@phdthesis{Birke2024, author = {Birke, Claudius B.}, title = {Low Mach and Well-Balanced Numerical Methods for Compressible Euler and Ideal MHD Equations with Gravity}, doi = {10.25972/OPUS-36330}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-363303}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2024}, abstract = {Physical regimes characterized by low Mach numbers and steep stratifications pose severe challenges to standard finite volume methods. We present three new methods specifically designed to navigate these challenges by being both low Mach compliant and well-balanced. These properties are crucial for numerical methods to efficiently and accurately compute solutions in the regimes considered. First, we concentrate on the construction of an approximate Riemann solver within Godunov-type finite volume methods. A new relaxation system gives rise to a two-speed relaxation solver for the Euler equations with gravity. Derived from fundamental mathematical principles, this solver reduces the artificial dissipation in the subsonic regime and preserves hydrostatic equilibria. The solver is particularly stable as it satisfies a discrete entropy inequality, preserves positivity of density and internal energy, and suppresses checkerboard modes. The second scheme is designed to solve the equations of ideal MHD and combines different approaches. In order to deal with low Mach numbers, it makes use of a low-dissipation version of the HLLD solver and a partially implicit time discretization to relax the CFL time step constraint. A Deviation Well-Balancing method is employed to preserve a priori known magnetohydrostatic equilibria and thereby reduces the magnitude of spatial discretization errors in strongly stratified setups. The third scheme relies on an IMEX approach based on a splitting of the MHD equations. The slow scale part of the system is discretized by a time-explicit Godunov-type method, whereas the fast scale part is discretized implicitly by central finite differences. Numerical dissipation terms and CFL time step restriction of the method depend solely on the slow waves of the explicit part, making the method particularly suited for subsonic regimes. Deviation Well-Balancing ensures the preservation of a priori known magnetohydrostatic equilibria. The three schemes are applied to various numerical experiments for the compressible Euler and ideal MHD equations, demonstrating their ability to accurately simulate flows in regimes with low Mach numbers and strong stratification even on coarse grids.}, subject = {Magnetohydrodynamik}, language = {en} } @phdthesis{Bossert2024, author = {Bossert, Patrick}, title = {Statistical structure and inference methods for discrete high-frequency observations of SPDEs in one and multiple space dimensions}, doi = {10.25972/OPUS-36113}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-361130}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2024}, abstract = {The focus of this thesis is on analysing a linear stochastic partial differential equation (SPDE) with a bounded domain. The first part of the thesis commences with an examination of a one-dimensional SPDE. In this context, we construct estimators for the parameters of a parabolic SPDE based on discrete observations of a solution in time and space on a bounded domain. We establish central limit theorems for a high-frequency asymptotic regime, showing substantially smaller asymptotic variances compared to existing estimation methods. Moreover, asymptotic confidence intervals are directly feasible. Our approach builds upon realized volatilities and their asymptotic illustration as the response of a log-linear model with a spatial explanatory variable. This yields efficient estimators based on realized volatilities with optimal rates of convergence and minimal variances. We demonstrate our results by Monte Carlo simulations. Extending this framework, we analyse a second-order SPDE model in multiple space dimensions in the second part of this thesis and develop estimators for the parameters of this model based on discrete observations in time and space on a bounded domain. While parameter estimation for one and two spatial dimensions was established in recent literature, this is the first work that generalizes the theory to a general, multi-dimensional framework. Our methodology enables the construction of an oracle estimator for volatility within the underlying model. For proving central limit theorems, we use a high-frequency observation scheme. To showcase our results, we conduct a Monte Carlo simulation, highlighting the advantages of our novel approach in a multi-dimensional context.}, subject = {Stochastische partielle Differentialgleichung}, language = {en} } @phdthesis{Koerner2024, author = {K{\"o}rner, Jacob}, title = {Theoretical and numerical analysis of Fokker-Planck optimal control problems by first- and second-order optimality conditions}, doi = {10.25972/OPUS-36299}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-362997}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2024}, abstract = {In this thesis, a variety of Fokker--Planck (FP) optimal control problems are investigated. Main emphasis is put on a first-- and second--order analysis of different optimal control problems, characterizing optimal controls, establishing regularity results for optimal controls, and providing a numerical analysis for a Galerkin--based numerical scheme. The Fokker--Planck equation is a partial differential equation (PDE) of linear parabolic type deeply connected to the theory of stochastic processes and stochastic differential equations. In essence, it describes the evolution over time of the probability distribution of the state of an object or system of objects under the influence of both deterministic and stochastic forces. The FP equation is a cornerstone in understanding and modeling phenomena ranging from the diffusion and motion of molecules in a fluid to the fluctuations in financial markets. Two different types of optimal control problems are analyzed in this thesis. On the one hand, Fokker--Planck ensemble optimal control problems are considered that have a wide range of applications in controlling a system of multiple non--interacting objects. In this framework, the goal is to collectively drive each object into a desired state. On the other hand, tracking--type control problems are investigated, commonly used in parameter identification problems or stemming from the field of inverse problems. In this framework, the aim is to determine certain parameters or functions of the FP equation, such that the resulting probability distribution function takes a desired form, possibly observed by measurements. In both cases, we consider FP models where the control functions are part of the drift, arising only from the deterministic forces of the system. Therefore, the FP optimal control problem has a bilinear control structure. Box constraints on the controls may be present, and the focus is on time--space dependent controls for ensemble--type problems and on only time--dependent controls for tracking--type optimal control problems. In the first chapter of the thesis, a proof of the connection between the FP equation and stochastic differential equations is provided. Additionally, stochastic optimal control problems, aiming to minimize an expected cost value, are introduced, and the corresponding formulation within a deterministic FP control framework is established. For the analysis of this PDE--constrained optimal control problem, the existence, and regularity of solutions to the FP problem are investigated. New \$L^\infty\$--estimates for solutions are established for low space dimensions under mild assumptions on the drift. Furthermore, based on the theory of Bessel potential spaces, new smoothness properties are derived for solutions to the FP problem in the case of only time--dependent controls. Due to these properties, the control--to--state map, which associates the control functions with the corresponding solution of the FP problem, is well--defined, Fr{\´e}chet differentiable and compact for suitable Lebesgue spaces or Sobolev spaces. The existence of optimal controls is proven under various assumptions on the space of admissible controls and objective functionals. First--order optimality conditions are derived using the adjoint system. The resulting characterization of optimal controls is exploited to achieve higher regularity of optimal controls, as well as their state and co--state functions. Since the FP optimal control problem is non--convex due to its bilinear structure, a first--order analysis should be complemented by a second--order analysis. Therefore, a second--order analysis for the ensemble--type control problem in the case of \$H^1\$--controls in time and space is performed, and sufficient second--order conditions are provided. Analogous results are obtained for the tracking--type problem for only time--dependent controls. The developed theory on the control problem and the first-- and second--order optimality conditions is applied to perform a numerical analysis for a Galerkin discretization of the FP optimal control problem. The main focus is on tracking-type problems with only time--dependent controls. The idea of the presented Galerkin scheme is to first approximate the PDE--constrained optimization problem by a system of ODE--constrained optimization problems. Then, conditions on the problem are presented such that the convergence of optimal controls from one problem to the other can be guaranteed. For this purpose, a class of bilinear ODE--constrained optimal control problems arising from the Galerkin discretization of the FP problem is analyzed. First-- and second--order optimality conditions are established, and a numerical analysis is performed. A discretization with linear finite elements for the state and co--state problem is investigated, while the control functions are approximated by piecewise constant or piecewise quadratic continuous polynomials. The latter choice is motivated by the bilinear structure of the optimal control problem, allowing to overcome the discrepancies between a discretize--then--optimize and optimize--then--discretize approach. Moreover, second--order accuracy results are shown using the space of continuous, piecewise quadratic polynomials as the discrete space of controls. Lastly, the theoretical results and the second--order convergence rates are numerically verified.}, subject = {Parabolische Differentialgleichung}, language = {en} } @phdthesis{Biersack2024, author = {Biersack, Florian}, title = {Topological Properties of Quasiconformal Automorphism Groups}, doi = {10.25972/OPUS-35917}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-359177}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2024}, abstract = {The goal of this thesis is to study the topological and algebraic properties of the quasiconformal automorphism groups of simply and multiply connected domains in the complex plain, in which the quasiconformal automorphism groups are endowed with the supremum metric on the underlying domain. More precisely, questions concerning central topological properties such as (local) compactness, (path)-connectedness and separability and their dependence on the boundary of the corresponding domains are studied, as well as completeness with respect to the supremum metric. Moreover, special subsets of the quasiconformal automorphism group of the unit disk are investigated, and concrete quasiconformal automorphisms are constructed. Finally, a possible application of quasiconformal unit disk automorphisms to symmetric cryptography is presented, in which a quasiconformal cryptosystem is defined and studied.}, subject = {Quasikonforme Abbildung}, language = {en} } @phdthesis{Scherz2024, author = {Scherz, Jan}, title = {Weak Solutions to Mathematical Models of the Interaction between Fluids, Solids and Electromagnetic Fields}, doi = {10.25972/OPUS-34920}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-349205}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2024}, abstract = {We analyze the mathematical models of two classes of physical phenomena. The first class of phenomena we consider is the interaction between one or more insulating rigid bodies and an electrically conducting fluid, inside of which the bodies are contained, as well as the electromagnetic fields trespassing both of the materials. We take into account both the cases of incompressible and compressible fluids. In both cases our main result yields the existence of weak solutions to the associated system of partial differential equations, respectively. The proofs of these results are built upon hybrid discrete-continuous approximation schemes: Parts of the systems are discretized with respect to time in order to deal with the solution-dependent test functions in the induction equation. The remaining parts are treated as continuous equations on the small intervals between consecutive discrete time points, allowing us to employ techniques which do not transfer to the discretized setting. Moreover, the solution-dependent test functions in the momentum equation are handled via the use of classical penalization methods. The second class of phenomena we consider is the evolution of a magnetoelastic material. Here too, our main result proves the existence of weak solutions to the corresponding system of partial differential equations. Its proof is based on De Giorgi's minimizing movements method, in which the system is discretized in time and, at each discrete time point, a minimization problem is solved, the associated Euler-Lagrange equations of which constitute a suitable approximation of the original equation of motion and magnetic force balance. The construction of such a minimization problem is made possible by the realization that, already on the continuous level, both of these equations can be written in terms of the same energy and dissipation potentials. The functional for the discrete minimization problem can then be constructed on the basis of these potentials.}, subject = {Fluid-Struktur-Wechselwirkung}, language = {en} } @phdthesis{Jia2023, author = {Jia, Xiaoxi}, title = {Augmented Lagrangian Methods invoking (Proximal) Gradient-type Methods for (Composite) Structured Optimization Problems}, doi = {10.25972/OPUS-32374}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-323745}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2023}, abstract = {This thesis, first, is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints, subsequently, as well as constrained structured optimization problems featuring a composite objective function and set-membership constraints. It is then concerned to convergence and rate-of-convergence analysis of proximal gradient methods for the composite optimization problems in the presence of the Kurdyka--{\L}ojasiewicz property without global Lipschitz assumption.}, subject = {Optimierung}, language = {en} } @phdthesis{Dippell2023, author = {Dippell, Marvin}, title = {Constraint Reduction in Algebra, Geometry and Deformation Theory}, doi = {10.25972/OPUS-30167}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-301670}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2023}, abstract = {To study coisotropic reduction in the context of deformation quantization we introduce constraint manifolds and constraint algebras as the basic objects encoding the additional information needed to define a reduction. General properties of various categories of constraint objects and their compatiblity with reduction are examined. A constraint Serre-Swan theorem, identifying constraint vector bundles with certain finitely generated projective constraint modules, as well as a constraint symbol calculus are proved. After developing the general deformation theory of constraint algebras, including constraint Hochschild cohomology and constraint differential graded Lie algebras, the second constraint Hochschild cohomology for the constraint algebra of functions on a constraint flat space is computed.}, subject = {Differentialgeometrie}, language = {en} } @phdthesis{Stumpf2022, author = {Stumpf, Pascal}, title = {On coverings and reduced residues in combinatorial number theory}, doi = {10.25972/OPUS-29350}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-293504}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2022}, abstract = {Our starting point is the Jacobsthal function \(j(m)\), defined for each positive integer \(m\) as the smallest number such that every \(j(m)\) consecutive integers contain at least one integer relatively prime to \(m\). It has turned out that improving on upper bounds for \(j(m)\) would also lead to advances in understanding the distribution of prime numbers among arithmetic progressions. If \(P_r\) denotes the product of the first \(r\) prime numbers, then a conjecture of Montgomery states that \(j(P_r)\) can be bounded from above by \(r (\log r)^2\) up to some constant factor. However, the until now very promising sieve methods seem to have reached a limit here, and the main goal of this work is to develop other combinatorial methods in hope of coming a bit closer to prove the conjecture of Montgomery. Alongside, we solve a problem of Recam{\´a}n about the maximum possible length among arithmetic progressions in the least (positive) reduced residue system modulo \(m\). Lastly, we turn towards three additive representation functions as introduced by Erdős, S{\´a}rk{\"o}zy and S{\´o}s who studied their surprising different monotonicity behavior. By an alternative approach, we answer a question of S{\´a}rk{\"o}zy and demostrate that another conjecture does not hold.}, subject = {Kombinatorische Zahlentheorie}, language = {en} } @phdthesis{Lechner2022, author = {Lechner, Theresa}, title = {Proximal Methods for Nonconvex Composite Optimization Problems}, doi = {10.25972/OPUS-28907}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-289073}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2022}, abstract = {Optimization problems with composite functions deal with the minimization of the sum of a smooth function and a convex nonsmooth function. In this thesis several numerical methods for solving such problems in finite-dimensional spaces are discussed, which are based on proximity operators. After some basic results from convex and nonsmooth analysis are summarized, a first-order method, the proximal gradient method, is presented and its convergence properties are discussed in detail. Known results from the literature are summarized and supplemented by additional ones. Subsequently, the main part of the thesis is the derivation of two methods which, in addition, make use of second-order information and are based on proximal Newton and proximal quasi-Newton methods, respectively. The difference between the two methods is that the first one uses a classical line search, while the second one uses a regularization parameter instead. Both techniques lead to the advantage that, in contrast to many similar methods, in the respective detailed convergence analysis global convergence to stationary points can be proved without any restricting precondition. Furthermore, comprehensive results show the local convergence properties as well as convergence rates of these algorithms, which are based on rather weak assumptions. Also a method for the solution of the arising proximal subproblems is investigated. In addition, the thesis contains an extensive collection of application examples and a detailed discussion of the related numerical results.}, subject = {Optimierung}, language = {en} } @phdthesis{Kortum2022, author = {Kortum, Joshua}, title = {Global Existence and Uniqueness Results for Nematic Liquid Crystal and Magnetoviscoelastic Flows}, doi = {10.25972/OPUS-27827}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-278271}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2022}, abstract = {Liquid crystals and polymeric fluids are found in many technical applications with liquid crystal displays probably being the most prominent one. Ferromagnetic materials are well established in industrial and everyday use, e.g. as magnets in generators, transformers and hard drive disks. Among ferromagnetic materials, we find a subclass which undergoes deformations if an external magnetic field is applied. This effect is exploited in actuators, magnetoelastic sensors, and new fluid materials have been produced which retain their induced magnetization during the flow. A central issue consists of a proper modelling for those materials. Several models exist regarding liquid crystals and liquid crystal flows, but up to now, none of them has provided a full insight into all observed effects. On materials encompassing magnetic, elastic and perhaps even fluid dynamic effects, the mathematical literature seems sparse in terms of models. To some extent, one can unify the modeling of nematic liquid crystals and magnetoviscoelastic materials employing a so-called energetic variational approach. Using the least action principle from theoretical physics, the actual task reduces to finding appropriate energies describing the observed behavior. The procedure leads to systems of evolutionary partial differential equations, which are analyzed in this work. From the mathematical point of view, fundamental questions on existence, uniqueness and stability of solutions remain unsolved. Concerning the Ericksen-Leslie system modelling nematic liquid crystal flows, an approximation to this model is given by the so-called Ginzburg-Landau approximation. Solutions to the latter are intended to approximately represent solutions to the Ericksen-Leslie system. Indeed, we verify this presumption in two spatial dimensions. More precisely, it is shown that weak solutions of the Ginzburg-Landau approximation converge to solutions of the Ericksen-Leslie system in the energy space for all positive times of evolution. In order to do so, theory for the Euler equations invented by DiPerna and Majda on weak compactness and concentration measures is used. The second part of the work deals with a system of partial differential equations modelling magnetoviscoelastic fluids. We provide a well-posedness result in two spatial dimensions for large energies and large times. Along the verification of that conclusion, existing theory on the Ericksen-Leslie system and the harmonic map flow is deployed and suitably extended.}, subject = {Magnetoelastizit{\"a}t}, language = {en} } @phdthesis{Barth2022, author = {Barth, Dominik}, title = {Computation of multi-branch-point covers and applications in Galois theory}, doi = {10.25972/OPUS-27702}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-277025}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2022}, abstract = {We present a technique for computing multi-branch-point covers with prescribed ramification and demonstrate the applicability of our method in relatively large degrees by computing several families of polynomials with symplectic and linear Galois groups. As a first application, we present polynomials over \(\mathbb{Q}(\alpha,t)\) for the primitive rank-3 groups \(PSp_4(3)\) and \(PSp_4(3).C_2\) of degree 27 and for the 2-transitive group \(PSp_6(2)\) in its actions on 28 and 36 points, respectively. Moreover, the degree-28 polynomial for \(PSp_6(2)\) admits infinitely many totally real specializations. Next, we present the first (to the best of our knowledge) explicit polynomials for the 2-transitive linear groups \(PSL_4(3)\) and \(PGL_4(3)\) of degree 40, and the imprimitive group \(Aut(PGL_4(3))\) of degree 80. Additionally, we negatively answer a question by K{\"o}nig whether there exists a degree-63 rational function with rational coefficients and monodromy group \(PSL_6(2)\) ramified over at least four points. This is achieved due to the explicit computation of the corresponding hyperelliptic genus-3 Hurwitz curve parameterizing this family, followed by a search for rational points on it. As a byproduct of our calculations we obtain the first explicit \(Aut(PSL_6(2))\)-realizations over \(\mathbb{Q}(t)\). At last, we present a technique by Elkies for bounding the transitivity degree of Galois groups. This provides an alternative way to verify the Galois groups from the previous chapters and also yields a proof that the monodromy group of a degree-276 cover computed by Monien is isomorphic to the sporadic 2-transitive Conway group \(Co_3\).}, subject = {Galois-Theorie}, language = {en} } @phdthesis{Schmeller2022, author = {Schmeller, Christof}, title = {Uniform distribution of zero ordinates of Epstein zeta-functions}, doi = {10.25972/OPUS-25199}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-251999}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2022}, abstract = {The dissertation investigates the wide class of Epstein zeta-functions in terms of uniform distribution modulo one of the ordinates of their nontrivial zeros. Main results are a proof of a Landau type theorem for all Epstein zeta-functions as well as uniform distribution modulo one for the zero ordinates of all Epstein zeta-functions asscoiated with binary quadratic forms.}, subject = {Zetafunktion}, language = {en} } @phdthesis{Bartsch2021, author = {Bartsch, Jan}, title = {Theoretical and numerical investigation of optimal control problems governed by kinetic models}, doi = {10.25972/OPUS-24906}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-249066}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {This thesis is devoted to the numerical and theoretical analysis of ensemble optimal control problems governed by kinetic models. The formulation and study of these problems have been put forward in recent years by R.W. Brockett with the motivation that ensemble control may provide a more general and robust control framework for dynamical systems. Following this formulation, a Liouville (or continuity) equation with an unbounded drift function is considered together with a class of cost functionals that include tracking of ensembles of trajectories of dynamical systems and different control costs. Specifically, \$L^2\$, \$H^1\$ and \$L^1\$ control costs are taken into account which leads to non--smooth optimization problems. For the theoretical investigation of the resulting optimal control problems, a well--posedness theory in weighted Sobolev spaces is presented for Liouville and related transport equations. Specifically, existence and uniqueness results for these equations and energy estimates in suitable norms are provided; in particular norms in weighted Sobolev spaces. Then, non--smooth optimal control problems governed by the Liouville equation are formulated with a control mechanism in the drift function. Further, box--constraints on the control are imposed. The control--to--state map is introduced, that associates to any control the unique solution of the corresponding Liouville equation. Important properties of this map are investigated, specifically, that it is well--defined, continuous and Frechet differentiable. Using the first two properties, the existence of solutions to the optimal control problems is shown. While proving the differentiability, a loss of regularity is encountered, that is natural to hyperbolic equations. This leads to the need of the investigation of the control--to--state map in the topology of weighted Sobolev spaces. Exploiting the Frechet differentiability, it is possible to characterize solutions to the optimal control problem as solutions to an optimality system. This system consists of the Liouville equation, its optimization adjoint in the form of a transport equation, and a gradient inequality. Numerical methodologies for solving Liouville and transport equations are presented that are based on a non--smooth Lagrange optimization framework. For this purpose, approximation and solution schemes for such equations are developed and analyzed. For the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov--Tadmor method, a Runge--Kutta scheme, and a Strang splitting method are discussed. Stability and second--order accuracy of these resulting schemes are proven in the discrete \$L^1\$ norm. In addition, conservation of mass and positivity preservation are confirmed for the solution method of the Liouville model. As numerical optimization strategy, an adapted Krylow--Newton method is applied. Since the control is considered to be an element of \$H^1\$ and to obey certain box--constraints, a method for calculating a \$H^1\$ projection is presented. Since the optimal control problem is non-smooth, a semi-smooth adaption of Newton's method is taken into account. Results of numerical experiments are presented that successfully validate the proposed deterministic framework. After the discussion of deterministic schemes, the linear space--homogeneous Keilson--Storer master equation is investigated. This equation was originally developed for the modelling of Brownian motion of particles immersed in a fluid and is a representative model of the class of linear Boltzmann equations. The well--posedness of the Keilson--Storer master equation is investigated and energy estimates in different topologies are derived. To solve this equation numerically, Monte Carlo methods are considered. Such methods take advantage of the kinetic formulation of the Liouville equation and directly implement the behaviour of the system of particles under consideration. This includes the probabilistic behaviour of the collisions between particles. Optimal control problems are formulated with an objective that is constituted of certain expected values in velocity space and the \$L^2\$ and \$H^1\$ costs of the control. The problems are governed by the Keilson--Storer master equation and the control mechanism is considered to be within the collision kernel. The objective of the optimal control of this model is to drive an ensemble of particles to acquire a desired mean velocity and to achieve a desired final velocity configuration. Existence of solutions of the optimal control problem is proven and a Keilson--Storer optimality system characterizing the solution of the proposed optimal control problem is obtained. The optimality system is used to construct a gradient--based optimization strategy in the framework of Monte--Carlo methods. This task requires to accommodate the resulting adjoint Keilson--Storer model in a form that is consistent with the kinetic formulation. For this reason, we derive an adjoint Keilson--Storer collision kernel and an additional source term. A similar approach is presented in the case of a linear space--inhomogeneous kinetic model with external forces and with Keilson--Storer collision term. In this framework, a control mechanism in the form of an external space--dependent force is investigated. The purpose of this control is to steer the multi--particle system to follow a desired mean velocity and position and to reach a desired final configuration in phase space. An optimal control problem using the formulation of ensemble controls is stated with an objective that is constituted of expected values in phase space and \$H^1\$ costs of the control. For solving the optimal control problems, a gradient--based computational strategy in the framework of Monte Carlo methods is developed. Part of this is the denoising of the distribution functions calculated by Monte Carlo algorithms using methods of the realm of partial differential equations. A standalone C++ code is presented that implements the developed non--linear conjugated gradient strategy. Results of numerical experiments confirm the ability of the designed probabilistic control framework to operate as desired. An outlook section about optimal control problems governed by non--linear space--inhomogeneous kinetic models completes this thesis.}, subject = {Optimale Kontrolle}, language = {en} } @phdthesis{Meyer2021, author = {Meyer, Michael}, title = {Practical isogeny-based cryptography}, doi = {10.25972/OPUS-24682}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-246821}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {This thesis aims at providing efficient and side-channel protected implementations of isogeny-based primitives, and at their application in threshold protocols. It is based on a sequence of academic papers. Chapter 3 reviews the original variable-time implementation of CSIDH and introduces several optimizations, e.g. a significant improvement of isogeny computations by using both Montgomery and Edwards curves. In total, our improvements yield a speedup of 25\% compared to the original implementation. Chapter 4 presents the first practical constant-time implementation of CSIDH. We describe how variable-time implementations of CSIDH leak information on private keys, and describe ways to mitigate this. Further, we present several techniques to speed up the implementation. In total, our constant-time implementation achieves a rather small slowdown by a factor of 3.03. Chapter 5 reviews practical fault injection attacks on CSIDH and presents countermeasures. We evaluate different attack models theoretically and practically, using low-budget equipment. Moreover, we present countermeasures that mitigate the proposed fault injection attacks, only leading to a small performance overhead of 7\%. Chapter 6 initiates the study of threshold schemes based on the Hard Homogeneous Spaces (HHS) framework of Couveignes. Using the HHS equivalent of Shamir's secret sharing in the exponents, we adapt isogeny based schemes to the threshold setting. In particular, we present threshold versions of the CSIDH public key encryption and the CSI-FiSh signature scheme. Chapter 7 gives a sieving algorithm for finding pairs of consecutive smooth numbers that utilizes solutions to the Prouhet-Tarry-Escott (PTE) problem. Recent compact isogeny-based protocols, namely B-SIDH and SQISign, both require large primes that lie between two smooth integers. Finding such a prime can be seen as a special case of finding twin smooth integers under the additional stipulation that their sum is a prime.}, subject = {Kryptologie}, language = {en} } @phdthesis{CalaCampana2021, author = {Cal{\`a} Campana, Francesca}, title = {Numerical methods for solving open-loop non zero-sum differential Nash games}, doi = {10.25972/OPUS-24590}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-245900}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {This thesis is devoted to a theoretical and numerical investigation of methods to solve open-loop non zero-sum differential Nash games. These problems arise in many applications, e.g., biology, economics, physics, where competition between different agents appears. In this case, the goal of each agent is in contrast with those of the others, and a competition game can be interpreted as a coupled optimization problem for which, in general, an optimal solution does not exist. In fact, an optimal strategy for one player may be unsatisfactory for the others. For this reason, a solution of a game is sought as an equilibrium and among the solutions concepts proposed in the literature, that of Nash equilibrium (NE) is the focus of this thesis. The building blocks of the resulting differential Nash games are a dynamical model with different control functions associated with different players that pursue non-cooperative objectives. In particular, the aim of this thesis is on differential models having linear or bilinear state-strategy structures. In this framework, in the first chapter, some well-known results are recalled, especially for non-cooperative linear-quadratic differential Nash games. Then, a bilinear Nash game is formulated and analysed. The main achievement in this chapter is Theorem 1.4.2 concerning existence of Nash equilibria for non-cooperative differential bilinear games. This result is obtained assuming a sufficiently small time horizon T, and an estimate of T is provided in Lemma 1.4.8 using specific properties of the regularized Nikaido-Isoda function. In Chapter 2, in order to solve a bilinear Nash game, a semi-smooth Newton (SSN) scheme combined with a relaxation method is investigated, where the choice of a SSN scheme is motivated by the presence of constraints on the players' actions that make the problem non-smooth. The resulting method is proved to be locally convergent in Theorem 2.1, and an estimate on the relaxation parameter is also obtained that relates the relaxation factor to the time horizon of a Nash equilibrium and to the other parameters of the game. For the bilinear Nash game, a Nash bargaining problem is also introduced and discussed, aiming at determining an improvement of all players' objectives with respect to the Nash equilibrium. A characterization of a bargaining solution is given in Theorem 2.2.1 and a numerical scheme based on this result is presented that allows to compute this solution on the Pareto frontier. Results of numerical experiments based on a quantum model of two spin-particles and on a population dynamics model with two competing species are presented that successfully validate the proposed algorithms. In Chapter 3 a functional formulation of the classical homicidal chauffeur (HC) Nash game is introduced and a new numerical framework for its solution in a time-optimal formulation is discussed. This methodology combines a Hamiltonian based scheme, with proximal penalty to determine the time horizon where the game takes place, with a Lagrangian optimal control approach and relaxation to solve the Nash game at a fixed end-time. The resulting numerical optimization scheme has a bilevel structure, which aims at decoupling the computation of the end-time from the solution of the pursuit-evader game. Several numerical experiments are performed to show the ability of the proposed algorithm to solve the HC game. Focusing on the case where a collision may occur, the time for this event is determined. The last part of this thesis deals with the analysis of a novel sequential quadratic Hamiltonian (SQH) scheme for solving open-loop differential Nash games. This method is formulated in the framework of Pontryagin's maximum principle and represents an efficient and robust extension of the successive approximations strategy in the realm of Nash games. In the SQH method, the Hamilton-Pontryagin functions are augmented by a quadratic penalty term and the Nikaido-Isoda function is used as a selection criterion. Based on this fact, the key idea of this SQH scheme is that the PMP characterization of Nash games leads to a finite-dimensional Nash game for any fixed time. A class of problems for which this finite-dimensional game admits a unique solution is identified and for this class of games theoretical results are presented that prove the well-posedness of the proposed scheme. In particular, Proposition 4.2.1 is proved to show that the selection criterion on the Nikaido-Isoda function is fulfilled. A comparison of the computational performances of the SQH scheme and the SSN-relaxation method previously discussed is shown. Applications to linear-quadratic Nash games and variants with control constraints, weighted L1 costs of the players' actions and tracking objectives are presented that corroborate the theoretical statements.}, subject = {Differential Games}, language = {en} } @phdthesis{Raharja2021, author = {Raharja, Andreas Budi}, title = {Optimisation Problems with Sparsity Terms: Theory and Algorithms}, doi = {10.25972/OPUS-24195}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-241955}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {The present thesis deals with optimisation problems with sparsity terms, either in the constraints which lead to cardinality-constrained problems or in the objective function which in turn lead to sparse optimisation problems. One of the primary aims of this work is to extend the so-called sequential optimality conditions to these two classes of problems. In recent years sequential optimality conditions have become increasingly popular in the realm of standard nonlinear programming. In contrast to the more well-known Karush-Kuhn-Tucker condition, they are genuine optimality conditions in the sense that every local minimiser satisfies these conditions without any further assumption. Lately they have also been extended to mathematical programmes with complementarity constraints. At around the same time it was also shown that optimisation problems with sparsity terms can be reformulated into problems which possess similar structures to mathematical programmes with complementarity constraints. These recent developments have become the impetus of the present work. But rather than working with the aforementioned reformulations which involve an artifical variable we shall first directly look at the problems themselves and derive sequential optimality conditions which are independent of any artificial variable. Afterwards we shall derive the weakest constraint qualifications associated with these conditions which relate them to the Karush-Kuhn-Tucker-type conditions. Another equally important aim of this work is to then consider the practicability of the derived sequential optimality conditions. The previously mentioned reformulations open up the possibilities to adapt methods which have been proven successful to handle mathematical programmes with complementarity constraints. We will show that the safeguarded augmented Lagrangian method and some regularisation methods may generate a point satisfying the derived conditions.}, subject = {Optimierungsproblem}, language = {en} } @phdthesis{Wenz2021, author = {Wenz, Andreas}, title = {Computation of Belyi maps with prescribed ramification and applications in Galois theory}, doi = {10.25972/OPUS-24083}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-240838}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {We compute genus-0 Belyi maps with prescribed monodromy and strictly verify the computed results. Among the computed examples are almost simple primitive groups that satisfy the rational rigidity criterion yielding polynomials with prescribed Galois groups over Q(t). We also give an explicit version of a theorem of Magaard, which lists all sporadic groups occurring as composition factors of monodromy groups of rational functions.}, subject = {Galois-Theorie}, language = {en} } @phdthesis{Herrmann2021, author = {Herrmann, Marc}, title = {The Total Variation on Surfaces and of Surfaces}, doi = {10.25972/OPUS-24073}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-240736}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {This thesis is concerned with applying the total variation (TV) regularizer to surfaces and different types of shape optimization problems. The resulting problems are challenging since they suffer from the non-differentiability of the TV-seminorm, but unlike most other priors it favors piecewise constant solutions, which results in piecewise flat geometries for shape optimization problems.The first part of this thesis deals with an analogue of the TV image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. A rigorous analytical framework is developed for this model and its Fenchel predual, which is a quadratic optimization problem with pointwise inequality constraints on the surface. A function space interior point method is proposed to solve it. Afterwards, a discrete variant (DTV) based on a nodal quadrature formula is defined for piecewise polynomial, globally discontinuous and continuous finite element functions on triangulated surface meshes. DTV has favorable properties, which include a convenient dual representation. Next, an analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. Its analysis is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. Shape calculus is used to characterize the relevant derivatives and an variant of the split Bregman method for manifold valued functions is proposed. This is followed by an extension of the total variation prior for the normal vector field for piecewise flat surfaces and the previous variant of split Bregman method is adapted. Numerical experiments confirm that the new prior favours polyhedral shapes.}, subject = {Gestaltoptimierung}, language = {en} } @phdthesis{Reinwand2021, author = {Reinwand, Simon}, title = {Functions of Bounded Variation: Theory, Methods, Applications}, publisher = {Cuvillier-Verlag, G{\"o}ttingen}, isbn = {9783736974036}, doi = {10.25972/OPUS-23515}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-235153}, school = {Universit{\"a}t W{\"u}rzburg}, pages = {326}, year = {2021}, abstract = {Functions of bounded variation are most important in many fields of mathematics. This thesis investigates spaces of functions of bounded variation with one variable of various types, compares them to other classical function spaces and reveals natural "habitats" of BV-functions. New and almost comprehensive results concerning mapping properties like surjectivity and injectivity, several kinds of continuity and compactness of both linear and nonlinear operators between such spaces are given. A new theory about different types of convergence of sequences of such operators is presented in full detail and applied to a new proof for the continuity of the composition operator in the classical BV-space. The abstract results serve as ingredients to solve Hammerstein and Volterra integral equations using fixed point theory. Many criteria guaranteeing the existence and uniqueness of solutions in BV-type spaces are given and later applied to solve boundary and initial value problems in a nonclassical setting. A big emphasis is put on a clear and detailed discussion. Many pictures and synoptic tables help to visualize and summarize the most important ideas. Over 160 examples and counterexamples illustrate the many abstract results and how delicate some of them are.}, subject = {Funktion von beschr{\"a}nkter Variation}, language = {en} } @phdthesis{Moenius2021, author = {M{\"o}nius, Katja}, title = {Algebraic and Arithmetic Properties of Graph Spectra}, doi = {10.25972/OPUS-23085}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-230850}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {In the present thesis we investigate algebraic and arithmetic properties of graph spectra. In particular, we study the algebraic degree of a graph, that is the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals, and examine the question whether there is a relation between the algebraic degree of a graph and its structural properties. This generalizes the yet open question ``Which graphs have integral spectra?'' stated by Harary and Schwenk in 1974. We provide an overview of graph products since they are useful to study graph spectra and, in particular, to construct families of integral graphs. Moreover, we present a relation between the diameter, the maximum vertex degree and the algebraic degree of a graph, and construct a potential family of graphs of maximum algebraic degree. Furthermore, we determine precisely the algebraic degree of circulant graphs and find new criteria for isospectrality of circulant graphs. Moreover, we solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Those results generalize a theorem of So who characterized all integral circulant graphs. For our proofs we exploit the theory of Schur rings which was already used in order to solve the isomorphism problem for circulant graphs. Besides that, we study spectra of zero-divisor graphs over finite commutative rings. Given a ring \(R\), the zero-divisor graph over \(R\) is defined as the graph with vertex set being the set of non-zero zero-divisors of \(R\) where two vertices \(x,y\) are adjacent if and only if \(xy=0\). We investigate relations between the eigenvalues of a zero-divisor graph, its structural properties and the algebraic properties of the respective ring.}, subject = {Algebraische Zahlentheorie}, language = {en} } @phdthesis{Berberich2021, author = {Berberich, Jonas Philipp}, title = {Fluids in Gravitational Fields - Well-Balanced Modifications for Astrophysical Finite-Volume Codes}, doi = {10.25972/OPUS-21967}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-219679}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2021}, abstract = {Stellar structure can -- in good approximation -- be described as a hydrostatic state, which which arises due to a balance between gravitational force and pressure gradient. Hydrostatic states are static solutions of the full compressible Euler system with gravitational source term, which can be used to model the stellar interior. In order to carry out simulations of dynamical processes occurring in stars, it is vital for the numerical method to accurately maintain the hydrostatic state over a long time period. In this thesis we present different methods to modify astrophysical finite volume codes in order to make them \emph{well-balanced}, preventing them from introducing significant discretization errors close to hydrostatic states. Our well-balanced modifications are constructed so that they can meet the requirements for methods applied in the astrophysical context: They can well-balance arbitrary hydrostatic states with any equation of state that is applied to model thermodynamical relations and they are simple to implement in existing astrophysical finite volume codes. One of our well-balanced modifications follows given solutions exactly and can be applied on any grid geometry. The other methods we introduce, which do no require any a priori knowledge, balance local high order approximations of arbitrary hydrostatic states on a Cartesian grid. All of our modifications allow for high order accuracy of the method. The improved accuracy close to hydrostatic states is verified in various numerical experiments.}, subject = {Fluid}, language = {en} } @phdthesis{Boergens2020, author = {B{\"o}rgens, Eike Alexander Lars Guido}, title = {ADMM-Type Methods for Optimization and Generalized Nash Equilibrium Problems in Hilbert Spaces}, doi = {10.25972/OPUS-21877}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-218777}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {This thesis is concerned with a certain class of algorithms for the solution of constrained optimization problems and generalized Nash equilibrium problems in Hilbert spaces. This class of algorithms is inspired by the alternating direction method of multipliers (ADMM) and eliminates the constraints using an augmented Lagrangian approach. The alternating direction method consists of splitting the augmented Lagrangian subproblem into smaller and more easily manageable parts. Before the algorithms are discussed, a substantial amount of background material, including the theory of Banach and Hilbert spaces, fixed-point iterations as well as convex and monotone set-valued analysis, is presented. Thereafter, certain optimization problems and generalized Nash equilibrium problems are reformulated and analyzed using variational inequalities and set-valued mappings. The analysis of the algorithms developed in the course of this thesis is rooted in these reformulations as variational inequalities and set-valued mappings. The first algorithms discussed and analyzed are one weakly and one strongly convergent ADMM-type algorithm for convex, linearly constrained optimization. By equipping the associated Hilbert space with the correct weighted scalar product, the analysis of these two methods is accomplished using the proximal point method and the Halpern method. The rest of the thesis is concerned with the development and analysis of ADMM-type algorithms for generalized Nash equilibrium problems that jointly share a linear equality constraint. The first class of these algorithms is completely parallelizable and uses a forward-backward idea for the analysis, whereas the second class of algorithms can be interpreted as a direct extension of the classical ADMM-method to generalized Nash equilibrium problems. At the end of this thesis, the numerical behavior of the discussed algorithms is demonstrated on a collection of examples.}, subject = {Constrained optimization}, language = {en} } @phdthesis{Lauerbach2020, author = {Lauerbach, Laura}, title = {Stochastic Homogenization in the Passage from Discrete to Continuous Systems - Fracture in Composite Materials}, doi = {10.25972/OPUS-21453}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-214534}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {The work in this thesis contains three main topics. These are the passage from discrete to continuous models by means of \$\Gamma\$-convergence, random as well as periodic homogenization and fracture enabled by non-convex Lennard-Jones type interaction potentials. Each of them is discussed in the following. We consider a discrete model given by a one-dimensional chain of particles with randomly distributed interaction potentials. Our interest lies in the continuum limit, which yields the effective behaviour of the system. This limit is achieved as the number of atoms tends to infinity, which corresponds to a vanishing distance between the particles. The starting point of our analysis is an energy functional in a discrete system; its continuum limit is obtained by variational \$\Gamma\$-convergence. The \$\Gamma\$-convergence methods are combined with a homogenization process in the framework of ergodic theory, which allows to focus on heterogeneous systems. On the one hand, composite materials or materials with impurities are modelled by a stochastic or periodic distribution of particles or interaction potentials. On the other hand, systems of one species of particles can be considered as random in cases when the orientation of particles matters. Nanomaterials, like chains of atoms, molecules or polymers, are an application of the heterogeneous chains in experimental sciences. A special interest is in fracture in such heterogeneous systems. We consider interaction potentials of Lennard-Jones type. The non-standard growth conditions and the convex-concave structure of the Lennard-Jones type interactions yield mathematical difficulties, but allow for fracture. The interaction potentials are long-range in the sense that their modulus decays slower than exponential. Further, we allow for interactions beyond nearest neighbours, which is also referred to as long-range. The main mathematical issue is to bring together the Lennard-Jones type interactions with ergodic theorems in the limiting process as the number of particles tends to infinity. The blow up at zero of the potentials prevents from using standard extensions of the Akcoglu-Krengel subadditive ergodic theorem. We overcome this difficulty by an approximation of the interaction potentials which shows suitable Lipschitz and H{\"o}lder regularity. Beyond that, allowing for continuous probability distributions instead of only finitely many different potentials leads to a further challenge. The limiting integral functional of the energy by means of \$\Gamma\$-convergence involves a homogenized energy density and allows for fracture, but without a fracture contribution in the energy. In order to refine this result, we rescale our model and consider its \$\Gamma\$-limit, which is of Griffith's type consisting of an elastic part and a jump contribution. In a further approach we study fracture at the level of the discrete energies. With an appropriate definition of fracture in the discrete setting, we define a fracture threshold separating the region of elasticity from that of fracture and consider the pointwise convergence of this threshold. This limit turns out to coincide with the one obtained in the variational \$\Gamma\$-convergence approach.}, subject = {Homogenisierung }, language = {en} } @phdthesis{Karl2020, author = {Karl, Veronika}, title = {Augmented Lagrangian Methods for State Constrained Optimal Control Problems}, doi = {10.25972/OPUS-21384}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-213846}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {This thesis is concerned with the solution of control and state constrained optimal control problems, which are governed by elliptic partial differential equations. Problems of this type are challenging since they suffer from the low regularity of the multiplier corresponding to the state constraint. Applying an augmented Lagrangian method we overcome these difficulties by working with multiplier approximations in \$L^2(\Omega)\$. For each problem class, we introduce the solution algorithm, carry out a thoroughly convergence analysis and illustrate our theoretical findings with numerical examples. The thesis is divided into two parts. The first part focuses on classical PDE constrained optimal control problems. We start by studying linear-quadratic objective functionals, which include the standard tracking type term and an additional regularization term as well as the case, where the regularization term is replaced by an \$L^1(\Omega)\$-norm term, which makes the problem ill-posed. We deepen our study of the augmented Lagrangian algorithm by examining the more complicated class of optimal control problems that are governed by a semilinear partial differential equation. The second part investigates the broader class of multi-player control problems. While the examination of jointly convex generalized Nash equilibrium problems (GNEP) is a simple extension of the linear elliptic optimal control case, the complexity is increased significantly for pure GNEPs. The existence of solutions of jointly convex GNEPs is well-studied. However, solution algorithms may suffer from non-uniqueness of solutions. Therefore, the last part of this thesis is devoted to the analysis of the uniqueness of normalized equilibria.}, subject = {Optimale Kontrolle}, language = {en} } @phdthesis{Wisheckel2020, author = {Wisheckel, Florian}, title = {Some Applications of D-Norms to Probability and Statistics}, doi = {10.25972/OPUS-21214}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-212140}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {This cumulative dissertation is organized as follows: After the introduction, the second chapter, based on "Asymptotic independence of bivariate order statistics" (2017) by Falk and Wisheckel, is an investigation of the asymptotic dependence behavior of the components of bivariate order statistics. We find that the two components of the order statistics become asymptotically independent for certain combinations of (sequences of) indices that are selected, and it turns out that no further assumptions on the dependence of the two components in the underlying sample are necessary. To establish this, an explicit representation of the conditional distribution of bivariate order statistics is derived. Chapter 3 is from "Conditional tail independence in archimedean copula models" (2019) by Falk, Padoan and Wisheckel and deals with the conditional distribution of an Archimedean copula, conditioned on one of its components. We show that its tails are independent under minor conditions on the generator function, even if the unconditional tails were dependent. The theoretical findings are underlined by a simulation study and can be generalized to Archimax copulas. "Generalized pareto copulas: A key to multivariate extremes" (2019) by Falk, Padoan and Wisheckel lead to Chapter 4 where we introduce a nonparametric approach to estimate the probability that a random vector exceeds a fixed threshold if it follows a Generalized Pareto copula. To this end, some theory underlying the concept of Generalized Pareto distributions is presented first, the estimation procedure is tested using a simulation and finally applied to a dataset of air pollution parameters in Milan, Italy, from 2002 until 2017. The fifth chapter collects some additional results on derivatives of D-norms, in particular a condition for the existence of directional derivatives, and multivariate spacings, specifically an explicit formula for the second-to-last bivariate spacing.}, subject = {Kopula }, language = {en} } @phdthesis{Suttner2020, author = {Suttner, Raik}, title = {Output Optimization by Lie Bracket Approximations}, doi = {10.25972/OPUS-21177}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-211776}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {In this dissertation, we develop and analyze novel optimizing feedback laws for control-affine systems with real-valued state-dependent output (or objective) functions. Given a control-affine system, our goal is to derive an output-feedback law that asymptotically stabilizes the closed-loop system around states at which the output function attains a minimum value. The control strategy has to be designed in such a way that an implementation only requires real-time measurements of the output value. Additional information, like the current system state or the gradient vector of the output function, is not assumed to be known. A method that meets all these criteria is called an extremum seeking control law. We follow a recently established approach to extremum seeking control, which is based on approximations of Lie brackets. For this purpose, the measured output is modulated by suitable highly oscillatory signals and is then fed back into the system. Averaging techniques for control-affine systems with highly oscillatory inputs reveal that the closed-loop system is driven, at least approximately, into the directions of certain Lie brackets. A suitable design of the control law ensures that these Lie brackets point into descent directions of the output function. Under suitable assumptions, this method leads to the effect that minima of the output function are practically uniformly asymptotically stable for the closed-loop system. The present document extends and improves this approach in various ways. One of the novelties is a control strategy that does not only lead to practical asymptotic stability, but in fact to asymptotic and even exponential stability. In this context, we focus on the application of distance-based formation control in autonomous multi-agent system in which only distance measurements are available. This means that the target formations as well as the sensed variables are determined by distances. We propose a fully distributed control law, which only involves distance measurements for each individual agent to stabilize a desired formation shape, while a storage of measured data is not required. The approach is applicable to point agents in the Euclidean space of arbitrary (but finite) dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic (and even exponential) stability. A similar statement is also derived for nonholonomic unicycle agents with all-to-all communication. We also show how the findings can be used to solve extremum seeking control problems. Another contribution is an extremum seeking control law with an adaptive dither signal. We present an output-feedback law that steers a fully actuated control-affine system with general drift vector field to a minimum of the output function. A key novelty of the approach is an adaptive choice of the frequency parameter. In this way, the task of determining a sufficiently large frequency parameter becomes obsolete. The adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift. The proposed control law does not only lead to convergence into a neighborhood of a minimum, but leads to exact convergence. For the case of an output function with a global minimum and no other critical point, we prove global convergence. Finally, we present an extremum seeking control law for a class of nonholonomic systems. A detailed averaging analysis reveals that the closed-loop system is driven approximately into descent directions of the output function along Lie brackets of the control vector fields. Those descent directions also originate from an approximation of suitably chosen Lie brackets. This requires a two-fold approximation of Lie brackets on different time scales. The proposed method can lead to practical asymptotic stability even if the control vector fields do not span the entire tangent space. It suffices instead that the tangent space is spanned by the elements in the Lie algebra generated by the control vector fields. This novel feature extends extremum seeking by Lie bracket approximations from the class of fully actuated systems to a larger class of nonholonomic systems.}, subject = {Extremwertregelung}, language = {en} } @phdthesis{Rehberg2020, author = {Rehberg, Martin}, title = {Weighted uniform distribution related to primes and the Selberg Class}, doi = {10.25972/OPUS-20925}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-209252}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {In the thesis at hand, several sequences of number theoretic interest will be studied in the context of uniform distribution modulo one.

In the first part we deduce for positive and real \(z\not=1\) a discrepancy estimate for the sequence \( \left((2\pi )^{-1}(\log z)\gamma_a\right) \), where \(\gamma_a\) runs through the positive imaginary parts of the nontrivial \(a\)-points of the Riemann zeta-function. If the considered imaginary parts are bounded by \(T\), the discrepancy of the sequence \( \left((2\pi )^{-1}(\log z)\gamma_a\right) \) tends to zero like \( (\log\log\log T)^{-1} \) as \(T\rightarrow \infty\). The proof is related to the proof of Hlawka, who determined a discrepancy estimate for the sequence containing the positive imaginary parts of the nontrivial zeros of the Riemann zeta-function.

The second part of this thesis is about a sequence whose asymptotic behaviour is motivated by the sequence of primes. If \( \alpha\not=0\) is real and \(f\) is a function of logarithmic growth, we specify several conditions such that the sequence \( (\alpha f(q_n)) \) is uniformly distributed modulo one. The corresponding discrepancy estimates will be stated. The sequence \( (q_n)\) of real numbers is strictly increasing and the conditions on its counting function \( Q(x)=\\#\lbrace q_n \leq x \rbrace \) are satisfied by primes and primes in arithmetic progessions. As an application we obtain that the sequence \( \left( (\log q_n)^K\right)\) is uniformly distributed modulo one for arbitrary \(K>1\), if the \(q_n\) are primes or primes in arithmetic progessions. The special case that \(q_n\) equals the \(\textit{n}\)th prime number \(p_n\) was studied by Too, Goto and Kano.

In the last part of this thesis we study for irrational \(\alpha\) the sequence \( (\alpha p_n)\) of irrational multiples of primes in the context of weighted uniform distribution modulo one. A result of Vinogradov concerning exponential sums states that this sequence is uniformly distributed modulo one. An alternative proof due to Vaaler uses L-functions. We extend this approach in the context of the Selberg class with polynomial Euler product. By doing so, we obtain two weighted versions of Vinogradov's result: The sequence \( (\alpha p_n)\) is \( (1+\chi_{D}(p_n))\log p_n\)-uniformly distributed modulo one, where \( \chi_D\) denotes the Legendre-Kronecker character. In the proof we use the Dedekind zeta-function of the quadratic number field \( \Bbb Q (\sqrt{D})\). As an application we obtain in case of \(D=-1\), that \( (\alpha p_n)\) is uniformly distributed modulo one, if the considered primes are congruent to one modulo four. Assuming additional conditions on the functions from the Selberg class we prove that the sequence \( (\alpha p_n) \) is also \( (\sum_{j=1}^{\nu_F}{\alpha_j(p_n)})\log p_n\)-uniformly distributed modulo one, where the weights are related to the Euler product of the function.}, subject = {Zahlentheorie}, language = {en} } @phdthesis{Fuller2020, author = {Fuller, Timo}, title = {Contributions to the Multivariate Max-Domain of Attraction}, doi = {10.25972/OPUS-20742}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-207422}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {This thesis covers a wide range of results for when a random vector is in the max-domain of attraction of max-stable random vector. It states some new theoretical results in D-norm terminology, but also gives an explaination why most approaches to multivariate extremes are equivalent to this specific approach. Then it covers new methods to deal with high-dimensional extremes, ranging from dimension reduction to exploratory methods and explaining why the Huessler-Reiss model is a powerful parametric model in multivariate extremes on par with the multivariate Gaussian distribution in multivariate regular statistics. It also gives new results for estimating and inferring the multivariate extremal dependence structure, strategies for choosing thresholds and compares the behavior of local and global threshold approaches. The methods are demonstrated in an artifical simulation study, but also on German weather data.}, subject = {Extremwertstatistik}, language = {en} } @phdthesis{Seifert2020, author = {Seifert, Bastian}, title = {Multivariate Chebyshev polynomials and FFT-like algorithms}, doi = {10.25972/OPUS-20684}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-206845}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {This dissertation investigates the application of multivariate Chebyshev polynomials in the algebraic signal processing theory for the development of FFT-like algorithms for discrete cosine transforms on weight lattices of compact Lie groups. After an introduction of the algebraic signal processing theory, a multivariate Gauss-Jacobi procedure for the development of orthogonal transforms is proven. Two theorems on fast algorithms in algebraic signal processing, one based on a decomposition property of certain polynomials and the other based on induced modules, are proven as multivariate generalizations of prior theorems. The definition of multivariate Chebyshev polynomials based on the theory of root systems is recalled. It is shown how to use these polynomials to define discrete cosine transforms on weight lattices of compact Lie groups. Furthermore it is shown how to develop FFT-like algorithms for these transforms. Then the theory of matrix-valued, multivariate Chebyshev polynomials is developed based on prior ideas. Under an existence assumption a formula for generating functions of these matrix-valued Chebyshev polynomials is deduced.}, subject = {Schnelle Fourier-Transformation}, language = {en} } @phdthesis{Kann2020, author = {Kann, Lennart}, title = {Statistical Failure Prediction with an Account for Prior Information}, doi = {10.25972/OPUS-20504}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-205049}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {Prediction intervals are needed in many industrial applications. Frequently in mass production, small subgroups of unknown size with a lifetime behavior differing from the remainder of the population exist. A risk assessment for such a subgroup consists of two steps: i) the estimation of the subgroup size, and ii) the estimation of the lifetime behavior of this subgroup. This thesis covers both steps. An efficient practical method to estimate the size of a subgroup is presented and benchmarked against other methods. A prediction interval procedure which includes prior information in form of a Beta distribution is provided. This scheme is applied to the prediction of binomial and negative binomial counts. The effect of the population size on the prediction of the future number of failures is considered for a Weibull lifetime distribution, whose parameters are estimated from censored field data. Methods to obtain a prediction interval for the future number of failures with unknown sample size are presented. In many applications, failures are reported with a delay. The effects of such a reporting delay on the coverage properties of prediction intervals for the future number of failures are studied. The total failure probability of the two steps can be decomposed as a product probability. One-sided confidence intervals for such a product probability are presented.}, subject = {Konfidenzintervall}, language = {en} } @phdthesis{Sourmelidis2020, author = {Sourmelidis, Athanasios}, title = {Universality and Hypertranscendence of Zeta-Functions}, doi = {10.25972/OPUS-19369}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-193699}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2020}, abstract = {The starting point of the thesis is the {\it universality} property of the Riemann Zeta-function \$\zeta(s)\$ which was proved by Voronin in 1975: {\it Given a positive number \$\varepsilon>0\$ and an analytic non-vanishing function \$f\$ defined on a compact subset \$\mathcal{K}\$ of the strip \$\left\{s\in\mathbb{C}:1/2 < \Re s< 1\right\}\$ with connected complement, there exists a real number \$\tau\$ such that \begin{align}\label{continuous} \max\limits_{s\in \mathcal{K}}|\zeta(s+i\tau)-f(s)|<\varepsilon. \end{align} } In 1980, Reich proved a discrete analogue of Voronin's theorem, also known as {\it discrete universality theorem} for \$\zeta(s)\$: {\it If \$\mathcal{K}\$, \$f\$ and \$\varepsilon\$ are as before, then \begin{align}\label{discretee} \liminf\limits_{N\to\infty}\dfrac{1}{N}\sharp\left\{1\leq n\leq N:\max\limits_{s\in \mathcal{K}}|\zeta(s+i\Delta n)-f(s)|<\varepsilon\right\}>0, \end{align} where \$\Delta\$ is an arbitrary but fixed positive number. } We aim at developing a theory which can be applied to prove the majority of all so far existing discrete universality theorems in the case of Dirichlet \$L\$-functions \$L(s,\chi)\$ and Hurwitz zeta-functions \$\zeta(s;\alpha)\$, where \$\chi\$ is a Dirichlet character and \$\alpha\in(0,1]\$, respectively. Both of the aforementioned classes of functions are generalizations of \$\zeta(s)\$, since \$\zeta(s)=L(s,\chi_0)=\zeta(s;1)\$, where \$\chi_0\$ is the principal Dirichlet character mod 1. Amongst others, we prove statement (2) where instead of \$\zeta(s)\$ we have \$L(s,\chi)\$ for some Dirichlet character \$\chi\$ or \$\zeta(s;\alpha)\$ for some transcendental or rational number \$\alpha\in(0,1]\$, and instead of \$(\Delta n)_{n\in\mathbb{N}}\$ we can have: \begin{enumerate} \item \textit{Beatty sequences,} \item \textit{sequences of ordinates of \$c\$-points of zeta-functions from the Selberg class,} \item \textit{sequences which are generated by polynomials.} \end{enumerate} In all the preceding cases, the notion of {\it uniformly distributed sequences} plays an important role and we draw attention to it wherever we can. Moreover, for the case of polynomials, we employ more advanced techniques from Analytic Number Theory such as bounds of exponential sums and zero-density estimates for Dirichlet \$L\$-functions. This will allow us to prove the existence of discrete second moments of \$L(s,\chi)\$ and \$\zeta(s;\alpha)\$ on the left of the vertical line \$1+i\mathbb{R}\$, with respect to polynomials. In the case of the Hurwitz Zeta-function \$\zeta(s;\alpha)\$, where \$\alpha\$ is transcendental or rational but not equal to \$1/2\$ or 1, the target function \$f\$ in (1) or (2), where \$\zeta(\cdot)\$ is replaced by \$\zeta(\cdot;\alpha)\$, is also allowed to have zeros. Until recently there was no result regarding the universality of \$\zeta(s;\alpha)\$ in the literature whenever \$\alpha\$ is an algebraic irrational. In the second half of the thesis, we prove that a weak version of statement \eqref{continuous} for \$\zeta(s;\alpha)\$ holds for all but finitely many algebraic irrational \$\alpha\$ in \$[A,1]\$, where \$A\in(0,1]\$ is an arbitrary but fixed real number. Lastly, we prove that the ordinary Dirichlet series \$\zeta(s;f)=\sum_{n\geq1}f(n)n^{-s}\$ and \$\zeta_\alpha(s)=\sum_{n\geq1}\lfloor P(\alpha n+\beta)\rfloor^{-s}\$ are hypertranscendental, where \$f:\mathbb{N}\to\mathbb{C}\$ is a {\it Besicovitch almost periodic arithmetical function}, \$\alpha,\beta>0\$ are such that \$\lfloor\alpha+\beta\rfloor>1\$ and \$P\in\mathbb{Z}[X]\$ is such that \$P(\mathbb{N})\subseteq\mathbb{N}\$.}, subject = {Analytische Zahlentheorie}, language = {en} } @phdthesis{Promkam2019, author = {Promkam, Ratthaprom}, title = {Hybrid Dynamical Systems: Modeling, Stability and Interconnection}, doi = {10.25972/OPUS-19099}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-190993}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2019}, abstract = {This work deals with a class of nonlinear dynamical systems exhibiting both continuous and discrete dynamics, which is called as hybrid dynamical system. We provide a broader framework of generalized hybrid dynamical systems allowing us to handle issues on modeling, stability and interconnections. Various sufficient stability conditions are proposed by extensions of direct Lyapunov method. We also explicitly show Lyapunov formulations of the nonlinear small-gain theorems for interconnected input-to-state stable hybrid dynamical systems. Applications on modeling and stability of hybrid dynamical systems are given by effective strategies of vaccination programs to control a spread of disease in epidemic systems.}, subject = {Dynamical system}, language = {en} } @phdthesis{Breitenbach2019, author = {Breitenbach, Tim}, title = {A sequential quadratic Hamiltonian scheme for solving optimal control problems with non-smooth cost functionals}, doi = {10.25972/OPUS-18217}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-182170}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2019}, abstract = {This thesis deals with a new so-called sequential quadratic Hamiltonian (SQH) iterative scheme to solve optimal control problems with differential models and cost functionals ranging from smooth to discontinuous and non-convex. This scheme is based on the Pontryagin maximum principle (PMP) that provides necessary optimality conditions for an optimal solution. In this framework, a Hamiltonian function is defined that attains its minimum pointwise at the optimal solution of the corresponding optimal control problem. In the SQH scheme, this Hamiltonian function is augmented by a quadratic penalty term consisting of the current control function and the control function from the previous iteration. The heart of the SQH scheme is to minimize this augmented Hamiltonian function pointwise in order to determine a control update. Since the PMP does not require any differ- entiability with respect to the control argument, the SQH scheme can be used to solve optimal control problems with both smooth and non-convex or even discontinuous cost functionals. The main achievement of the thesis is the formulation of a robust and efficient SQH scheme and a framework in which the convergence analysis of the SQH scheme can be carried out. In this framework, convergence of the scheme means that the calculated solution fulfills the PMP condition. The governing differential models of the considered optimal control problems are ordinary differential equations (ODEs) and partial differential equations (PDEs). In the PDE case, elliptic and parabolic equations as well as the Fokker-Planck (FP) equation are considered. For both the ODE and the PDE cases, assumptions are formulated for which it can be proved that a solution to an optimal control problem has to fulfill the PMP. The obtained results are essential for the discussion of the convergence analysis of the SQH scheme. This analysis has two parts. The first one is the well-posedness of the scheme which means that all steps of the scheme can be carried out and provide a result in finite time. The second part part is the PMP consistency of the solution. This means that the solution of the SQH scheme fulfills the PMP conditions. In the ODE case, the following results are obtained that state well-posedness of the SQH scheme and the PMP consistency of the corresponding solution. Lemma 7 states the existence of a pointwise minimum of the augmented Hamiltonian. Lemma 11 proves the existence of a weight of the quadratic penalty term such that the minimization of the corresponding augmented Hamiltonian results in a control updated that reduces the value of the cost functional. Lemma 12 states that the SQH scheme stops if an iterate is PMP optimal. Theorem 13 proves the cost functional reducing properties of the SQH control updates. The main result is given in Theorem 14, which states the pointwise convergence of the SQH scheme towards a PMP consistent solution. In this ODE framework, the SQH method is applied to two optimal control problems. The first one is an optimal quantum control problem where it is shown that the SQH method converges much faster to an optimal solution than a globalized Newton method. The second optimal control problem is an optimal tumor treatment problem with a system of coupled highly non-linear state equations that describe the tumor growth. It is shown that the framework in which the convergence of the SQH scheme is proved is applicable for this highly non-linear case. Next, the case of PDE control problems is considered. First a general framework is discussed in which a solution to the corresponding optimal control problem fulfills the PMP conditions. In this case, many theoretical estimates are presented in Theorem 59 and Theorem 64 to prove in particular the essential boundedness of the state and adjoint variables. The steps for the convergence analysis of the SQH scheme are analogous to that of the ODE case and result in Theorem 27 that states the PMP consistency of the solution obtained with the SQH scheme. This framework is applied to different elliptic and parabolic optimal control problems, including linear and bilinear control mechanisms, as well as non-linear state equations. Moreover, the SQH method is discussed for solving a state-constrained optimal control problem in an augmented formulation. In this case, it is shown in Theorem 30 that for increasing the weight of the augmentation term, which penalizes the violation of the state constraint, the measure of this state constraint violation by the corresponding solution converges to zero. Furthermore, an optimal control problem with a non-smooth L\(^1\)-tracking term and a non-smooth state equation is investigated. For this purpose, an adjoint equation is defined and the SQH method is used to solve the corresponding optimal control problem. The final part of this thesis is devoted to a class of FP models related to specific stochastic processes. The discussion starts with a focus on random walks where also jumps are included. This framework allows a derivation of a discrete FP model corresponding to a continuous FP model with jumps and boundary conditions ranging from absorbing to totally reflecting. This discussion allows the consideration of the drift-control resulting from an anisotropic probability of the steps of the random walk. Thereafter, in the PMP framework, two drift-diffusion processes and the corresponding FP models with two different control strategies for an optimal control problem with an expectation functional are considered. In the first strategy, the controls depend on time and in the second one, the controls depend on space and time. In both cases a solution to the corresponding optimal control problem is characterized with the PMP conditions, stated in Theorem 48 and Theorem 49. The well-posedness of the SQH scheme is shown in both cases and further conditions are discussed that ensure the convergence of the SQH scheme to a PMP consistent solution. The case of a space and time dependent control strategy results in a special structure of the corresponding PMP conditions that is exploited in another solution method, the so-called direct Hamiltonian (DH) method.}, subject = {Optimale Kontrolle}, language = {en} } @phdthesis{Pohl2019, author = {Pohl, Daniel}, title = {Universal Locally Univalent Functions and Universal Conformal Metrics}, doi = {10.25972/OPUS-17717}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-177174}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2019}, abstract = {The work at hand discusses various universality results for locally univalent and conformal metrics. In Chapter 2 several interesting approximation results are discussed. Runge-type Theorems for holomorphic and meromorphic locally univalent functions are shown. A well-known local approximation theorem for harmonic functions due to Keldysh is generalized to solutions of the curvature equation. In Chapter 3 and 4 these approximation theorems are used to establish universality results for locally univalent functions and conformal metrics. In particular locally univalent analogues for well-known universality results due Birkhoff, Seidel \& Walsh and Heins are shown.}, subject = {Schlichte Funktion}, language = {en} } @phdthesis{Sans2019, author = {Sans, Wolfgang}, title = {Monotonic Probability Distribution : Characterisation, Measurements under Prior Information, and Application}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-175194}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2019}, abstract = {Statistical Procedures for modelling a random phenomenon heavily depend on the choice of a certain family of probability distributions. Frequently, this choice is governed by a good mathematical feasibility, but disregards that some distribution properties may contradict reality. At most, the choosen distribution may be considered as an approximation. The present thesis starts with a construction of distributions, which uses solely available information and yields distributions having greatest uncertainty in the sense of the maximum entropy principle. One of such distributions is the monotonic distribution, which is solely determined by its support and the mean. Although classical frequentist statistics provides estimation procedures which may incorporate prior information, such procedures are rarely considered. A general frequentist scheme for the construction of shortest confidence intervals for distribution parameters under prior information is presented. In particular, the scheme is used for establishing confidence intervals for the mean of the monotonic distribution and compared to classical procedures. Additionally, an approximative procedure for the upper bound of the support of the monotonic distribution is proposed. A core purpose of auditing sampling is the determination of confidence intervals for the mean of zero-inflated populations. The monotonic distribution is used for modelling such a population and is utilised for the procedure of a confidence interval under prior information for the mean. The results are compared to two-sided intervals of Stringer-type.}, subject = {Mathematik}, 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{Schoetz2018, author = {Sch{\"o}tz, Matthias}, title = {Convergent Star Products and Abstract O*-Algebras}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-174355}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {Diese Dissertation behandelt ein Problem aus der Deformationsquantisierung: Nachdem man die Quantisierung eines klassischen Systems konstruiert hat, w{\"u}rde man gerne ihre mathematischen Eigenschaften verstehen (sowohl die des klassischen Systems als auch die des Quantensystems). Falls beide Systeme durch *-Algebren {\"u}ber dem K{\"o}rper der komplexen Zahlen beschrieben werden, bedeutet dies dass man die Eigenschaften bestimmter *-Algebren verstehen muss: Welche Darstellungen gibt es? Was sind deren Eigenschaften? Wie k{\"o}nnen die Zust{\"a}nde in diesen Darstellungen beschrieben werden? Wie kann das Spektrum der Observablen beschrieben werden? Um eine hinreichend allgemeine Behandlung dieser Fragen zu erm{\"o}glichen, wird das Konzept von abstrakten O*-Algebren entwickelt. Dies sind im Wesentlichen *-Algebren zusammen mit einem Kegel positiver linearer Funktionale darauf (z.B. die stetigen positiven linearen Funktionale wenn man mit einer *-Algebra startet, die mit einer gutartigen Topologie versehen ist). Im Anschluss daran wird dieser Ansatz dann auf zwei Beispiele aus der Deformationsquantisierung angewandt, die im Detail untersucht werden.}, subject = {Deformationsquantisierung}, language = {en} } @phdthesis{Sapozhnikova2018, author = {Sapozhnikova, Kateryna}, title = {Robust Stability of Differential Equations with Maximum}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-173945}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {In this thesis stability and robustness properties of systems of functional differential equations which dynamics depends on the maximum of a solution over a prehistory time interval is studied. Max-operator is analyzed and it is proved that due to its presence such kind of systems are particular case of state dependent delay differential equations with piecewise continuous delay function. They are nonlinear, infinite-dimensional and may reduce to one-dimensional along its solution. Stability analysis with respect to input is accomplished by trajectory estimate and via averaging method. Numerical method is proposed.}, subject = {Differentialgleichung}, language = {en} } @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{Poerner2018, author = {P{\"o}rner, Frank}, title = {Regularization Methods for Ill-Posed Optimal Control Problems}, edition = {1. Auflage}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-086-3 (Print)}, doi = {10.25972/WUP-978-3-95826-087-0}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-163153}, school = {W{\"u}rzburg University Press}, pages = {xiii, 166}, year = {2018}, abstract = {This thesis deals with the construction and analysis of solution methods for a class of ill-posed optimal control problems involving elliptic partial differential equations as well as inequality constraints for the control and state variables. The objective functional is of tracking type, without any additional \(L^2\)-regularization terms. This makes the problem ill-posed and numerically challenging. We split this thesis in two parts. The first part deals with linear elliptic partial differential equations. In this case, the resulting solution operator of the partial differential equation is linear, making the objective functional linear-quadratic. To cope with additional control constraints we introduce and analyse an iterative regularization method based on Bregman distances. This method reduces to the proximal point method for a specific choice of the regularization functional. It turns out that this is an efficient method for the solution of ill-posed optimal control problems. We derive regularization error estimates under a regularity assumption which is a combination of a source condition and a structural assumption on the active sets. If additional state constraints are present we combine an augmented Lagrange approach with a Tikhonov regularization scheme to solve this problem. The second part deals with non-linear elliptic partial differential equations. This significantly increases the complexity of the optimal control as the associated solution operator of the partial differential equation is now non-linear. In order to regularize and solve this problem we apply a Tikhonov regularization method and analyse this problem with the help of a suitable second order condition. Regularization error estimates are again derived under a regularity assumption. These results are then extended to a sparsity promoting objective functional.}, subject = {Optimale Steuerung}, language = {en} } @phdthesis{Technau2018, author = {Technau, Marc}, title = {On Beatty sets and some generalisations thereof}, edition = {1. Auflage}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-088-7 (Print)}, doi = {10.25972/WUP-978-3-95826-089-4}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-163303}, school = {W{\"u}rzburg University Press}, pages = {xv, 88}, year = {2018}, abstract = {Beatty sets (also called Beatty sequences) have appeared as early as 1772 in the astronomical studies of Johann III Bernoulli as a tool for easing manual calculations and - as Elwin Bruno Christoffel pointed out in 1888 - lend themselves to exposing intricate properties of the real irrationals. Since then, numerous researchers have explored a multitude of arithmetic properties of Beatty sets; the interrelation between Beatty sets and modular inversion, as well as Beatty sets and the set of rational primes, being the central topic of this book. The inquiry into the relation to rational primes is complemented by considering a natural generalisation to imaginary quadratic number fields.}, subject = {Zahlentheorie}, language = {en} } @phdthesis{Pirner2018, author = {Pirner, Marlies}, title = {Kinetic modelling of gas mixtures}, edition = {1. Auflage}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-080-1 (Print)}, doi = {10.25972/WUP-978-3-95826-081-8}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-161077}, school = {W{\"u}rzburg University Press}, pages = {xi, 222}, year = {2018}, abstract = {This book deals with the kinetic modelling of gas mixtures. It extends the existing literature in mathematics for one species of gas to the case of gasmixtures. This is more realistic in applications. Thepresentedmodel for gas mixtures is proven to be consistentmeaning it satisfies theconservation laws, it admitsanentropy and an equilibriumstate. Furthermore, we can guarantee the existence, uniqueness and positivity of solutions. Moreover, the model is used for different applications, for example inplasma physics, for fluids with a small deviation from equilibrium and in the case of polyatomic gases.}, subject = {Polyatomare Verbindungen}, language = {en} } @phdthesis{Zenk2018, author = {Zenk, Markus}, title = {On Numerical Methods for Astrophysical Applications}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-162669}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {Diese Arbeit befasst sich mit der Approximation der L{\"o}sungen von Modellen zur Beschreibung des Str{\"o}mungsverhaltens in Atmosph{\"a}ren. Im Speziellen umfassen die hier behandelten Modelle die kompressiblen Euler Gleichungen der Gasdynamik mit einem Quellterm bez{\"u}glich der Gravitation und die Flachwassergleichungen mit einem nicht konstanten Bodenprofil. Verschiedene Methoden wurden bereits entwickelt um die L{\"o}sungen dieser Gleichungen zu approximieren. Im Speziellen geht diese Arbeit auf die Approximation von L{\"o}sungen nahe des Gleichgewichts und, im Falle der Euler Gleichungen, bei kleinen Mach Zahlen ein. Die meisten numerischen Methoden haben die Eigenschaft, dass die Qualit{\"a}t der Approximation sich mit der Anzahl der Freiheitsgrade verbessert. In der Praxis werden deswegen diese numerischen Methoden auf großen Computern implementiert um eine m{\"o}glichst hohe Approximationsg{\"u}te zu erreichen. Jedoch sind auch manchmal diese großen Maschinen nicht ausreichend, um die gew{\"u}nschte Qualit{\"a}t zu erreichen. Das Hauptaugenmerk dieser Arbeit ist darauf gerichtet, die Qualit{\"a}t der Approximation bei gleicher Anzahl von Freiheitsgrade zu verbessern. Diese Arbeit ist im Zusammenhang einer Kollaboration zwischen Prof. Klingenberg des Mathemaitschen Instituts in W{\"u}rzburg und Prof. R{\"o}pke des Astrophysikalischen Instituts in W{\"u}rzburg entstanden. Das Ziel dieser Kollaboration ist es, Methoden zur Berechnung von stellarer Atmosph{\"a}ren zu entwickeln. In dieser Arbeit werden vor allem zwei Problemstellungen behandelt. Die erste Problemstellung bezieht sich auf die akkurate Approximation des Quellterms, was zu den so genannten well-balanced Schemata f{\"u}hrt. Diese erlauben genaue Approximationen von L{\"o}sungen nahe des Gleichgewichts. Die zweite Problemstellung bezieht sich auf die Approximation von Str{\"o}mungen bei kleinen Mach Zahlen. Es ist bekannt, dass L{\"o}sungen der kompressiblen Euler Gleichungen zu L{\"o}sungen der inkompressiblen Euler Gleichungen konvergieren, wenn die Mach Zahl gegen null geht. Klassische numerische Schemata zeigen ein stark diffusives Verhalten bei kleinen Mach Zahlen. Das hier entwickelte Schema f{\"a}llt in die Kategorie der asymptotic preserving Schematas, d.h. das numerische Schema ist auf einem diskrete Level kompatibel mit dem auf dem Kontinuum gezeigten verhalten. Zus{\"a}tzlich wird gezeigt, dass die Diffusion des hier entwickelten Schemas unabh{\"a}ngig von der Mach Zahl ist. In Kapitel 3 wird ein HLL approximativer Riemann L{\"o}ser f{\"u}r die Approximation der L{\"o}sungen der Flachwassergleichungen mit einem nicht konstanten Bodenprofil angewendet und ein well-balanced Schema entwickelt. Die meisten well-balanced Schemata f{\"u}r die Flachwassergleichungen behandeln nur den Fall eines Fluids im Ruhezustand, die so genannten Lake at Rest L{\"o}sungen. Hier wird ein Schema entwickelt, welches sich mit allen Gleichgewichten befasst. Zudem wird eine zweiter Ordnung Methode entwickelt, welche im Gegensatz zu anderen in der Literatur nicht auf einem iterativen Verfahren basiert. Numerische Experimente werden durchgef{\"u}hrt um die Vorteile des neuen Verfahrens zu zeigen. In Kapitel 4 wird ein Suliciu Relaxations L{\"o}ser angepasst um die hydrostatischen Gleichgewichte der Euler Gleichungen mit einem Gravitationspotential aufzul{\"o}sen. Die Gleichungen der hydrostatischen Gleichgewichte sind unterbestimmt und lassen deshalb keine Eindeutigen L{\"o}sungen zu. Es wird jedoch gezeigt, dass das neue Schema f{\"u}r eine große Klasse dieser L{\"o}sungen die well-balanced Eigenschaft besitzt. F{\"u}r bestimmte Klassen werden Quadraturformeln zur Approximation des Quellterms entwickelt. Es wird auch gezeigt, dass das Schema robust, d.h. es erh{\"a}lt die Positivit{\"a}t der Masse und Energie, und stabil bez{\"u}glich der Entropieungleichung ist. Die numerischen Experimente konzentrieren sich vor allem auf den Einfluss der Quadraturformeln auf die well-balanced Eigenschaften. In Kapitel 5 wird ein Suliciu Relaxations Schema angepasst f{\"u}r Simulationen im Bereich kleiner Mach Zahlen. Es wird gezeigt, dass das neue Schema asymptotic preserving und die Diffusion kontrolliert ist. Zudem wird gezeigt, dass das Schema f{\"u}r bestimmte Parameter robust ist. Eine Stabilit{\"a}t wird aus einer Chapman-Enskog Analyse abgeleitet. Resultate numerische Experimente werden gezeigt um die Vorteile des neuen Verfahrens zu zeigen. In Kapitel 6 werden die Schemata aus den Kapiteln 4 und 5 kombiniert um das Verhalten des numerischen Schemas bei Fl{\"u}ssen mit kleiner Mach Zahl in durch die Gravitation geschichteten Atmosph{\"a}ren zu untersuchen. Es wird gezeigt, dass das Schema well-balanced ist. Die Robustheit und die Stabilit{\"a}t werden analog zu Kapitel 5 behandelt. Auch hier werden numerische Tests durchgef{\"u}hrt. Es zeigt sich, dass das neu entwickelte Schema in der Lage ist, die Dynamiken besser Aufzul{\"o}sen als vor der Anpassung. Das Kapitel 7 besch{\"a}ftigt sich mit der Entwicklung eines multidimensionalen Schemas basierend auf der Suliciu Relaxation. Jedoch ist die Arbeit an diesem Ansatz noch nicht beendet und numerische Resultate k{\"o}nnen nicht pr{\"a}sentiert werden. Es wird aufgezeigt, wo sich die Schw{\"a}chen dieses Ansatzes befinden und weiterer Entwicklungsbedarf besteht.}, subject = {Str{\"o}mung}, language = {en} } @phdthesis{Barsukow2018, author = {Barsukow, Wasilij}, title = {Low Mach number finite volume methods for the acoustic and Euler equations}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-159965}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {Finite volume methods for compressible Euler equations suffer from an excessive diffusion in the limit of low Mach numbers. This PhD thesis explores new approaches to overcome this. The analysis of a simpler set of equations that also possess a low Mach number limit is found to give valuable insights. These equations are the acoustic equations obtained as a linearization of the Euler equations. For both systems the limit is characterized by a divergencefree velocity. This constraint is nontrivial only in multiple spatial dimensions. As the Jacobians of the acoustic system do not commute, acoustics cannot be reduced to some kind of multi-dimensional advection. Therefore first an exact solution in multiple spatial dimensions is obtained. It is shown that the low Mach number limit can be interpreted as a limit of long times. It is found that the origin of the inability of a scheme to resolve the low Mach number limit is the lack a discrete counterpart to the limit of long times. Numerical schemes whose discrete stationary states discretize all the analytic stationary states of the PDE are called stationarity preserving. It is shown that for the acoustic equations, stationarity preserving schemes are vorticity preserving and are those that are able to resolve the low Mach limit (low Mach compliant). This establishes a new link between these three concepts. Stationarity preservation is studied in detail for both dimensionally split and multi-dimensional schemes for linear acoustics. In particular it is explained why the same multi-dimensional stencils appear in literature in very different contexts: These stencils are unique discretizations of the divergence that allow for stabilizing stationarity preserving diffusion. Stationarity preservation can also be generalized to nonlinear systems such as the Euler equations. Several ways how such numerical schemes can be constructed for the Euler equations are presented. In particular a low Mach compliant numerical scheme is derived that uses a novel construction idea. Its diffusion is chosen such that it depends on the velocity divergence rather than just derivatives of the different velocity components. This is demonstrated to overcome the low Mach number problem. The scheme shows satisfactory results in numerical simulations and has been found to be stable under explicit time integration.}, subject = {Finite-Volumen-Methode}, language = {en} } @phdthesis{Gathungu2018, author = {Gathungu, Duncan Kioi}, title = {On Multigrid and H-Matrix Methods for Partial Integro-Differential Equations}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-156430}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {The main theme of this thesis is the development of multigrid and hierarchical matrix solution procedures with almost linear computational complexity for classes of partial integro-differential problems. An elliptic partial integro-differential equation, a convection-diffusion partial integro-differential equation and a convection-diffusion partial integro-differential optimality system are investigated. In the first part of this work, an efficient multigrid finite-differences scheme for solving an elliptic Fredholm partial integro-differential equation (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization and a Simpson's quadrature rule to approximate the PIDE problem and a multigrid scheme and a fast multilevel integration method of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework that includes numerical experiments for elliptic PIDE problems with singular kernels. The experience gained in this part of the work is used for the investigation of convection diffusion partial-integro differential equations in the second part of this thesis. Convection-diffusion PIDE problems are discretized using a finite volume scheme referred to as the Chang and Cooper (CC) scheme and a quadrature rule. Also for this class of PIDE problems and this numerical setting, a stability and accuracy analysis of the CC scheme combined with a Simpson's quadrature rule is presented proving second-order accuracy of the numerical solution. To extend and investigate the proposed approximation and solution strategy to the case of systems of convection-diffusion PIDE, an optimal control problem governed by this model is considered. In this case the research focus is the CC-Simpson's discretization of the optimality system and its solution by the proposed multigrid strategy. Second-order accuracy of the optimization solution is proved and results of local Fourier analysis are presented that provide sharp convergence estimates of the optimal computational complexity of the multigrid-fast integration technique. While (geometric) multigrid techniques require ad-hoc implementation depending on the structure of the PIDE problem and on the dimensionality of the domain where the problem is considered, the hierarchical matrix framework allows a more general treatment that exploits the algebraic structure of the problem at hand. In this thesis, this framework is extended to the case of combined differential and integral problems considering the case of a convection-diffusion PIDE. In this case, the starting point is the CC discretization of the convection-diffusion operator combined with the trapezoidal quadrature rule. The hierarchical matrix approach exploits the algebraic nature of the hierarchical matrices for blockwise approximations by low-rank matrices of the sparse convection-diffusion approximation and enables data sparse representation of the fully populated matrix where all essential matrix operations are performed with at most logarithmic optimal complexity. The factorization of part of or the whole coefficient matrix is used as a preconditioner to the solution of the PIDE problem using a generalized minimum residual (GMRes) procedure as a solver. Numerical analysis estimates of the accuracy of the finite-volume and trapezoidal rule approximation are presented and combined with estimates of the hierarchical matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical estimates and the optimal computational complexity of the proposed hierarchical matrix solution procedure. These results include an extension to higher dimensions and an application to the time evolution of the probability density function of a jump diffusion process.}, subject = {Mehrgitterverfahren}, language = {en} } @phdthesis{Reichert2017, author = {Reichert, Thorsten}, title = {Classification and Reduction of Equivariant Star Products on Symplectic Manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-153623}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2017}, abstract = {This doctoral thesis provides a classification of equivariant star products (star products together with quantum momentum maps) in terms of equivariant de Rham cohomology. This classification result is then used to construct an analogon of the Kirwan map from which one can directly obtain the characteristic class of certain reduced star products on Marsden-Weinstein reduced symplectic manifolds from the equivariant characteristic class of their corresponding unreduced equivariant star product. From the surjectivity of this map one can conclude that every star product on Marsden-Weinstein reduced symplectic manifolds can (up to equivalence) be obtained as a reduced equivariant star product.}, subject = {Homologische Algebra}, language = {en} } @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{Sprengel2017, author = {Sprengel, Martin}, title = {A Theoretical and Numerical Analysis of a Kohn-Sham Equation and Related Control Problems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-153545}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2017}, abstract = {In this work, multi-particle quantum optimal control problems are studied in the framework of time-dependent density functional theory (TDDFT). Quantum control problems are of great importance in both fundamental research and application of atomic and molecular systems. Typical applications are laser induced chemical reactions, nuclear magnetic resonance experiments, and quantum computing. Theoretically, the problem of how to describe a non-relativistic system of multiple particles is solved by the Schr{\"o}dinger equation (SE). However, due to the exponential increase in numerical complexity with the number of particles, it is impossible to directly solve the Schr{\"o}dinger equation for large systems of interest. An efficient and successful approach to overcome this difficulty is the framework of TDDFT and the use of the time-dependent Kohn-Sham (TDKS) equations therein. This is done by replacing the multi-particle SE with a set of nonlinear single-particle Schr{\"o}dinger equations that are coupled through an additional potential. Despite the fact that TDDFT is widely used for physical and quantum chemical calculation and software packages for its use are readily available, its mathematical foundation is still under active development and even fundamental issues remain unproven today. The main purpose of this thesis is to provide a consistent and rigorous setting for the TDKS equations and of the related optimal control problems. In the first part of the thesis, the framework of density functional theory (DFT) and TDDFT are introduced. This includes a detailed presentation of the different functional sets forming DFT. Furthermore, the known equivalence of the TDKS system to the original SE problem is further discussed. To implement the TDDFT framework for multi-particle computations, the TDKS equations provide one of the most successful approaches nowadays. However, only few mathematical results concerning these equations are available and these results do not cover all issues that arise in the formulation of optimal control problems governed by the TDKS model. It is the purpose of the second part of this thesis to address these issues such as higher regularity of TDKS solutions and the case of weaker requirements on external (control) potentials that are instrumental for the formulation of well-posed TDKS control problems. For this purpose, in this work, existence and uniqueness of TDKS solutions are investigated in the Galerkin framework and using energy estimates for the nonlinear TDKS equations. In the third part of this thesis, optimal control problems governed by the TDKS model are formulated and investigated. For this purpose, relevant cost functionals that model the purpose of the control are discussed. Henceforth, TDKS control problems result from the requirement of optimising the given cost functionals subject to the differential constraint given by the TDKS equations. The analysis of these problems is novel and represents one of the main contributions of the present thesis. In particular, existence of minimizers is proved and their characterization by TDKS optimality systems is discussed in detail. To this end, Fr{\´e}chet differentiability of the TDKS model and of the cost functionals is addressed considering \(H^1\) cost of the control. This part is concluded by deriving the reduced gradient in the \(L^2\) and \(H^1\) inner product. While the \(L^2\) optimization is widespread in the literature, the choice of the \(H^1\) gradient is motivated in this work by theoretical consideration and by resulting numerical advantages. The last part of the thesis is devoted to the numerical approximation of the TDKS optimality systems and to their solution by gradient-based optimization techniques. For the former purpose, Strang time-splitting pseudo-spectral schemes are discussed including a review of some recent theoretical estimates for these schemes and a numerical validation of these estimates. For the latter purpose, nonlinear (projected) conjugate gradient methods are implemented and are used to validate the theoretical analysis of this thesis with results of numerical experiments with different cost functional settings.}, subject = {Optimale Kontrolle}, 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} } @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} }