@phdthesis{Klotzky2018,
  author    = {Klotzky, Jens},
  title     = {Well-posedness of a fluid-particle interaction model},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-169009},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2018},
  abstract  = {This thesis considers a model of a scalar partial differential equation in the presence of a singular source term, modeling the interaction between an inviscid fluid represented by the Burgers equation and an arbitrary, finite amount of particles moving inside the fluid, each one acting as a point-wise drag force with a particle related friction constant. \begin{align*} \partial_t u + \partial_x (u^2/2) \&= \sum_{i \in N(t)} \lambda_i \Big(h_i'(t)-u(t,h_i(t)\Big)\delta(x-h_i(t)) \end{align*} The model was introduced for the case of a single particle by Lagouti{\`e}re, Seguin and Takahashi, is a first step towards a better understanding of interaction between fluids and solids on the level of partial differential equations and has the unique property of considering entropy admissible solutions and the interaction with shockwaves. The model is extended to an arbitrary, finite number of particles and interactions like merging, splitting and crossing of particle paths are considered. The theory of entropy admissibility is revisited for the cases of interfaces and discontinuous flux conservation laws, existing results are summarized and compared, and adapted for regions of particle interactions. To this goal, the theory of germs introduced by Andreianov, Karlsen and Risebro is extended to this case of non-conservative interface coupling. Exact solutions for the Riemann Problem of particles drifting apart are computed and analysis on the behavior of entropy solutions across the particle related interfaces is used to determine physically relevant and consistent behavior for merging and splitting of particles. Well-posedness of entropy solutions to the Cauchy problem is proven, using an explicit construction method, L-infinity bounds, an approximation of the particle paths and compactness arguments to obtain existence of entropy solutions. Uniqueness is shown in the class of weak entropy solutions using almost classical Kruzkov-type analysis and the notion of L1-dissipative germs. Necessary fundamentals of hyperbolic conservation laws, including weak solutions, shocks and rarefaction waves and the Rankine-Hugoniot condition are briefly recapitulated.},
  subject      = {Hyperbolische Differentialgleichung},
  language  = {en}
}
@phdthesis{Ruppert2017,
  author    = {Ruppert, Markus},
  title     = {Wege der Analogiebildung - Eine qualitative Studie {\"u}ber den Prozess der Analogiebildung beim L{\"o}sen von Aufgaben},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-155910},
  school      = {Universit{\"a}t W{\"u}rzburg},
  pages     = {311},
  year      = {2017},
  abstract  = {{\"U}ber die besondere Bedeutung von Analogiebildungsprozessen beim Lernen im Allgemeinen und beim Lernen von Mathematik im Speziellen besteht ein breiter wissenschaftlicher Konsens. Es liegt deshalb nahe, von einem lernf{\"o}rderlichen Mathematikunterricht zu verlangen, dass er im Bewusstsein dieser Bedeutung entwickelt ist - dass er also einerseits Analogien aufzeigt und sich diese beim Lehren von Mathematik zunutze macht, dass er andererseits aber auch dem Lernenden Gelegenheiten bietet, Analogien zu erkennen und zu entwickeln. Kurz: Die F{\"a}higkeit zum Bilden von Analogien soll durch den Unterricht gezielt gef{\"o}rdert werden. Um diesem Anspruch gerecht werden zu k{\"o}nnen, m{\"u}ssen ausreichende Kenntnisse dar{\"u}ber vorliegen, wie Analogiebildungsprozesse beim Lernen von Mathematik und beim L{\"o}sen mathematischer Aufgaben ablaufen, wodurch sich erfolgreiche Analogiebildungsprozesse auszeichnen und an welchen Stellen m{\"o}glicherweise Schwierigkeiten bestehen. Der Autor zeigt auf, wie Prozesse der Analogiebildung beim L{\"o}sen mathematischer Aufgaben initiiert, beobachtet, beschrieben und interpretiert werden k{\"o}nnen, um auf dieser Grundlage Ansatzpunkte f{\"u}r geeignete F{\"o}rdermaßnahmen zu identifizieren, bestehende Ideen zur F{\"o}rderung der Analogiebildungsf{\"a}higkeit zu beurteilen und neue Ideen zu entwickeln. Es werden dabei Wege der Analogiebildung nachgezeichnet und untersucht, die auf der Verschr{\"a}nkung zweier Dimensionen der Analogiebildung im Rahmen des zugrundeliegenden theoretischen Modells beruhen. So k{\"o}nnen verschiedene Vorgehensweisen ebenso kontrastiert werden, wie kritische Punkte im Verlauf eines Analogiebildungsprozesses. Es ergeben sich daraus Unterrichtsvorschl{\"a}ge, die auf den Ideen zum beispielbasierten Lernen aufbauen.},
  subject      = {Analogie},
  language  = {de}
}
@phdthesis{Lieb2017,
  author    = {Lieb, Julia},
  title     = {Counting Polynomial Matrices over Finite Fields : Matrices with Certain Primeness Properties and Applications to Linear Systems and Coding Theory},
  edition   = {1. Auflage},
  publisher = {W{\"u}rzburg University Press},
  address   = {W{\"u}rzburg},
  isbn      = {978-3-95826-064-1 (print)},
  doi       = {10.25972/WUP-978-3-95826-065-8},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-151303},
  school      = {W{\"u}rzburg University Press},
  pages     = {164},
  year      = {2017},
  abstract  = {This dissertation is dealing with three mathematical areas, namely polynomial matrices over finite fields, linear systems and coding theory. Coprimeness properties of polynomial matrices provide criteria for the reachability and observability of interconnected linear systems. Since time-discrete linear systems over finite fields and convolutional codes are basically the same objects, these results could be transfered to criteria for non-catastrophicity of convolutional codes. We calculate the probability that specially structured polynomial matrices are right prime. In particular, formulas for the number of pairwise coprime polynomials and for the number of mutually left coprime polynomial matrices are calculated. This leads to the probability that a parallel connected linear system is reachable and that a parallel connected convolutional codes is non-catastrophic. Moreover, the corresponding probabilities are calculated for other networks of linear systems and convolutional codes, such as series connection. Furthermore, the probabilities that a convolutional codes is MDP and that a clock code is MDS are approximated. Finally, we consider the probability of finding a solution for a linear network coding problem.},
  subject      = {Lineares System},
  language  = {en}
}
@phdthesis{Steck2018,
  author    = {Steck, Daniel},
  title     = {Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-174444},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2018},
  abstract  = {This thesis is concerned with a class of general-purpose algorithms for constrained minimization problems, variational inequalities, and quasi-variational inequalities in Banach spaces. A substantial amount of background material from Banach space theory, convex analysis, variational analysis, and optimization theory is presented, including some results which are refinements of those existing in the literature. This basis is used to formulate an augmented Lagrangian algorithm with multiplier safeguarding for the solution of constrained optimization problems in Banach spaces. The method is analyzed in terms of local and global convergence, and many popular problem classes such as nonlinear programming, semidefinite programming, and function space optimization are shown to be included as special cases of the general setting. The algorithmic framework is then extended to variational and quasi-variational inequalities, which include, by extension, Nash and generalized Nash equilibrium problems. For these problem classes, the convergence is analyzed in detail. The thesis then presents a rich collection of application examples for all problem classes, including implementation details and numerical results.},
  subject      = {Optimierung},
  language  = {en}
}
@phdthesis{Forster2016,
  author    = {Forster, Johannes},
  title     = {Variational Approach to the Modeling and Analysis of Magnetoelastic Materials},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-147226},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {This doctoral thesis is concerned with the mathematical modeling of magnetoelastic materials and the analysis of PDE systems describing these materials and obtained from a variational approach. The purpose is to capture the behavior of elastic particles that are not only magnetic but exhibit a magnetic domain structure which is well described by the micromagnetic energy and the Landau-Lifshitz-Gilbert equation of the magnetization. The equation of motion for the material's velocity is derived in a continuum mechanical setting from an energy ansatz. In the modeling process, the focus is on the interplay between Lagrangian and Eulerian coordinate systems to combine elasticity and magnetism in one model without the assumption of small deformations. The resulting general PDE system is simplified using special assumptions. Existence of weak solutions is proved for two variants of the PDE system, one including gradient flow dynamics on the magnetization, and the other featuring the Landau-Lifshitz-Gilbert equation. The proof is based on a Galerkin method and a fixed point argument. The analysis of the PDE system with the Landau-Lifshitz-Gilbert equation uses a more involved approach to obtain weak solutions based on G. Carbou and P. Fabrie 2001.},
  subject      = {Magnetoelastizit{\"a}t},
  language  = {en}
}
@article{KasangKalluvyaMajingeetal.2016,
  author    = {Kasang, Christa and Kalluvya, Samuel and Majinge, Charles and Kongola, Gilbert and Mlewa, Mathias and Massawe, Irene and Kabyemera, Rogatus and Magambo, Kinanga and Ulmer, Albrecht and Klinker, Hartwig and Gschmack, Eva and Horn, Anne and Koutsilieri, Eleni and Preiser, Wolfgang and Hofmann, Daniela and Hain, Johannes and M{\"u}ller, Andreas and D{\"o}lken, Lars and Weissbrich, Benedikt and Rethwilm, Axel and Stich, August and Scheller, Carsten},
  title     = {Effects of Prednisolone on Disease Progression in Antiretroviral-Untreated HIV Infection: A 2-Year Randomized, Double-Blind Placebo-Controlled Clinical Trial},
  series = {PLoS One},
  volume    = {11},
  journal   = {PLoS One},
  number    = {1},
  doi       = {10.1371/journal.pone.0146678},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-146479},
  pages     = {e0146678},
  year      = {2016},
  abstract  = {Background HIV-disease progression correlates with immune activation. Here we investigated whether corticosteroid treatment can attenuate HIV disease progression in antiretroviral-untreated patients. Methods Double-blind, placebo-controlled randomized clinical trial including 326 HIV-patients in a resource-limited setting in Tanzania (clinicaltrials.gov NCT01299948). Inclusion criteria were a CD4 count above 300 cells/μl, the absence of AIDS-defining symptoms and an ART-na{\"i}ve therapy status. Study participants received 5 mg prednisolone per day or placebo for 2 years. Primary endpoint was time to progression to an AIDS-defining condition or to a CD4-count below 200 cells/μl. Results No significant change in progression towards the primary endpoint was observed in the intent-to-treat (ITT) analysis (19 cases with prednisolone versus 28 cases with placebo, p = 0.1407). In a per-protocol (PP)-analysis, 13 versus 24 study participants progressed to the primary study endpoint (p = 0.0741). Secondary endpoints: Prednisolone-treatment decreased immune activation (sCD14, suPAR, CD38/HLA-DR/CD8+) and increased CD4-counts (+77.42 ± 5.70 cells/μl compared to -37.42 ± 10.77 cells/μl under placebo, p < 0.0001). Treatment with prednisolone was associated with a 3.2-fold increase in HIV viral load (p < 0.0001). In a post-hoc analysis stratifying for sex, females treated with prednisolone progressed significantly slower to the primary study endpoint than females treated with placebo (ITT-analysis: 11 versus 21 cases, p = 0.0567; PP-analysis: 5 versus 18 cases, p = 0.0051): No changes in disease progression were observed in men. Conclusions This study could not detect any significant effects of prednisolone on disease progression in antiretroviral-untreated HIV infection within the intent-to-treat population. However, significant effects were observed on CD4 counts, immune activation and HIV viral load. This study contributes to a better understanding of the role of immune activation in the pathogenesis of HIV infection.},
  language  = {en}
}
@phdthesis{Koch2016,
  author    = {Koch, Julia Diana},
  title     = {Value Ranges for Schlicht Functions},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-144978},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {This thesis deals with value sets, i.e. the question of what the set of values that a set of functions can take in a prescribed point looks like. Interest in such problems has been around for a long time; a first answer was given by the Schwarz lemma in the 19th century, and soon various refinements were proven. Since the 1930s, a powerful method for solving such problems has been developed, namely Loewner theory. We make extensive use of this tool, as well as variation methods which go back to Schiffer to examine the following questions: We describe the set of values a schlicht normalised function on the unit disc with prescribed derivative at the origin can take by applying Pontryagin's maximum principle to the radial Loewner equation. We then determine the value ranges for the set of holomorphic, normalised, and bounded functions that have only real coefficients in their power series expansion around 0, and for the smaller set of functions which are additionally typically real. Furthermore, we describe the values a univalent self-mapping of the upper half-plane with hydrodynamical normalization which is symmetric with respect to the imaginary axis can take. Lastly, we give a necessary condition for a schlicht bounded function f on the unit disc to have extremal derivative in a point z where its value f(z) is fixed by using variation methods.},
  subject      = {Pontrjagin-Maximumprinzip},
  language  = {en}
}
@article{SchindeleBorzi2016,
  author    = {Schindele, Andreas and Borz{\`i}, Alfio},
  title     = {Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional},
  series = {Applied Mathematics},
  volume    = {7},
  journal   = {Applied Mathematics},
  number    = {9},
  doi       = {10.4236/am.2016.79086},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145850},
  pages     = {967-992},
  year      = {2016},
  abstract  = {First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.},
  language  = {en}
}
@phdthesis{GallegoValencia2017,
  author    = {Gallego Valencia, Juan Pablo},
  title     = {On Runge-Kutta discontinuous Galerkin methods for compressible Euler equations and the ideal magneto-hydrodynamical model},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-148874},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2017},
  abstract  = {An explicit Runge-Kutta discontinuous Galerkin (RKDG) method is used to device numerical schemes for both the compressible Euler equations of gas dynamics and the ideal magneto- hydrodynamical (MHD) model. These systems of conservation laws are known to have discontinuous solutions. Discontinuities are the source of spurious oscillations in the solution profile of the numerical approximation, when a high order accurate numerical method is used. Different techniques are reviewed in order to control spurious oscillations. A shock detection technique is shown to be useful in order to determine the regions where the spurious oscillations appear such that a Limiter can be used to eliminate these numeric artifacts. To guarantee the positivity of specific variables like the density and the pressure, a positivity preserving limiter is used. Furthermore, a numerical flux, proven to preserve the entropy stability of the semi-discrete DG scheme for the MHD system is used. Finally, the numerical schemes are implemented using the deal.II C++ libraries in the dflo code. The solution of common test cases show the capability of the method.},
  subject      = {Eulersche Differentialgleichung},
  language  = {en}
}
@unpublished{BreitenbachBorzi2019,
  author    = {Breitenbach, Tim and Borz{\`i}, Alfio},
  title     = {On the SQH scheme to solve non-smooth PDE optimal control problems},
  series = {Numerical Functional Analysis and Optimization},
  journal   = {Numerical Functional Analysis and Optimization},
  doi       = {10.1080/01630563.2019.1599911},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-180936},
  year      = {2019},
  abstract  = {A sequential quadratic Hamiltonian (SQH) scheme for solving different classes of non-smooth and non-convex PDE optimal control problems is investigated considering seven different benchmark problems with increasing difficulty. These problems include linear and nonlinear PDEs with linear and bilinear control mechanisms, non-convex and discontinuous costs of the controls, L\(^1\) tracking terms, and the case of state constraints. The SQH method is based on the characterisation of optimality of PDE optimal control problems by the Pontryagin's maximum principle (PMP). For each problem, a theoretical discussion of the PMP optimality condition is given and results of numerical experiments are presented that demonstrate the large range of applicability of the SQH scheme.},
  language  = {en}
}