@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{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{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{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{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} } @article{Weishaeupl2023, author = {Weish{\"a}upl, Sebastian}, title = {The weak Gram law for Hecke \(L\)-functions}, series = {The Ramanujan Journal}, volume = {60}, journal = {The Ramanujan Journal}, number = {4}, issn = {1382-4090}, doi = {10.1007/s11139-022-00638-5}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324404}, pages = {981-997}, year = {2023}, abstract = {We generalize a theorem by Titchmarsh about the mean value of Hardy's \(Z\)-function at the Gram points to the Hecke \(L\)-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration.}, language = {en} } @article{LuMoenius2023, author = {Lu, Lu and M{\"o}nius, Katja}, title = {Algebraic degree of Cayley graphs over abelian groups and dihedral groups}, series = {Journal of Algebraic Combinatorics}, volume = {57}, journal = {Journal of Algebraic Combinatorics}, number = {3}, issn = {0925-9899}, doi = {10.1007/s10801-022-01190-7}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324380}, pages = {753-761}, year = {2023}, abstract = {For a graph \(\Gamma\) , let K be the smallest field containing all eigenvalues of the adjacency matrix of \(\Gamma\) . The algebraic degree \(\deg (\Gamma )\) is the extension degree \([K:\mathbb {Q}]\). In this paper, we completely determine the algebraic degrees of Cayley graphs over abelian groups and dihedral groups.}, language = {en} } @article{GreefrathOldenburgSilleretal.2023, author = {Greefrath, Gilbert and Oldenburg, Reinhard and Siller, Hans-Stefan and Ulm, Volker and Weigand, Hans-Georg}, title = {Mathematics students' characteristics of basic mental models of the derivative}, series = {Journal f{\"u}r Mathematik-Didaktik}, volume = {44}, journal = {Journal f{\"u}r Mathematik-Didaktik}, number = {1}, issn = {0173-5322}, doi = {10.1007/s13138-022-00207-9}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324317}, pages = {143-169}, year = {2023}, abstract = {The concept of derivative is characterised with reference to four basic mental models. These are described as theoretical constructs based on theoretical considerations. The four basic mental models—local rate of change, tangent slope, local linearity and amplification factor—are not only quantified empirically but are also validated. To this end, a test instrument for measuring students' characteristics of basic mental models is presented and analysed regarding quality criteria. Mathematics students (n = 266) were tested with this instrument. The test results show that the four basic mental models of the derivative can be reconstructed among the students with different characteristics. The tangent slope has the highest agreement values across all tasks. The agreement on explanations based on the basic mental model of rate of change is not as strongly established among students as one would expect due to framework settings in the school system by means of curricula and educational standards. The basic mental model of local linearity plays a rather subordinate role. The amplification factor achieves the lowest agreement values. In addition, cluster analysis was conducted to identify different subgroups of the student population. Moreover, the test results can be attributed to characteristics of the task types as well as to the students' previous experiences from mathematics classes by means of qualitative interpretation. These and other results of students' basic mental models of the derivative are presented and discussed in detail.}, language = {en} } @article{SillerElschenbroichGreefrathetal.2023, author = {Siller, Hans-Stefan and Elschenbroich, Hans-J{\"u}rgen and Greefrath, Gilbert and Vorh{\"o}lter, Katrin}, title = {Mathematical modelling of exponential growth as a rich learning environment for mathematics classrooms}, series = {ZDM Mathematics Education}, volume = {55}, journal = {ZDM Mathematics Education}, number = {1}, issn = {1863-9690}, doi = {10.1007/s11858-022-01433-8}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324393}, pages = {17-33}, year = {2023}, abstract = {Mathematical concepts are regularly used in media reports concerning the Covid-19 pandemic. These include growth models, which attempt to explain or predict the effectiveness of interventions and developments, as well as the reproductive factor. Our contribution has the aim of showing that basic mental models about exponential growth are important for understanding media reports of Covid-19. Furthermore, we highlight how the coronavirus pandemic can be used as a context in mathematics classrooms to help students understand that they can and should question media reports on their own, using their mathematical knowledge. Therefore, we first present the role of mathematical modelling in achieving these goals in general. The same relevance applies to the necessary basic mental models of exponential growth. Following this description, based on three topics, namely, investigating the type of growth, questioning given course models, and determining exponential factors at different times, we show how the presented theoretical aspects manifest themselves in teaching examples when students are given the task of reflecting critically on existing media reports. Finally, the value of the three topics regarding the intended goals is discussed and conclusions concerning the possibilities and limits of their use in schools are drawn.}, language = {en} } @article{SteudingTongsomporn2023, author = {Steuding, J{\"o}rn and Tongsomporn, Janyarak}, title = {On the order of growth of Lerch zeta functions}, series = {Mathematics}, volume = {11}, journal = {Mathematics}, number = {3}, issn = {2227-7390}, doi = {10.3390/math11030723}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-303981}, year = {2023}, abstract = {We extend Bourgain's bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t\(^{13/84+ϵ}\) as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t\(^ϵ\) (which is the so-called Lindel{\"o}f hypothesis). The growth of an analytic function is closely related to the distribution of its zeros.}, language = {en} } @article{HeinsRothWaldmann2023, author = {Heins, Michael and Roth, Oliver and Waldmann, Stefan}, title = {Convergent star products on cotangent bundles of Lie groups}, series = {Mathematische Annalen}, volume = {386}, journal = {Mathematische Annalen}, number = {1-2}, issn = {0025-5831}, doi = {10.1007/s00208-022-02384-x}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324324}, pages = {151-206}, year = {2023}, abstract = {For a connected real Lie group G we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of G. This star product trivially converges on polynomial functions on T\(^*\)G thanks to its homogeneity. We define a nuclear Fr{\´e}chet algebra of certain analytic functions on T\(^*\)G, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter \(\hbar\). This nuclear Fr{\´e}chet algebra is realized as the completed (projective) tensor product of a nuclear Fr{\´e}chet algebra of entire functions on G with an appropriate nuclear Fr{\´e}chet algebra of functions on \({\mathfrak {g}}^*\). The passage to the Weyl-ordered star product, i.e. the Gutt star product on T\(^*\)G, is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on \(\hbar\).}, language = {en} } @article{JotzMehtaPapantonis2023, author = {Jotz, M. and Mehta, R. A. and Papantonis, T.}, title = {Modules and representations up to homotopy of Lie n-algebroids}, series = {Journal of Homotopy and Related Structures}, volume = {18}, journal = {Journal of Homotopy and Related Structures}, number = {1}, issn = {2193-8407}, doi = {10.1007/s40062-022-00322-x}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324333}, pages = {23-70}, year = {2023}, abstract = {This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general \(n\in {\mathbb {N}}\). The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles.}, language = {en} } @article{GerberQuarderGreefrathetal.2023, author = {Gerber, Sebastian and Quarder, Jascha and Greefrath, Gilbert and Siller, Hans-Stefan}, title = {Promoting adaptive intervention competence for teaching simulations and mathematical modelling with digital tools}, series = {Frontiers in Education}, volume = {8}, journal = {Frontiers in Education}, issn = {2504-284X}, doi = {10.3389/feduc.2023.1141063}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-323701}, year = {2023}, abstract = {Providing adaptive, independence-preserving and theory-guided support to students in dealing with real-world problems in mathematics lessons is a major challenge for teachers in their professional practice. This paper examines this challenge in the context of simulations and mathematical modelling with digital tools: in addition to mathematical difficulties when autonomously working out individual solutions, students may also experience challenges when using digital tools. These challenges need to be closely examined and diagnosed, and might - if necessary - have to be overcome by intervention in such a way that the students can subsequently continue working independently. Thus, if a difficulty arises in the working process, two knowledge dimensions are necessary in order to provide adapted support to students. For teaching simulations and mathematical modelling with digital tools, more specifically, these knowledge dimensions are: pedagogical content knowledge about simulation and modelling processes supported by digital tools (this includes knowledge about phases and difficulties in the working process) and pedagogical content knowledge about interventions during the mentioned processes (focussing on characteristics of suitable interventions as well as their implementation and effects on the students' working process). The two knowledge dimensions represent cognitive dispositions as the basis for the conceptualisation and operationalisation of a so-called adaptive intervention competence for teaching simulations and mathematical modelling with digital tools. In our article, we present a domain-specific process model and distinguish different types of teacher interventions. Then we describe the design and content of a university course at two German universities aiming to promote this domain-specific professional adaptive intervention competence, among others. In a study using a quasi-experimental pre-post design (N = 146), we confirm that the structure of cognitive dispositions of adaptive intervention competence for teaching simulations and mathematical modelling with digital tools can be described empirically by a two-dimensional model. In addition, the effectiveness of the course is examined and confirmed quantitatively. Finally, the results are discussed, especially against the background of the sample and the research design, and conclusions are derived for possibilities of promoting professional adaptive intervention competence in university courses.}, 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{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} } @article{KakaroumpasSoleriGibert2022, author = {Kakaroumpas, Spyridon and Soler i Gibert, Od{\´i}}, title = {Dyadic product BMO in the Bloom setting}, series = {Journal of the London Mathematical Society}, volume = {106}, journal = {Journal of the London Mathematical Society}, number = {2}, issn = {0024-6107}, doi = {10.1112/jlms.12588}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-318602}, pages = {899 -- 935}, year = {2022}, abstract = {{\´O}. Blasco and S. Pott showed that the supremum of operator norms over L\(^{2}\) of all bicommutators (with the same symbol) of one-parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1 < p < ∞. The main tool is a new characterization in terms of paraproducts and two-weight John-Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multiparameter setting, with the appropriate iterated commutator in each case.}, 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} } @article{DashkovskiySlynko2022, author = {Dashkovskiy, Sergey and Slynko, Vitalii}, title = {Stability conditions for impulsive dynamical systems}, series = {Mathematics of Control, Signals, and Systems}, volume = {34}, journal = {Mathematics of Control, Signals, and Systems}, number = {1}, issn = {1435-568X}, doi = {10.1007/s00498-021-00305-y}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-268390}, pages = {95-128}, year = {2022}, abstract = {In this work, we consider impulsive dynamical systems evolving on an infinite-dimensional space and subjected to external perturbations. We look for stability conditions that guarantee the input-to-state stability for such systems. Our new dwell-time conditions allow the situation, where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov like methods are developed for this purpose. Illustrative finite and infinite dimensional examples are provided to demonstrate the application of the main results. These examples cannot be treated by any other published approach and demonstrate the effectiveness of our results.}, 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{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} } @article{HelinKretschmann2022, author = {Helin, Tapio and Kretschmann, Remo}, title = {Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems}, series = {Numerische Mathematik}, volume = {150}, journal = {Numerische Mathematik}, number = {2}, doi = {10.1007/s00211-021-01266-9}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-265399}, pages = {521-549}, year = {2022}, abstract = {In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915-971, 2020. https://doi.org/10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.}, language = {en} } @techreport{GerberQuarder2022, author = {Gerber, Sebastian and Quarder, Jascha}, title = {Erfassung von Aspekten professioneller Kompetenz zum Lehren des Simulierens und mathematischen Modellierens mit digitalen Werkzeugen. Ein Testinstrument}, doi = {10.25972/OPUS-27359}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-273597}, pages = {42}, year = {2022}, abstract = {Die Auseinandersetzung mit Simulations- und Modellierungsaufgaben, die mit digitalen Werkzeugen zu bearbeiten sind, stellt ver{\"a}nderte Anforderungen an Mathematiklehrkr{\"a}fte in der Unterrichtsplanung und -durchf{\"u}hrung. Werden digitale Werkzeuge sinnvoll eingesetzt, so unterst{\"u}tzen sie Simulations- und Modellierungsprozesse und erm{\"o}glichen realit{\"a}tsn{\"a}here Sachkontexte im Mathematikunterricht. F{\"u}r die empirische Untersuchung professioneller Kompetenzen zum Lehren des Simulierens und mathematischen Modellierens mit digitalen Werkzeugen ist es notwendig, Aspekte globaler Lehrkompetenzen von (angehenden) Mathematiklehrkr{\"a}ften bereichsspezifisch auszudeuten. Daher haben wir ein Testinstrument entwickelt, das die {\"U}berzeugungen, die Selbstwirksamkeitserwartungen und das fachdidaktische Wissen zum Lehren des Simulierens und mathematischen Modellierens mit digitalen Werkzeugen erfasst. Erg{\"a}nzt wird das Testinstrument durch selbstberichtete Vorerfahrungen zum eigenen Gebrauch digitaler Werkzeuge sowie zur Verwendung digitaler Werkzeuge in Unterrichtsplanung und -durchf{\"u}hrung. Das Testinstrument ist geeignet, um mittels Analysen von Veranstaltungsgruppen im Pr{\"a}-Post-Design den Zuwachs der oben beschriebenen Kompetenz von (angehenden) Mathematiklehrkr{\"a}ften zu messen. Somit k{\"o}nnen in Zukunft anhand der Ergebnisse die Wirksamkeit von Lehrveranstaltungen, die diese Kompetenz f{\"o}rdern (sollen), untersucht und evaluiert werden. Der Beitrag gliedert sich in zwei Teile: Zun{\"a}chst werden in der Testbeschreibung das zugrundeliegende Konstrukt und der Anwendungsbereich des Testinstruments sowie dessen Aufbau und Hinweise zur Durchf{\"u}hrung beschrieben. Zudem wird die Testg{\"u}te anhand der Pilotierungsergebnisse {\"u}berpr{\"u}ft. Im zweiten Teil befindet sich das vollst{\"a}ndige Testinstrument.}, subject = {GeoGebra}, language = {de} } @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} } @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} } @article{FallMinlendRatzkin2022, author = {Fall, Mouhammed Moustapha and Minlend, Ignace Aristide and Ratzkin, Jesse}, title = {Foliation of an Asymptotically Flat End by Critical Capacitors}, series = {The Journal of Geometric Analysis}, volume = {32}, journal = {The Journal of Geometric Analysis}, number = {2}, issn = {1559-002X}, doi = {10.1007/s12220-021-00746-6}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269997}, year = {2022}, abstract = {We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace-Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem.}, language = {en} } @article{EastonvanDalenGoebetal.2022, author = {Easton, Andrew and van Dalen, Okki and Goeb, Rainer and Di Bucchianico, Alessandro}, title = {Bivariate copula monitoring}, series = {Quality and Reliability Engineering International}, volume = {38}, journal = {Quality and Reliability Engineering International}, number = {3}, doi = {10.1002/qre.3034}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-276501}, pages = {1272 -- 1288}, year = {2022}, abstract = {The assumption of multivariate normality underlying the Hotelling T\(^{2}\) chart is often violated for process data. The multivariate dependency structure can be separated from marginals with the help of copula theory, which permits to model association structures beyond the covariance matrix. Copula-based estimation and testing routines have reached maturity regarding a variety of practical applications. We have constructed a rich design matrix for the comparison of the Hotelling T\(^{2}\) chart with the copula test by Verdier and the copula test by Vuong, which allows for weighting the observations adaptively. Based on the design matrix, we have conducted a large and computationally intensive simulation study. The results show that the copula test by Verdier performs better than Hotelling T\(^{2}\) in a large variety of out-of-control cases, whereas the weighted Vuong scheme often fails to provide an improvement.}, language = {en} } @article{CampanaBorzi2022, author = {Campana, Francesca Cal{\`a} and Borz{\`i}, Alfio}, title = {On the SQH Method for Solving Differential Nash Games}, series = {Journal of Dynamical and Control Systems}, volume = {28}, journal = {Journal of Dynamical and Control Systems}, issn = {1573-8698}, doi = {10.1007/s10883-021-09546-1}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269111}, pages = {739-755}, year = {2022}, abstract = {A sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed scheme, and results of numerical experiments are reported that successfully validate its computational performance.}, language = {en} } @article{KrausMouchaRoth2022, author = {Kraus, Daniela and Moucha, Annika and Roth, Oliver}, title = {A sharp Bernstein-type inequality and application to the Carleson embedding theorem with matrix weights}, series = {Analysis and Mathematical Physics}, volume = {12}, journal = {Analysis and Mathematical Physics}, number = {1}, issn = {1664-235X}, doi = {10.1007/s13324-021-00639-5}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-270485}, year = {2022}, abstract = {We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil (Int. Math. Res. Not. 2019: 3301-3312, 2019) on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.}, language = {en} } @article{JustSiller2022, author = {Just, Janina and Siller, Hans-Stefan}, title = {The role of mathematics in STEM secondary classrooms: a systematic literature review}, series = {Education Sciences}, volume = {12}, journal = {Education Sciences}, number = {9}, issn = {2227-7102}, doi = {10.3390/educsci12090629}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-288075}, year = {2022}, abstract = {Nowadays, science, technology, engineering, and mathematics (STEM) play a critical role in a nation's global competitiveness and prosperity. Thus, there is a need to educate students in these subjects to meet the current and future demands of personal life and society. While applications, especially in science, engineering, and technology, are directly obvious, mathematics underpins the other STEM disciplines. It is recognized that mathematics is the foundation for all other STEM disciplines; the role of mathematics in classrooms is not clear yet. Therefore, the question arises: What is the current role of mathematics in secondary STEM classrooms? To answer this question, we conducted a systematic literature review based on three publication databases (Web of Science, ERIC, and EBSCO Teacher Referral Center). This literature review paper is intended to contribute to the current state of the role of mathematics in STEM education in secondary classrooms. Through the search, starting with 1910 documents, only 14 eligible documents were found. In these, mathematics is often seen as a minor matter and a means to an end in the eyes of science educators. From this, we conclude that the role of mathematics in the STEM classroom should be further strengthened. Overall, the paper highlights a major research gap, and proposes possible initial solutions to close it.}, language = {en} } @article{AppellBritoReinwand2022, author = {Appell, J{\"u}rgen and Brito, Bel{\´e}n L{\´o}pez and Reinwand, Simon}, title = {Counterexamples on compositions}, series = {Mathematische Semesterberichte}, volume = {70}, journal = {Mathematische Semesterberichte}, number = {1}, issn = {0720-728X}, doi = {10.1007/s00591-022-00318-x}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324306}, pages = {43-56}, year = {2022}, abstract = {We give a collection of 16 examples which show that compositions \(g\) \(\circ\) \(f\) of well-behaved functions \(f\) and \(g\) can be badly behaved. Remarkably, in 10 of the 16 examples it suffices to take as outer function \(g\) simply a power-type or characteristic function. Such a collection of examples may serve as a source of exercises for a calculus course.}, language = {en} } @misc{Kanzow2022, author = {Kanzow, Christian}, title = {Y. Cui, J.-S. Pang: "Modern Nonconvex Nondifferentiable Optimization"}, series = {Jahresbericht der Deutschen Mathematiker-Vereinigung}, volume = {124}, journal = {Jahresbericht der Deutschen Mathematiker-Vereinigung}, number = {2}, issn = {0012-0456}, doi = {10.1365/s13291-022-00250-y}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324346}, pages = {137-143}, year = {2022}, abstract = {No abstract available.}, language = {en} } @article{KanzowMehlitz2022, author = {Kanzow, Christian and Mehlitz, Patrick}, title = {Convergence properties of monotone and nonmonotone proximal gradient methods revisited}, series = {Journal of Optimization Theory and Applications}, volume = {195}, journal = {Journal of Optimization Theory and Applications}, number = {2}, issn = {0022-3239}, doi = {10.1007/s10957-022-02101-3}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324351}, pages = {624-646}, year = {2022}, abstract = {Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis.}, language = {en} } @article{KourouZarvalis2022, author = {Kourou, Maria and Zarvalis, Konstantinos}, title = {Compact sets in petals and their backward orbits under semigroups of holomorphic functions}, series = {Potential Analysis}, volume = {59}, journal = {Potential Analysis}, number = {4}, issn = {0926-2601}, doi = {10.1007/s11118-022-10036-7}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-324368}, pages = {1913-1939}, year = {2022}, abstract = {Let (ϕ\(_t\))\(_{t≥0}\) be a semigroup of holomorphic functions in the unit disk \(\mathbb {D}\) and K a compact subset of \(\mathbb {D}\). We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk.}, language = {en} } @article{DippellEspositoWaldmann2022, author = {Dippell, Marvin and Esposito, Chiara and Waldmann, Stefan}, title = {Deformation and Hochschild cohomology of coisotropic algebras}, series = {Annali di Matematica Pura ed Applicata}, volume = {201}, journal = {Annali di Matematica Pura ed Applicata}, number = {3}, doi = {10.1007/s10231-021-01158-7}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-329069}, pages = {1295-1323}, year = {2022}, abstract = {Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper, we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.}, language = {en} } @article{GreefrathSillerKlocketal.2022, author = {Greefrath, Gilbert and Siller, Hans-Stefan and Klock, Heiner and Wess, Raphael}, title = {Pre-service secondary teachers' pedagogical content knowledge for the teaching of mathematical modelling}, series = {Educational Studies in Mathematics}, volume = {109}, journal = {Educational Studies in Mathematics}, number = {2}, issn = {0013-1954}, doi = {10.1007/s10649-021-10038-z}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-308259}, pages = {383-407}, year = {2022}, abstract = {The article deals with the pedagogical content knowledge of mathematical modelling as part of the professional competence of pre-service teachers. With the help of a test developed for this purpose from a conceptual model, we examine whether this pedagogical content knowledge can be promoted in its different facets—especially knowledge about modelling tasks and about interventions—by suitable university seminars. For this purpose, the test was administered to three groups in a seminar for the teaching of mathematical modelling: (1) to those respondents who created their own modelling tasks for use with students, (2) to those trained to intervene in mathematical modelling processes, and (3) participating students who are not required to address mathematical modelling. The findings of the study—based on variance analysis—indicate that certain facets (knowledge of modelling tasks, modelling processes, and interventions) have increased significantly in both experimental groups but to varying degrees. By contrast, pre-service teachers in the control group demonstrated no significant change to their level of pedagogical content knowledge.}, 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{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} } @article{KanzowRaharjaSchwartz2021, author = {Kanzow, Christian and Raharja, Andreas B. and Schwartz, Alexandra}, title = {An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems}, series = {Journal of Optimization Theory and Applications}, volume = {189}, journal = {Journal of Optimization Theory and Applications}, number = {3}, issn = {1573-2878}, doi = {10.1007/s10957-021-01854-7}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269166}, pages = {793-813}, year = {2021}, abstract = {A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.}, language = {en} } @article{HellmuthKlingenbergLietal.2021, author = {Hellmuth, Kathrin and Klingenberg, Christian and Li, Qin and Tang, Min}, title = {Multiscale convergence of the inverse problem for chemotaxis in the Bayesian setting}, series = {Computation}, volume = {9}, journal = {Computation}, number = {11}, issn = {2079-3197}, doi = {10.3390/computation9110119}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-250216}, year = {2021}, abstract = {Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller-Segel equation and a chemotaxis kinetic equation. These two equations describe the organism's movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller-Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms' population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions.}, language = {en} } @article{TongsompornWananiyakulSteuding2021, author = {Tongsomporn, Janyarak and Wananiyakul, Saeree and Steuding, J{\"o}rn}, title = {The values of the periodic zeta-function at the nontrivial zeros of Riemann's zeta-function}, series = {Symmetry}, volume = {13}, journal = {Symmetry}, number = {12}, issn = {2073-8994}, doi = {10.3390/sym13122410}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-252261}, year = {2021}, abstract = {In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa's approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.}, language = {en} } @article{GreefrathOldenburgSilleretal.2021, author = {Greefrath, Gilbert and Oldenburg, Reinhard and Siller, Hans-Stefan and Ulm, Volker and Weigand, Hans-Georg}, title = {Basic Mental Models of Integrals - Theoretical Conception, Development of a Test Instrument, and first Results}, series = {ZDM - Mathematics Education}, volume = {53}, journal = {ZDM - Mathematics Education}, issn = {1863-9690}, doi = {10.1007/s11858-020-01207-0}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232830}, pages = {649-661}, year = {2021}, abstract = {A basic mental model (BMM—in German 'Grundvorstellung') of a mathematical concept is a content-related interpretation that gives meaning to this concept. This paper defines normative and individual BMMs and concretizes them using the integral as an example. Four BMMs are developed about the concept of definite integral, sometimes used in specific teaching approaches: the BMMs of area, reconstruction, average, and accumulation. Based on theoretical work, in this paper we ask how these BMMs could be identified empirically. A test instrument was developed, piloted, validated and applied with 428 students in first-year mathematics courses. The test results show that the four normative BMMs of the integral can be detected and separated empirically. Moreover, the results allow a comparison of the existing individual BMMs and the requested normative BMMs. Consequences for future developments are discussed.}, language = {en} } @article{HaackHauckKlingenbergetal.2021, author = {Haack, J. and Hauck, C. and Klingenberg, C. and Pirner, M. and Warnecke, S.}, title = {A Consistent BGK Model with Velocity-Dependent Collision Frequency for Gas Mixtures}, series = {Journal of Statistical Physics}, volume = {184}, journal = {Journal of Statistical Physics}, number = {3}, issn = {1572-9613}, doi = {10.1007/s10955-021-02821-2}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269146}, pages = {31}, year = {2021}, abstract = {We derive a multi-species BGK model with velocity-dependent collision frequency for a non-reactive, multi-component gas mixture. The model is derived by minimizing a weighted entropy under the constraint that the number of particles of each species, total momentum, and total energy are conserved. We prove that this minimization problem admits a unique solution for very general collision frequencies. Moreover, we prove that the model satisfies an H-Theorem and characterize the form of equilibrium.}, language = {en} } @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{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{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{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} } @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} } @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{AlmeidaHristovaDashkovskiy2021, author = {Almeida, R. and Hristova, S. and Dashkovskiy, S.}, title = {Uniform bounded input bounded output stability of fractional-order delay nonlinear systems with input}, series = {International Journal of Robust and Nonlinear Control}, volume = {31}, journal = {International Journal of Robust and Nonlinear Control}, number = {1}, doi = {10.1002/rnc.5273}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-218554}, pages = {225 -- 249}, year = {2021}, abstract = {The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time-varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system's dynamic response, and demonstrate the effectiveness of our theoretical results.}, language = {en} } @article{HomburgWeissAlwanetal.2021, author = {Homburg, Annika and Weiß, Christian H. and Alwan, Layth C. and Frahm, Gabriel and G{\"o}b, Rainer}, title = {A performance analysis of prediction intervals for count time series}, series = {Journal of Forecasting}, volume = {40}, journal = {Journal of Forecasting}, number = {4}, doi = {10.1002/for.2729}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-217906}, pages = {603 -- 609}, year = {2021}, abstract = {One of the major motivations for the analysis and modeling of time series data is the forecasting of future outcomes. The use of interval forecasts instead of point forecasts allows us to incorporate the apparent forecast uncertainty. When forecasting count time series, one also has to account for the discreteness of the range, which is done by using coherent prediction intervals (PIs) relying on a count model. We provide a comprehensive performance analysis of coherent PIs for diverse types of count processes. We also compare them to approximate PIs that are computed based on a Gaussian approximation. Our analyses rely on an extensive simulation study. It turns out that the Gaussian approximations do considerably worse than the coherent PIs. Furthermore, special characteristics such as overdispersion, zero inflation, or trend clearly affect the PIs' performance. We conclude by presenting two empirical applications of PIs for count time series: the demand for blood bags in a hospital and the number of company liquidations in Germany.}, language = {en} } @article{NatemeyerWachsmuth2021, author = {Natemeyer, Carolin and Wachsmuth, Daniel}, title = {A proximal gradient method for control problems with non-smooth and non-convex control cost}, series = {Computational Optimization and Applications}, volume = {80}, journal = {Computational Optimization and Applications}, number = {2}, issn = {1573-2894}, doi = {10.1007/s10589-021-00308-0}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269069}, pages = {639-677}, year = {2021}, abstract = {We investigate the convergence of the proximal gradient method applied to control problems with non-smooth and non-convex control cost. Here, we focus on control cost functionals that promote sparsity, which includes functionals of L\(^{p}\)-type for p\in [0,1). We prove stationarity properties of weak limit points of the method. These properties are weaker than those provided by Pontryagin's maximum principle and weaker than L-stationarity.}, language = {en} } @article{FalkFuller2021, author = {Falk, Michael and Fuller, Timo}, title = {New characterizations of multivariate Max-domain of attraction and D-Norms}, series = {Extremes}, volume = {24}, journal = {Extremes}, number = {4}, issn = {1572-915X}, doi = {10.1007/s10687-021-00416-4}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269071}, pages = {849-879}, year = {2021}, abstract = {In this paper we derive new results on multivariate extremes and D-norms. In particular we establish new characterizations of the multivariate max-domain of attraction property. The limit distribution of certain multivariate exceedances above high thresholds is derived, and the distribution of that generator of a D-norm on R\(^{d}\), whose components sum up to d, is obtained. Finally we introduce exchangeable D-norms and show that the set of exchangeable D-norms is a simplex.}, language = {en} } @article{Moenius2021, author = {M{\"o}nius, Katja}, title = {Eigenvalues of zero-divisor graphs of finite commutative rings}, series = {Journal of Algebraic Combinatorics}, volume = {54}, journal = {Journal of Algebraic Combinatorics}, issn = {0925-9899}, doi = {10.1007/s10801-020-00989-6}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232792}, pages = {787-802}, year = {2021}, abstract = {We investigate eigenvalues of the zero-divisor graph Γ(R) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of Γ(R). The graph Γ(R) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever xy=0. We provide formulas for the nullity of Γ(R), i.e., the multiplicity of the eigenvalue 0 of Γ(R). Moreover, we precisely determine the spectra of \(\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)\) and \(\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)\) for a prime number p. We introduce a graph product ×Γ with the property that Γ(R)≅Γ(R\(_1\))×Γ⋯×ΓΓ(R\(_r\)) whenever R≅R\(_1\)×⋯×R\(_r\). With this product, we find relations between the number of vertices of the zero-divisor graph Γ(R), the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of Γ(R).}, language = {en} } @article{KalousekMitraSchloemerkemper2021, author = {Kalousek, Martin and Mitra, Sourav and Schl{\"o}merkemper, Anja}, title = {Existence of weak solutions of diffuse interface models for magnetic fluids}, series = {Proceedings in Applied Mathematics and Mechanics}, volume = {21}, journal = {Proceedings in Applied Mathematics and Mechanics}, number = {1}, doi = {10.1002/pamm.202100205}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-257642}, year = {2021}, abstract = {In this article we collect some recent results on the global existence of weak solutions for diffuse interface models involving incompressible magnetic fluids. We consider both the cases of matched and unmatched specific densities. For the model involving fluids with identical densities we consider the free energy density to be a double well potential whereas for the unmatched density case it is crucial to work with a singular free energy density.}, language = {en} } @article{BreitenbachHelfrichFoersterDandekar2021, author = {Breitenbach, Tim and Helfrich-F{\"o}rster, Charlotte and Dandekar, Thomas}, title = {An effective model of endogenous clocks and external stimuli determining circadian rhythms}, series = {Scientific Reports}, volume = {11}, journal = {Scientific Reports}, number = {1}, doi = {10.1038/s41598-021-95391-y}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-261655}, pages = {16165}, year = {2021}, abstract = {Circadian endogenous clocks of eukaryotic organisms are an established and rapidly developing research field. To investigate and simulate in an effective model the effect of external stimuli on such clocks and their components we developed a software framework for download and simulation. The application is useful to understand the different involved effects in a mathematical simple and effective model. This concerns the effects of Zeitgebers, feedback loops and further modifying components. We start from a known mathematical oscillator model, which is based on experimental molecular findings. This is extended with an effective framework that includes the impact of external stimuli on the circadian oscillations including high dose pharmacological treatment. In particular, the external stimuli framework defines a systematic procedure by input-output-interfaces to couple different oscillators. The framework is validated by providing phase response curves and ranges of entrainment. Furthermore, Aschoffs rule is computationally investigated. It is shown how the external stimuli framework can be used to study biological effects like points of singularity or oscillators integrating different signals at once. The mathematical framework and formalism is generic and allows to study in general the effect of external stimuli on oscillators and other biological processes. For an easy replication of each numerical experiment presented in this work and an easy implementation of the framework the corresponding Mathematica files are fully made available. They can be downloaded at the following link: https://www.biozentrum.uni-wuerzburg.de/bioinfo/computing/circadian/.}, language = {en} } @article{KanzowLechner2021, author = {Kanzow, Christian and Lechner, Theresa}, title = {Globalized inexact proximal Newton-type methods for nonconvex composite functions}, series = {Computational Optimization and Applications}, volume = {78}, journal = {Computational Optimization and Applications}, number = {2}, doi = {10.1007/s10589-020-00243-6}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-283715}, pages = {377-410}, year = {2021}, abstract = {Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their generalizations like proximal Newton and quasi-Newton methods. The current literature on these classes of methods almost exclusively considers the case where also the smooth term is convex. Here we present a globalized proximal Newton-type method which allows the smooth term to be nonconvex. The method is shown to have nice global and local convergence properties, and some numerical results indicate that this method is very promising also from a practical point of view.}, language = {en} } @misc{KanzowLechner2021, author = {Kanzow, Christian and Lechner, Theresa}, title = {Correction to: Globalized inexact proximal Newton-type methods for nonconvex composite functions}, series = {Computational Optimization and Applications}, volume = {80}, journal = {Computational Optimization and Applications}, number = {2}, doi = {10.1007/s10589-021-00302-6}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-348858}, pages = {679-680}, year = {2021}, abstract = {No abstract available.}, language = {en} } @article{CampanaCiaramellaBorzi2021, author = {Campana, Francesca Cal{\`a} and Ciaramella, Gabriele and Borz{\`i}, Alfio}, title = {Nash Equilibria and Bargaining Solutions of Differential Bilinear Games}, series = {Dynamic Games and Applications}, volume = {11}, journal = {Dynamic Games and Applications}, number = {1}, doi = {10.1007/s13235-020-00351-2}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-283897}, pages = {1-28}, year = {2021}, abstract = {This paper is devoted to a theoretical and numerical investigation of Nash equilibria and Nash bargaining problems governed by bilinear (input-affine) differential models. These systems with a bilinear state-control structure arise in many applications in, e.g., biology, economics, physics, where competition between different species, agents, and forces needs to be modelled. For this purpose, the concept of Nash equilibria (NE) appears appropriate, and the building blocks of the resulting differential Nash games are different control functions associated with different players that pursue different non-cooperative objectives. In this framework, existence of Nash equilibria is proved and computed with a semi-smooth Newton scheme combined with a relaxation method. Further, a related Nash bargaining (NB) problem is discussed. This aims at determining an improvement of all players' objectives with respect to the Nash equilibria. Results of numerical experiments successfully demonstrate the effectiveness of the proposed NE and NB computational framework.}, 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{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{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{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} } @article{GuensterWeigand2020, author = {G{\"u}nster, Stephan Michael and Weigand, Hans-Georg}, title = {Designing digital technology tasks for the development of functional thinking}, series = {ZDM - Mathematics Education}, volume = {52}, journal = {ZDM - Mathematics Education}, doi = {10.1007/s11858-020-01179-1}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-231913}, pages = {1259-1274}, year = {2020}, abstract = {In this paper we introduce a theoretical framework concerned with fostering functional thinking in Grade 8 students by utilizing digital technologies. This framework is meant to be used to guide the systematic variation of tasks for implementation in the classroom while using digital technologies. Examples of problems and tasks illustrate this process. Additionally, results of an empirical investigation with Grade 8 students, which focusses on the students' skills with digital technologies, how they utilize these tools when engaging with the developed tasks, and how they influence their functional thinking, are presented. The research aim is to investigate in which way tasks designed according to the theoretical framework could promote functional thinking while using digital technologies in the sense of the operative principle. The results show that the developed framework — Function-Operation-Matrix — is a sound basis for initiating students' actions in the sense of the operative principle, to foster the development of functional thinking in its three aspects, namely, assignment, co-variation and object, and that digital technologies can support this process in a meaningful way.}, language = {en} } @article{KarlNeitzelWachsmuth2020, author = {Karl, Veronika and Neitzel, Ira and Wachsmuth, Daniel}, title = {A Lagrange multiplier method for semilinear elliptic state constrained optimal control problems}, series = {Computational Optimization and Applications}, volume = {77}, journal = {Computational Optimization and Applications}, issn = {0926-6003}, doi = {10.1007/s10589-020-00223-w}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232811}, pages = {7831-869}, year = {2020}, abstract = {In this paper we apply an augmented Lagrange method to a class of semilinear ellip-tic optimal control problems with pointwise state constraints. We show strong con-vergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.}, 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{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} } @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} } @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} } @article{FeireislKlingenbergMarkfelder2020, author = {Feireisl, Eduard and Klingenberg, Christian and Markfelder, Simon}, title = {On the density of "wild" initial data for the compressible Euler system}, series = {Calculus of Variations and Partial Differential Equations}, volume = {59}, journal = {Calculus of Variations and Partial Differential Equations}, issn = {0944-2669}, doi = {10.1007/s00526-020-01806-5}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232607}, year = {2020}, abstract = {We consider a class of "wild" initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural L1-topology and show that its complement is rather large, specifically it is an open dense set.}, language = {en} } @article{Mitra2020, author = {Mitra, Sourav}, title = {Local Existence of Strong Solutions of a Fluid-Structure Interaction Model}, series = {Journal of Mathematical Fluid Mechanics}, volume = {22}, journal = {Journal of Mathematical Fluid Mechanics}, issn = {1422-6928}, doi = {10.1007/s00021-020-00520-8}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-235097}, year = {2020}, abstract = {We are interested in studying a system coupling the compressible Navier-Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler-Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.}, language = {en} } @article{Schoenlein2020, author = {Sch{\"o}nlein, Michael}, title = {Ensemble reachability of homogenous parameter-depedent systems}, series = {Proceedings in Applied Mathematics and Mechanics}, volume = {20}, journal = {Proceedings in Applied Mathematics and Mechanics}, number = {1}, doi = {10.1002/pamm.202000342}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-257637}, year = {2020}, abstract = {In this paper we consider the class (θA, B) of parameter-dependent linear systems given by matrices A ∈ ℂ\(^{nxn}\) and B ∈ ℂ\(^{nxm}\). This class is of interest for several applications and the frequently met task for such systems is to steer the origin toward a given target family f(θ) by using an input that is independent from the parameter. This paper provides a collection of necessary and sufficient conditions for ensemble reachability for these systems.}, language = {en} } @article{DashkovskiyKapustyanSchmid2020, author = {Dashkovskiy, Sergey and Kapustyan, Oleksiy and Schmid, Jochen}, title = {A local input-to-state stability result w.r.t. attractors of nonlinear reaction-diffusion equations}, series = {Mathematics of Control, Signals, and Systems}, volume = {32}, journal = {Mathematics of Control, Signals, and Systems}, number = {3}, doi = {10.1007/s00498-020-00256-w}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-281099}, pages = {309-326}, year = {2020}, abstract = {We establish the local input-to-state stability of a large class of disturbed nonlinear reaction-diffusion equations w.r.t. the global attractor of the respective undisturbed system.}, language = {en} } @unpublished{BreitenbachBorzi2019, author = {Breitenbach, Tim and Borz{\`i}, Alfio}, title = {On the SQH scheme to solve non-smooth PDE optimal control problems}, series = {Numerical Functional Analysis and Optimization}, journal = {Numerical Functional Analysis and Optimization}, doi = {10.1080/01630563.2019.1599911}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-180936}, year = {2019}, abstract = {A sequential quadratic Hamiltonian (SQH) scheme for solving different classes of non-smooth and non-convex PDE optimal control problems is investigated considering seven different benchmark problems with increasing difficulty. These problems include linear and nonlinear PDEs with linear and bilinear control mechanisms, non-convex and discontinuous costs of the controls, L\(^1\) tracking terms, and the case of state constraints. The SQH method is based on the characterisation of optimality of PDE optimal control problems by the Pontryagin's maximum principle (PMP). For each problem, a theoretical discussion of the PMP optimality condition is given and results of numerical experiments are presented that demonstrate the large range of applicability of the SQH scheme.}, language = {en} } @article{HomburgWeissAlwanetal.2019, author = {Homburg, Annika and Weiß, Christian H. and Alwan, Layth C. and Frahm, Gabriel and G{\"o}b, Rainer}, title = {Evaluating approximate point forecasting of count processes}, series = {Econometrics}, volume = {7}, journal = {Econometrics}, number = {3}, issn = {2225-1146}, doi = {10.3390/econometrics7030030}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-196929}, year = {2019}, abstract = {In forecasting count processes, practitioners often ignore the discreteness of counts and compute forecasts based on Gaussian approximations instead. For both central and non-central point forecasts, and for various types of count processes, the performance of such approximate point forecasts is analyzed. The considered data-generating processes include different autoregressive schemes with varying model orders, count models with overdispersion or zero inflation, counts with a bounded range, and counts exhibiting trend or seasonality. We conclude that Gaussian forecast approximations should be avoided.}, 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{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{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} } @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} } @article{LauerbachNeukammSchaeffneretal.2019, author = {Lauerbach, Laura and Neukamm, Stefan and Sch{\"a}ffner, Mathias and Schl{\"o}merkemper, Anja}, title = {Continuum Limit and Homogenization of Stochastic and Periodic Discrete Systems - Fracture in Composite Materials}, series = {Proceedings in Applied Mathematics \& Mechanics}, volume = {19}, journal = {Proceedings in Applied Mathematics \& Mechanics}, number = {1}, doi = {10.1002/pamm.201900070}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-211835}, pages = {e201900070}, year = {2019}, abstract = {The limiting behaviour of a one-dimensional discrete system is studied by means of Γ-convergence. We consider a toy model of a chain of atoms. The interaction potentials are of Lennard-Jones type and periodically or stochastically distributed. The energy of the system is considered in the discrete to continuum limit, i.e. as the number of atoms tends to infinity. During that limit, a homogenization process takes place. The limiting functional is discussed, especially with regard to fracture. Secondly, we consider a rescaled version of the problem, which yields a limiting energy of Griffith's type consisting of a quadratic integral term and a jump contribution. The periodic case can be found in [8], the stochastic case in [6,7].}, 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{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{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} } @article{SchaeffnerSchloemerkemper2018, author = {Sch{\"a}ffner, M. and Schl{\"o}merkemper, A.}, title = {On Lennard-Jones systems with finite range interactions and their asymptotic analysis}, series = {Networks and Heterogeneous Media}, volume = {13}, journal = {Networks and Heterogeneous Media}, number = {1}, doi = {10.3934/nhm.2018005}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-228428}, pages = {95-118}, year = {2018}, abstract = {The aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of Gamma-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formula for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, we provide an explicit expression for the continuum limit in the case of finite range interactions. The obtained homogenization formula is given by the convexification of a Cauchy-Born energy density. Furthermore, we study rescaled energies in which bulk and surface contributions scale in the same way. The related discrete-to-continuum limit yields a rigorous derivation of a one-dimensional version of Griffith' fracture energy and thus generalizes earlier derivations for nearest and next-to-nearest neighbors to the case of finite range interactions. A crucial ingredient to our proofs is a novel decomposition of the energy that allows for re fined estimates.}, 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{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{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} }