@article{KanzowRaharjaSchwartz2021, author = {Kanzow, Christian and Raharja, Andreas B. and Schwartz, Alexandra}, title = {Sequential optimality conditions for cardinality-constrained optimization problems with applications}, series = {Computational Optimization and Applications}, volume = {80}, journal = {Computational Optimization and Applications}, number = {1}, issn = {1573-2894}, doi = {10.1007/s10589-021-00298-z}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269052}, pages = {185-211}, year = {2021}, abstract = {Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka-Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.}, 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{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{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{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} } @misc{Breitenbach2019, author = {Breitenbach, Tim}, title = {Codes of examples for SQH method}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-178588}, year = {2019}, abstract = {Code examples for the paper "On the SQH Scheme to Solve Nonsmooth PDE Optimal Control Problems" by Tim Breitenbach and Alfio Borz{\`i} published in the journal "Numerical Functional Analysis and Optimization", in 2019, DOI: 10.1080/01630563.2019.1599911}, 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{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} } @article{BartschBorziFanellietal.2021, author = {Bartsch, Jan and Borz{\`i}, Alfio and Fanelli, Francesco and Roy, Souvik}, title = {A numerical investigation of Brockett's ensemble optimal control problems}, series = {Numerische Mathematik}, volume = {149}, journal = {Numerische Mathematik}, number = {1}, doi = {10.1007/s00211-021-01223-6}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-265352}, pages = {1-42}, year = {2021}, abstract = {This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, 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. The resulting optimality system is solved by a projected semi-smooth Krylov-Newton method. Results of numerical experiments are presented that successfully validate the proposed framework.}, language = {en} } @article{MatlachDhillonHainetal.2015, author = {Matlach, Juliane and Dhillon, Christine and Hain, Johannes and Schlunck, G{\"u}nther and Grehn, Franz and Klink, Thomas}, title = {Trabeculectomy versus canaloplasty (TVC study) in the treatment of patients with open-angle glaucoma: a prospective randomized clinical trial}, series = {Acta Ophthalmologica}, volume = {93}, journal = {Acta Ophthalmologica}, doi = {10.1111/aos.12722}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-149263}, pages = {753-761}, year = {2015}, abstract = {Purpose: To compare the outcomes of canaloplasty and trabeculectomy in open-angle glaucoma. Methods: This prospective, randomized clinical trial included 62 patients who randomly received trabeculectomy (n = 32) or canaloplasty (n = 30) and were followed up prospectively for 2 years. Primary endpoint was complete (without medication) and qualified success (with or without medication) defined as an intraocular pressure (IOP) of ≤18 mmHg (definition 1) or IOP ≤21 mmHg and ≥20\% IOP reduction (definition 2), IOP ≥5 mmHg, no vision loss and no further glaucoma surgery. Secondary endpoints were the absolute IOP reduction, visual acuity, medication, complications and second surgeries. Results: Surgical treatment significantly reduced IOP in both groups (p < 0.001). Complete success was achieved in 74.2\% and 39.1\% (definition 1, p = 0.01), and 67.7\% and 39.1\% (definition 2, p = 0.04) after 2 years in the trabeculectomy and canaloplasty group, respectively. Mean absolute IOP reduction was 10.8 ± 6.9 mmHg in the trabeculectomy and 9.3 ± 5.7 mmHg in the canaloplasty group after 2 years (p = 0.47). Mean IOP was 11.5 ± 3.4 mmHg in the trabeculectomy and 14.4 ± 4.2 mmHg in the canaloplasty group after 2 years. Following trabeculectomy, complications were more frequent including hypotony (37.5\%), choroidal detachment (12.5\%) and elevated IOP (25.0\%). Conclusions: Trabeculectomy is associated with a stronger IOP reduction and less need for medication at the cost of a higher rate of complications. If target pressure is attainable by moderate IOP reduction, canaloplasty may be considered for its relative ease of postoperative care and lack of complications.}, language = {en} } @article{HellmuthKlingenberg2022, author = {Hellmuth, Kathrin and Klingenberg, Christian}, title = {Computing Black Scholes with uncertain volatility — a machine learning approach}, series = {Mathematics}, volume = {10}, journal = {Mathematics}, number = {3}, issn = {2227-7390}, doi = {10.3390/math10030489}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-262280}, year = {2022}, abstract = {In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.}, language = {en} } @phdthesis{Mungenast2022, author = {Mungenast, Sebastian}, title = {Zur Bedeutung von Metakognition beim Umgang mit Mathematik - Dokumentation metakognitiver Aktivit{\"a}ten bei Studienanf{\"a}nger_innen, Entwicklung eines Kategoriensystems f{\"u}r Metakognition beim Umgang mit Mathematik und Er{\"o}rterung von Ansatzpunkten f{\"u}r Metakognition in der Analysis}, doi = {10.25972/OPUS-29311}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-293114}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2022}, abstract = {Die vorliegende Arbeit besch{\"a}ftigt sich explorativ mit Metakognition beim Umgang mit Mathematik. Aufbauend auf der vorgestellten Forschungsliteratur wird der Einsatz von Metakognition im Rahmen einer qualitativen Studie bei Studienanf{\"a}nger_innen aus verschiedenen Mathematik-(Lehramts-)Studieng{\"a}ngen dokumentiert. Unter Verwendung der Qualitativen Inhaltsanalyse nach Mayring erfolgt die Etablierung eines Kategoriensystems f{\"u}r den Begriff Metakognition im Hinblick auf den Einsatz in der Mathematik, das bisherige Systematisierungen erweitert. Schließlich wird der Einsatz der entsprechenden metakognitiven Aspekte am Beispiel verschiedener Begriffe und Verfahren aus dem Analysis-Unterricht exemplarisch aufgezeigt.}, subject = {Metakognition}, language = {de} } @phdthesis{Nedrenco2022, author = {Nedrenco, Dmitri}, title = {Axiomatisieren lernen mit Papierfalten : Entwicklung, Durchf{\"u}hrung und Auswertung eines Hochschulkurses f{\"u}r gymnasiale Lehramtsstudierende}, doi = {10.25972/OPUS-27938}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-279383}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2022}, abstract = {In dieser Arbeit wird mathematisches Papierfalten und speziell 1-fach-Origami im universitären Kontext untersucht. Die Arbeit besteht aus drei Teilen. Der erste Teil ist im Wesentlichen der Sachanalyse des 1-fach-Origami gewidmet. Im ersten Kapitel gehen wir auf die geschichtliche Einordnung des 1-fach-Origami, betrachten axiomatische Grundlagen und diskutieren, wie das Axiomatisieren von 1-fach-Origami zum Verständnis des Axiomenbegriffs beitragen könnte. Im zweiten Kapitel schildern wir das Design der zugehörigen explorativen Studie, beschreiben unsere Forschungsziele und -fragen. Im dritten Kapitel wird 1-fach-Origami mathematisiert, definiert und eingehend untersucht. Der zweite Teil beschäftigt sich mit den von uns gestalteten und durchgef{\"u}hrten Kursen »Axiomatisieren lernen mit Papierfalten«. Im vierten Kapitel beschreiben wir die Lehrmethodik und die Gestaltung der Kurse, das f{\"u}nfte Kapitel enthält ein Exzerpt der Kurse. Im dritten Teil werden die zugehörigen Tests beschrieben. Im sechsten Kapitel erläutern wir das Design der Tests sowie die Testmethodik. Im siebten Kapitel findet die Auswertung ebendieser Tests statt.}, subject = {Mathematikunterricht}, language = {de} } @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{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} } @article{delAlamoLiMunketal.2020, author = {del Alamo, Miguel and Li, Housen and Munk, Axel and Werner, Frank}, title = {Variational Multiscale Nonparametric Regression: Algorithms and Implementation}, series = {Algorithms}, volume = {13}, journal = {Algorithms}, number = {11}, issn = {1999-4893}, doi = {10.3390/a13110296}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-219332}, year = {2020}, abstract = {Many modern statistically efficient methods come with tremendous computational challenges, often leading to large-scale optimisation problems. In this work, we examine such computational issues for recently developed estimation methods in nonparametric regression with a specific view on image denoising. We consider in particular certain variational multiscale estimators which are statistically optimal in minimax sense, yet computationally intensive. Such an estimator is computed as the minimiser of a smoothness functional (e.g., TV norm) over the class of all estimators such that none of its coefficients with respect to a given multiscale dictionary is statistically significant. The so obtained multiscale Nemirowski-Dantzig estimator (MIND) can incorporate any convex smoothness functional and combine it with a proper dictionary including wavelets, curvelets and shearlets. The computation of MIND in general requires to solve a high-dimensional constrained convex optimisation problem with a specific structure of the constraints induced by the statistical multiscale testing criterion. To solve this explicitly, we discuss three different algorithmic approaches: the Chambolle-Pock, ADMM and semismooth Newton algorithms. Algorithmic details and an explicit implementation is presented and the solutions are then compared numerically in a simulation study and on various test images. We thereby recommend the Chambolle-Pock algorithm in most cases for its fast convergence. We stress that our analysis can also be transferred to signal recovery and other denoising problems to recover more general objects whenever it is possible to borrow statistical strength from data patches of similar object structure.}, 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{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} } @misc{Breitenbach2018, author = {Breitenbach, Tim}, title = {Codes of examples for SQH method}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-167587}, year = {2018}, abstract = {Code examples for the paper "On the SQH Scheme to Solve Nonsmooth PDE Optimal Control Problems" by Tim Breitenbach and Alfio Borz{\`i} published in the journal "Numerical Functional Analysis and Optimization", in 2019, DOI: 10.1080/01630563.2019.1599911}, 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} } @article{BreitenbachBorzi2020, author = {Breitenbach, Tim and Borz{\`i}, Alfio}, title = {The Pontryagin maximum principle for solving Fokker-Planck optimal control problems}, series = {Computational Optimization and Applications}, volume = {76}, journal = {Computational Optimization and Applications}, issn = {0926-6003}, doi = {10.1007/s10589-020-00187-x}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232665}, pages = {499-533}, year = {2020}, abstract = {The characterization and numerical solution of two non-smooth optimal control problems governed by a Fokker-Planck (FP) equation are investigated in the framework of the Pontryagin maximum principle (PMP). The two FP control problems are related to the problem of determining open- and closed-loop controls for a stochastic process whose probability density function is modelled by the FP equation. In both cases, existence and PMP characterisation of optimal controls are proved, and PMP-based numerical optimization schemes are implemented that solve the PMP optimality conditions to determine the controls sought. Results of experiments are presented that successfully validate the proposed computational framework and allow to compare the two control strategies.}, 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} } @article{HomburgWeissFrahmetal.2021, author = {Homburg, Annika and Weiß, Christian H. and Frahm, Gabriel and Alwan, Layth C. and G{\"o}b, Rainer}, title = {Analysis and forecasting of risk in count processes}, series = {Journal of Risk and Financial Management}, volume = {14}, journal = {Journal of Risk and Financial Management}, number = {4}, issn = {1911-8074}, doi = {10.3390/jrfm14040182}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-236692}, year = {2021}, abstract = {Risk measures are commonly used to prepare for a prospective occurrence of an adverse event. If we are concerned with discrete risk phenomena such as counts of natural disasters, counts of infections by a serious disease, or counts of certain economic events, then the required risk forecasts are to be computed for an underlying count process. In practice, however, the discrete nature of count data is sometimes ignored and risk forecasts are calculated based on Gaussian time series models. But even if methods from count time series analysis are used in an adequate manner, the performance of risk forecasting is affected by estimation uncertainty as well as certain discreteness phenomena. To get a thorough overview of the aforementioned issues in risk forecasting of count processes, a comprehensive simulation study was done considering a broad variety of risk measures and count time series models. It becomes clear that Gaussian approximate risk forecasts substantially distort risk assessment and, thus, should be avoided. In order to account for the apparent estimation uncertainty in risk forecasting, we use bootstrap approaches for count time series. The relevance and the application of the proposed approaches are illustrated by real data examples about counts of storm surges and counts of financial transactions.}, language = {en} } @article{Pirner2021, author = {Pirner, Marlies}, title = {A review on BGK models for gas mixtures of mono and polyatomic molecules}, series = {Fluids}, volume = {6}, journal = {Fluids}, number = {11}, issn = {2311-5521}, doi = {10.3390/fluids6110393}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-250161}, year = {2021}, abstract = {We consider the Bathnagar-Gross-Krook (BGK) model, an approximation of the Boltzmann equation, describing the time evolution of a single momoatomic rarefied gas and satisfying the same two main properties (conservation properties and entropy inequality). However, in practical applications, one often has to deal with two additional physical issues. First, a gas often does not consist of only one species, but it consists of a mixture of different species. Second, the particles can store energy not only in translational degrees of freedom but also in internal degrees of freedom such as rotations or vibrations (polyatomic molecules). Therefore, here, we will present recent BGK models for gas mixtures for mono- and polyatomic particles and the existing mathematical theory for these models.}, language = {en} } @article{SteudingSuriajaya2020, author = {Steuding, J{\"o}rn and Suriajaya, Ade Irma}, title = {Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines}, series = {Computational Methods and Function Theory}, volume = {20}, journal = {Computational Methods and Function Theory}, issn = {1617-9447}, doi = {10.1007/s40315-020-00316-x}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232621}, pages = {389-401}, year = {2020}, abstract = {For an arbitrary complex number a≠0 we consider the distribution of values of the Riemann zeta-function ζ at the a-points of the function Δ which appears in the functional equation ζ(s)=Δ(s)ζ(1-s). These a-points δa are clustered around the critical line 1/2+i\(\mathbb {R}\) which happens to be a Julia line for the essential singularity of ζ at infinity. We observe a remarkable average behaviour for the sequence of values ζ(δ\(_a\)).}, 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{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} }