Die Dissertation behandelt methodische und algorithmische Aspekte der Spektralelementemethode zur räumlichen Diskretisierung partieller Differentialgleichungen. Die Weiterentwicklung einer symmetriebasierten Faktorisierung ermöglicht effiziente Operatoren für Tetraederelemente. Auf Grundlage einer umfassenden Leistungsanalyse werden Engpässe in der Implementierung der Operatoren identifiziert und durch algorithmische Modifikationen der Methode eliminiert.

Hierarchical Markovian Models (HMMs) are composed of multiple low level models (LLMs) and high level model (HLM) that defines the interaction among LLMs. The essence of the HMM approach is to model the system at hand in the form of interacting components so that its (larger) underlying continous-time Markov chain (CTMC) is not generated but implicitly represented as a sum of Kronecker products of (smaller) component matrices. The Kronecker structure of an HMM induces nested block partitionings in its underlying CTMC. These partitionings may be used in block versions of classical iterative methods based on splittings, such as block SOR (BSOR), to solve the underlying CTMC for its stationary vector. Therein the problem becomes that of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of a particular partitioning. This paper shows that in each HLM state there may be diagonal blocks with identical off-diagonal parts and diagonals differing from each other by a multiple of the identity matrix. Such diagonal blocks are named candidate blocks. The paper explains how candidate blocks can be detected and how the can mutually benefit from a single real Schur factorization. It gives sufficient conditions for the existence of diagonal blocks with real eigenvalues and shows how these conditions can be checked using component matrices. It describes how the sparse real Schur factors of candidate blocks satisfying these conditions can be constructed from component matrices and their real Schur factors. It also demonstrates how fill in- of LU factorized (non-candidate) diagonal blocks can be reduced by using the column approximate minimum degree algorithm (COLAMD). Then it presents a three-level BSOR solver in which the diagonal blocks at the first level are solved using block Gauss-Seidel (BGS) at the second and the methods of real Schur and LU factorizations at the third level. Finally, on a set of numerical experiments it shows how these ideas can be used to reduce the storage required by the factors of the diagonal blocks at the third level and to improve the solution time compared to an all LU factorization implementation of the three-level BSOR solver.

This work entails efforts to understand effects of solvent choice on paracetamol crystallization. Various techniques have been developed and implemented to study aforementioned. A clear-cut, direct evidence of two-step nucleation mechanism is demonstrated using a bench top Raman spectrometer and a novel method named as OSANO. / Polymorphismus ist die Eigenschaft vieler anorganischer und insbesondere organischer Moleküle, in mehr als einer Struktur zu kristallisieren. Es ist wichtig, die Faktoren zu verstehen, die den Polymorphismus beeinflussen, da er viele physikochemische Eigenschaften wie Stabilität und Löslichkeit beeinflusst. Nahezu 80 % der vermarkteten Medikamente weisen Polymorphismus auf. In dieser Arbeit wurde der Einfluss der Wahl des organischen Lösungsmittels auf den Polymorphismus von Paracetamol untersucht und verschiedene Methoden entwickelt und angewandt, um den Einfluss genauer zu verstehen. Es wurde festgestellt, dass Ethanol viel stärker auf Paracetamol-Kristallisation als Methanol wirkt. Nichtgleichgewichts-Molekulardynamiksimulationen mit periodischer, simulierter Abkühlung (Simulated Annealing) wurden verwendet, um Vorläufer der metastabilen Zwischenprodukte im Kristallisationsprozess zu untersuchen. Es wurde festgestellt, dass die Strukturen der Bausteine der Paracetamol-Kristalle durch geometrische Wechselwirkungen zwischen Lösungsmittel und Paracetamol bestimmt werden. Die statistisch häufigsten Bausteine in der Selbstassemblierung definieren die finale Kristallstruktur. Ein speziell angefertigter akustischer Levitator hat die Proben zuverlässig gehalten, wodurch die Untersuchung des Einflusses von Lösungsmitteln ermöglicht, heterogene Keimbildung abgeschwächt und andere Umgebungsfaktoren stabilisiert wurden. Die Kristallisation wurde in diesem Aufbau mit zeitaufgelöster In-situ-Raman-Spektroskopie verfolgt und mit einer neuen Zielfunktion basierenden Methode der nichtnegativen Matrixfaktorisierung (NMF) analysiert. Orthogonale Zeitrafferfotografie wurde in Verbindung mit NMF verwendet, um eindeutige und genaue Faktoren zu erhalten, die sich auf die Spektren und Konzentrationen verschiedener Anteile der Paracetamol-Kristallisation beziehen, die als latente Komponenten in den unbehandelten Daten vorhanden sind. / Polymorphism is the property exhibited by many inorganic and organic molecules to crystallize in more than one crystal structure. There is a strong need for understanding the influencing factors on polymorphism, as it is responsible for differences in many physicochemical properties such as stability and solubility. Nearly 80 % of marketed drugs exhibit polymorphism. In this work, we took the model system of paracetamol to investigate the influence of solvent choice on its polymorphism. Different methods were developed and employed to understand the influence of small organic solvents on the crystallization of paracetamol. Non-equilibrium molecular dynamics simulations with periodic simulated annealing were used as a tool to probe the nature of precursors of the metastable intermediates occurring in the crystallization process. Using this method, it was found that the structures of the building blocks of crystals of paracetamol is governed by solvent-solute interactions. In situ Raman spectroscopy was used with a custom-made acoustic levitator to follow crystallization. This set-up is a reliable method for investigating solvent influence, attenuating heterogeneous nucleation and stabilizing other environmental factors. It was established that as a solvent, ethanol is much stronger than methanol in its effect of driving paracetamol solutions to their crystal form. The time-resolved Raman spectroscopy crystallization data was processed using a newly developed objective function based non-negative matrix factorization method (NMF). An orthogonal time-lapse photography was used in conjunction with NMF to get unique and accurate factors that pertain to the spectra and concentrations of different moieties of paracetamol crystallization existing as latent components in the untreated data.