \([Co_2Pt_1]^-\) and \([Co_1Pt_2]^-\) are special in causing effective reactions and single dehydrogenation through some kind of cooperativity while \([Co_nPt_{1,2}]^- (n \ge 3)\) do not react at all. Kinetic isotope effects KIE(n) in total reaction rates are inverse and - in part - large, dehydrogenation isotope effects DIE(n) are normal. A multistep model of adsorption and stepwise dehydrogenation from the precursor adsorbate proves suitable to rationalize the found KIEs and DIEs in principle. Particular insights into the effects of charge and of cluster size are largely beyond this model. Some DFT calculations - though preliminary - lend strong support to the otherwise assumed structures and enthalpies. More insights into the cause of the found effects of charge, size and composition of both pure and mixed clusters shall arise from ongoing high level ab initio modeling (of especially the \(n + m = 3\) case for mixed clusters).The influence of the methylester group in the molecules aspartame (Asp-PheOMe) and Asp-Phe has been explored. Therefore, their protonated and deprotonated species and their complexes with alkali metal ions attached were investigated with different techniques utilizing mass spectrometry.Gas phase H-/D-exchange with \(ND_3\) has proven that in both molecules all acidic NH and OH binding motifs do exchange their hydrogen atom and that simultaneous multi exchange is present. Kinetic studies revealed that with alkali metal ions attached the speed of the first exchange step decreases with increasing ion size. The additional OH of the carboxylic COOHPhe group in Asp-Phe increases the exchange speed by a constant value. CID experiments yielded water and the protonated Asp-Phe anhydride as main fragments out of the protonated molecules, neutral Asp anhydride and \([Phe M]^+ / [PheOMe M]^+\) for \(Li^+\) and \(Na^+\) attached, and neutral aspartame / Asp-Phe and ionic \(M^+\) for \(K^+\), \(Rb^+\) and \(Cs^+\) attached. The threshold energy \(E_{CID}\), indicating ion stability, decreases with increasing ion size. For aspartame fragmentation occurs at lower \(E_{CID}\) values for complexes with \(H^+\), \(Li^+\) and \(Na^+\) than for the Asp-Phe analoga. Complexes with \(K^+\), \(Rb^+\) and \(Cs^+\) give the same \(E_{CID}\) value for aspartame and Asp-Phe. IR-MPD investigations lead to the same fragments as the CID experiments. In combination with quantum mechanical calculations a change in the preferred structure from charge-solvated, tridentate type for complexes with small alkali metal ions (\(Li^+\)) to salt-bridge type structure for large alkali metal ions (\(Cs^+\)) could be confirmed. Calculations thereby reveal nearly no structural differences between aspartame and Asp-Phe for cationized species. The deprotonation of the additional COOHPhe group in Asp-Phe is preferred against other acidic positions. A better experimental distinction between possible (calculated) structure types would arise from additional FEL IR-MPD measurements in the energy range of 600 to 1800 \(cm^{-1}\). The comparison of the \(E_{CID}\) values with calculated fragmentation energy values proves that not only for alkali metal complexes with \(K^+\), \(Rb^+\) and \(Cs^+\), but also for \(Li^+\) and \(Na^+\) the bond breaking of all metal atom bonds is part of the transition state. The lower \(E_{CID}\) values for aspartame with small cations may be explained in terms of internal energy. Aspartame is a larger molecule, possesses more internal energy and can be recognized as the larger heat bath. Less energy is needed for fragmentation, if the Phe part with the additional methylester group is involved in the fragmentation process.