@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{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}
}
@misc{Breitenbach2018,
author = {Breitenbach, Tim},
title = {Codes of examples for SQH method},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-165669},
year = {2018},
abstract = {Code examples for the paper "On the SQH Scheme to Solve Nonsmooth PDE Optimal Control Problems" by Tim Breitenbach and Alfio Borz{\`i} published in the journal "Numerical Functional Analysis and Optimization", in 2019, DOI: 10.1080/01630563.2019.1599911},
language = {en}
}
@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}
}
@phdthesis{Gathungu2018,
author = {Gathungu, Duncan Kioi},
title = {On Multigrid and H-Matrix Methods for Partial Integro-Differential Equations},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-156430},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2018},
abstract = {The main theme of this thesis is the development of multigrid and hierarchical matrix solution procedures with almost linear computational complexity for classes of partial integro-differential problems. An elliptic partial integro-differential equation, a convection-diffusion partial integro-differential equation and a convection-diffusion partial integro-differential optimality system are investigated. In the first part of this work, an efficient multigrid finite-differences scheme for solving an elliptic Fredholm partial integro-differential equation (PIDE) is discussed. This scheme combines a second-order accurate finite difference discretization and a Simpson's quadrature rule to approximate the PIDE problem and a multigrid scheme and a fast multilevel integration method of the Fredholm operator allowing the fast solution of the PIDE problem. Theoretical estimates of second-order accuracy and results of local Fourier analysis of convergence of the proposed multigrid scheme are presented. Results of numerical experiments validate these estimates and demonstrate optimal computational complexity of the proposed framework that includes numerical experiments for elliptic PIDE problems with singular kernels. The experience gained in this part of the work is used for the investigation of convection diffusion partial-integro differential equations in the second part of this thesis. Convection-diffusion PIDE problems are discretized using a finite volume scheme referred to as the Chang and Cooper (CC) scheme and a quadrature rule. Also for this class of PIDE problems and this numerical setting, a stability and accuracy analysis of the CC scheme combined with a Simpson's quadrature rule is presented proving second-order accuracy of the numerical solution. To extend and investigate the proposed approximation and solution strategy to the case of systems of convection-diffusion PIDE, an optimal control problem governed by this model is considered. In this case the research focus is the CC-Simpson's discretization of the optimality system and its solution by the proposed multigrid strategy. Second-order accuracy of the optimization solution is proved and results of local Fourier analysis are presented that provide sharp convergence estimates of the optimal computational complexity of the multigrid-fast integration technique. While (geometric) multigrid techniques require ad-hoc implementation depending on the structure of the PIDE problem and on the dimensionality of the domain where the problem is considered, the hierarchical matrix framework allows a more general treatment that exploits the algebraic structure of the problem at hand. In this thesis, this framework is extended to the case of combined differential and integral problems considering the case of a convection-diffusion PIDE. In this case, the starting point is the CC discretization of the convection-diffusion operator combined with the trapezoidal quadrature rule. The hierarchical matrix approach exploits the algebraic nature of the hierarchical matrices for blockwise approximations by low-rank matrices of the sparse convection-diffusion approximation and enables data sparse representation of the fully populated matrix where all essential matrix operations are performed with at most logarithmic optimal complexity. The factorization of part of or the whole coefficient matrix is used as a preconditioner to the solution of the PIDE problem using a generalized minimum residual (GMRes) procedure as a solver. Numerical analysis estimates of the accuracy of the finite-volume and trapezoidal rule approximation are presented and combined with estimates of the hierarchical matrix approximation and with the accuracy of the GMRes iterates. Results of numerical experiments are reported that successfully validate the theoretical estimates and the optimal computational complexity of the proposed hierarchical matrix solution procedure. These results include an extension to higher dimensions and an application to the time evolution of the probability density function of a jump diffusion process.},
subject = {Mehrgitterverfahren},
language = {en}
}
@phdthesis{Technau2018,
author = {Technau, Marc},
title = {On Beatty sets and some generalisations thereof},
edition = {1. Auflage},
publisher = {W{\"u}rzburg University Press},
address = {W{\"u}rzburg},
isbn = {978-3-95826-088-7 (Print)},
doi = {10.25972/WUP-978-3-95826-089-4},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-163303},
school = {W{\"u}rzburg University Press},
pages = {xv, 88},
year = {2018},
abstract = {Beatty sets (also called Beatty sequences) have appeared as early as 1772 in the astronomical studies of Johann III Bernoulli as a tool for easing manual calculations and - as Elwin Bruno Christoffel pointed out in 1888 - lend themselves to exposing intricate properties of the real irrationals. Since then, numerous researchers have explored a multitude of arithmetic properties of Beatty sets; the interrelation between Beatty sets and modular inversion, as well as Beatty sets and the set of rational primes, being the central topic of this book. The inquiry into the relation to rational primes is complemented by considering a natural generalisation to imaginary quadratic number fields.},
subject = {Zahlentheorie},
language = {en}
}
@phdthesis{Barsukow2018,
author = {Barsukow, Wasilij},
title = {Low Mach number finite volume methods for the acoustic and Euler equations},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-159965},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2018},
abstract = {Finite volume methods for compressible Euler equations suffer from an excessive diffusion in the limit of low Mach numbers. This PhD thesis explores new approaches to overcome this. The analysis of a simpler set of equations that also possess a low Mach number limit is found to give valuable insights. These equations are the acoustic equations obtained as a linearization of the Euler equations. For both systems the limit is characterized by a divergencefree velocity. This constraint is nontrivial only in multiple spatial dimensions. As the Jacobians of the acoustic system do not commute, acoustics cannot be reduced to some kind of multi-dimensional advection. Therefore first an exact solution in multiple spatial dimensions is obtained. It is shown that the low Mach number limit can be interpreted as a limit of long times. It is found that the origin of the inability of a scheme to resolve the low Mach number limit is the lack a discrete counterpart to the limit of long times. Numerical schemes whose discrete stationary states discretize all the analytic stationary states of the PDE are called stationarity preserving. It is shown that for the acoustic equations, stationarity preserving schemes are vorticity preserving and are those that are able to resolve the low Mach limit (low Mach compliant). This establishes a new link between these three concepts. Stationarity preservation is studied in detail for both dimensionally split and multi-dimensional schemes for linear acoustics. In particular it is explained why the same multi-dimensional stencils appear in literature in very different contexts: These stencils are unique discretizations of the divergence that allow for stabilizing stationarity preserving diffusion. Stationarity preservation can also be generalized to nonlinear systems such as the Euler equations. Several ways how such numerical schemes can be constructed for the Euler equations are presented. In particular a low Mach compliant numerical scheme is derived that uses a novel construction idea. Its diffusion is chosen such that it depends on the velocity divergence rather than just derivatives of the different velocity components. This is demonstrated to overcome the low Mach number problem. The scheme shows satisfactory results in numerical simulations and has been found to be stable under explicit time integration.},
subject = {Finite-Volumen-Methode},
language = {en}
}
@phdthesis{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{Schoetz2018,
author = {Sch{\"o}tz, Matthias},
title = {Convergent Star Products and Abstract O*-Algebras},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-174355},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2018},
abstract = {Diese Dissertation behandelt ein Problem aus der Deformationsquantisierung: Nachdem man die Quantisierung eines klassischen Systems konstruiert hat, w{\"u}rde man gerne ihre mathematischen Eigenschaften verstehen (sowohl die des klassischen Systems als auch die des Quantensystems). Falls beide Systeme durch *-Algebren {\"u}ber dem K{\"o}rper der komplexen Zahlen beschrieben werden, bedeutet dies dass man die Eigenschaften bestimmter *-Algebren verstehen muss: Welche Darstellungen gibt es? Was sind deren Eigenschaften? Wie k{\"o}nnen die Zust{\"a}nde in diesen Darstellungen beschrieben werden? Wie kann das Spektrum der Observablen beschrieben werden? Um eine hinreichend allgemeine Behandlung dieser Fragen zu erm{\"o}glichen, wird das Konzept von abstrakten O*-Algebren entwickelt. Dies sind im Wesentlichen *-Algebren zusammen mit einem Kegel positiver linearer Funktionale darauf (z.B. die stetigen positiven linearen Funktionale wenn man mit einer *-Algebra startet, die mit einer gutartigen Topologie versehen ist). Im Anschluss daran wird dieser Ansatz dann auf zwei Beispiele aus der Deformationsquantisierung angewandt, die im Detail untersucht werden.},
subject = {Deformationsquantisierung},
language = {en}
}
@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}
}
@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{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}
}
@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}
}
@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{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}
}
@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}
}
@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{Sourmelidis2020,
author = {Sourmelidis, Athanasios},
title = {Universality and Hypertranscendence of Zeta-Functions},
doi = {10.25972/OPUS-19369},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-193699},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2020},
abstract = {The starting point of the thesis is the {\it universality} property of the Riemann Zeta-function \$\zeta(s)\$ which was proved by Voronin in 1975: {\it Given a positive number \$\varepsilon>0\$ and an analytic non-vanishing function \$f\$ defined on a compact subset \$\mathcal{K}\$ of the strip \$\left\{s\in\mathbb{C}:1/2 < \Re s< 1\right\}\$ with connected complement, there exists a real number \$\tau\$ such that \begin{align}\label{continuous} \max\limits_{s\in \mathcal{K}}|\zeta(s+i\tau)-f(s)|<\varepsilon. \end{align} } In 1980, Reich proved a discrete analogue of Voronin's theorem, also known as {\it discrete universality theorem} for \$\zeta(s)\$: {\it If \$\mathcal{K}\$, \$f\$ and \$\varepsilon\$ are as before, then \begin{align}\label{discretee} \liminf\limits_{N\to\infty}\dfrac{1}{N}\sharp\left\{1\leq n\leq N:\max\limits_{s\in \mathcal{K}}|\zeta(s+i\Delta n)-f(s)|<\varepsilon\right\}>0, \end{align} where \$\Delta\$ is an arbitrary but fixed positive number. } We aim at developing a theory which can be applied to prove the majority of all so far existing discrete universality theorems in the case of Dirichlet \$L\$-functions \$L(s,\chi)\$ and Hurwitz zeta-functions \$\zeta(s;\alpha)\$, where \$\chi\$ is a Dirichlet character and \$\alpha\in(0,1]\$, respectively. Both of the aforementioned classes of functions are generalizations of \$\zeta(s)\$, since \$\zeta(s)=L(s,\chi_0)=\zeta(s;1)\$, where \$\chi_0\$ is the principal Dirichlet character mod 1. Amongst others, we prove statement (2) where instead of \$\zeta(s)\$ we have \$L(s,\chi)\$ for some Dirichlet character \$\chi\$ or \$\zeta(s;\alpha)\$ for some transcendental or rational number \$\alpha\in(0,1]\$, and instead of \$(\Delta n)_{n\in\mathbb{N}}\$ we can have: \begin{enumerate} \item \textit{Beatty sequences,} \item \textit{sequences of ordinates of \$c\$-points of zeta-functions from the Selberg class,} \item \textit{sequences which are generated by polynomials.} \end{enumerate} In all the preceding cases, the notion of {\it uniformly distributed sequences} plays an important role and we draw attention to it wherever we can. Moreover, for the case of polynomials, we employ more advanced techniques from Analytic Number Theory such as bounds of exponential sums and zero-density estimates for Dirichlet \$L\$-functions. This will allow us to prove the existence of discrete second moments of \$L(s,\chi)\$ and \$\zeta(s;\alpha)\$ on the left of the vertical line \$1+i\mathbb{R}\$, with respect to polynomials. In the case of the Hurwitz Zeta-function \$\zeta(s;\alpha)\$, where \$\alpha\$ is transcendental or rational but not equal to \$1/2\$ or 1, the target function \$f\$ in (1) or (2), where \$\zeta(\cdot)\$ is replaced by \$\zeta(\cdot;\alpha)\$, is also allowed to have zeros. Until recently there was no result regarding the universality of \$\zeta(s;\alpha)\$ in the literature whenever \$\alpha\$ is an algebraic irrational. In the second half of the thesis, we prove that a weak version of statement \eqref{continuous} for \$\zeta(s;\alpha)\$ holds for all but finitely many algebraic irrational \$\alpha\$ in \$[A,1]\$, where \$A\in(0,1]\$ is an arbitrary but fixed real number. Lastly, we prove that the ordinary Dirichlet series \$\zeta(s;f)=\sum_{n\geq1}f(n)n^{-s}\$ and \$\zeta_\alpha(s)=\sum_{n\geq1}\lfloor P(\alpha n+\beta)\rfloor^{-s}\$ are hypertranscendental, where \$f:\mathbb{N}\to\mathbb{C}\$ is a {\it Besicovitch almost periodic arithmetical function}, \$\alpha,\beta>0\$ are such that \$\lfloor\alpha+\beta\rfloor>1\$ and \$P\in\mathbb{Z}[X]\$ is such that \$P(\mathbb{N})\subseteq\mathbb{N}\$.},
subject = {Analytische Zahlentheorie},
language = {en}
}
@phdthesis{Kortum2022,
author = {Kortum, Joshua},
title = {Global Existence and Uniqueness Results for Nematic Liquid Crystal and Magnetoviscoelastic Flows},
doi = {10.25972/OPUS-27827},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-278271},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2022},
abstract = {Liquid crystals and polymeric fluids are found in many technical applications with liquid crystal displays probably being the most prominent one. Ferromagnetic materials are well established in industrial and everyday use, e.g. as magnets in generators, transformers and hard drive disks. Among ferromagnetic materials, we find a subclass which undergoes deformations if an external magnetic field is applied. This effect is exploited in actuators, magnetoelastic sensors, and new fluid materials have been produced which retain their induced magnetization during the flow. A central issue consists of a proper modelling for those materials. Several models exist regarding liquid crystals and liquid crystal flows, but up to now, none of them has provided a full insight into all observed effects. On materials encompassing magnetic, elastic and perhaps even fluid dynamic effects, the mathematical literature seems sparse in terms of models. To some extent, one can unify the modeling of nematic liquid crystals and magnetoviscoelastic materials employing a so-called energetic variational approach. Using the least action principle from theoretical physics, the actual task reduces to finding appropriate energies describing the observed behavior. The procedure leads to systems of evolutionary partial differential equations, which are analyzed in this work. From the mathematical point of view, fundamental questions on existence, uniqueness and stability of solutions remain unsolved. Concerning the Ericksen-Leslie system modelling nematic liquid crystal flows, an approximation to this model is given by the so-called Ginzburg-Landau approximation. Solutions to the latter are intended to approximately represent solutions to the Ericksen-Leslie system. Indeed, we verify this presumption in two spatial dimensions. More precisely, it is shown that weak solutions of the Ginzburg-Landau approximation converge to solutions of the Ericksen-Leslie system in the energy space for all positive times of evolution. In order to do so, theory for the Euler equations invented by DiPerna and Majda on weak compactness and concentration measures is used. The second part of the work deals with a system of partial differential equations modelling magnetoviscoelastic fluids. We provide a well-posedness result in two spatial dimensions for large energies and large times. Along the verification of that conclusion, existing theory on the Ericksen-Leslie system and the harmonic map flow is deployed and suitably extended.},
subject = {Magnetoelastizit{\"a}t},
language = {en}
}
@phdthesis{Reichert2017,
author = {Reichert, Thorsten},
title = {Classification and Reduction of Equivariant Star Products on Symplectic Manifolds},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-153623},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2017},
abstract = {This doctoral thesis provides a classification of equivariant star products (star products together with quantum momentum maps) in terms of equivariant de Rham cohomology. This classification result is then used to construct an analogon of the Kirwan map from which one can directly obtain the characteristic class of certain reduced star products on Marsden-Weinstein reduced symplectic manifolds from the equivariant characteristic class of their corresponding unreduced equivariant star product. From the surjectivity of this map one can conclude that every star product on Marsden-Weinstein reduced symplectic manifolds can (up to equivalence) be obtained as a reduced equivariant star product.},
subject = {Homologische Algebra},
language = {en}
}
@phdthesis{Sprengel2017,
author = {Sprengel, Martin},
title = {A Theoretical and Numerical Analysis of a Kohn-Sham Equation and Related Control Problems},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-153545},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2017},
abstract = {In this work, multi-particle quantum optimal control problems are studied in the framework of time-dependent density functional theory (TDDFT). Quantum control problems are of great importance in both fundamental research and application of atomic and molecular systems. Typical applications are laser induced chemical reactions, nuclear magnetic resonance experiments, and quantum computing. Theoretically, the problem of how to describe a non-relativistic system of multiple particles is solved by the Schr{\"o}dinger equation (SE). However, due to the exponential increase in numerical complexity with the number of particles, it is impossible to directly solve the Schr{\"o}dinger equation for large systems of interest. An efficient and successful approach to overcome this difficulty is the framework of TDDFT and the use of the time-dependent Kohn-Sham (TDKS) equations therein. This is done by replacing the multi-particle SE with a set of nonlinear single-particle Schr{\"o}dinger equations that are coupled through an additional potential. Despite the fact that TDDFT is widely used for physical and quantum chemical calculation and software packages for its use are readily available, its mathematical foundation is still under active development and even fundamental issues remain unproven today. The main purpose of this thesis is to provide a consistent and rigorous setting for the TDKS equations and of the related optimal control problems. In the first part of the thesis, the framework of density functional theory (DFT) and TDDFT are introduced. This includes a detailed presentation of the different functional sets forming DFT. Furthermore, the known equivalence of the TDKS system to the original SE problem is further discussed. To implement the TDDFT framework for multi-particle computations, the TDKS equations provide one of the most successful approaches nowadays. However, only few mathematical results concerning these equations are available and these results do not cover all issues that arise in the formulation of optimal control problems governed by the TDKS model. It is the purpose of the second part of this thesis to address these issues such as higher regularity of TDKS solutions and the case of weaker requirements on external (control) potentials that are instrumental for the formulation of well-posed TDKS control problems. For this purpose, in this work, existence and uniqueness of TDKS solutions are investigated in the Galerkin framework and using energy estimates for the nonlinear TDKS equations. In the third part of this thesis, optimal control problems governed by the TDKS model are formulated and investigated. For this purpose, relevant cost functionals that model the purpose of the control are discussed. Henceforth, TDKS control problems result from the requirement of optimising the given cost functionals subject to the differential constraint given by the TDKS equations. The analysis of these problems is novel and represents one of the main contributions of the present thesis. In particular, existence of minimizers is proved and their characterization by TDKS optimality systems is discussed in detail. To this end, Fr{\´e}chet differentiability of the TDKS model and of the cost functionals is addressed considering \(H^1\) cost of the control. This part is concluded by deriving the reduced gradient in the \(L^2\) and \(H^1\) inner product. While the \(L^2\) optimization is widespread in the literature, the choice of the \(H^1\) gradient is motivated in this work by theoretical consideration and by resulting numerical advantages. The last part of the thesis is devoted to the numerical approximation of the TDKS optimality systems and to their solution by gradient-based optimization techniques. For the former purpose, Strang time-splitting pseudo-spectral schemes are discussed including a review of some recent theoretical estimates for these schemes and a numerical validation of these estimates. For the latter purpose, nonlinear (projected) conjugate gradient methods are implemented and are used to validate the theoretical analysis of this thesis with results of numerical experiments with different cost functional settings.},
subject = {Optimale Kontrolle},
language = {en}
}
@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{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{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{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}
}
@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}
}
@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}
}
@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}
}
@phdthesis{Promkam2019,
author = {Promkam, Ratthaprom},
title = {Hybrid Dynamical Systems: Modeling, Stability and Interconnection},
doi = {10.25972/OPUS-19099},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-190993},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2019},
abstract = {This work deals with a class of nonlinear dynamical systems exhibiting both continuous and discrete dynamics, which is called as hybrid dynamical system. We provide a broader framework of generalized hybrid dynamical systems allowing us to handle issues on modeling, stability and interconnections. Various sufficient stability conditions are proposed by extensions of direct Lyapunov method. We also explicitly show Lyapunov formulations of the nonlinear small-gain theorems for interconnected input-to-state stable hybrid dynamical systems. Applications on modeling and stability of hybrid dynamical systems are given by effective strategies of vaccination programs to control a spread of disease in epidemic systems.},
subject = {Dynamical system},
language = {en}
}
@phdthesis{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}
}
@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{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{KanzowRaharjaSchwartz2021,
author = {Kanzow, Christian and Raharja, Andreas B. and Schwartz, Alexandra},
title = {Sequential optimality conditions for cardinality-constrained optimization problems with applications},
series = {Computational Optimization and Applications},
volume = {80},
journal = {Computational Optimization and Applications},
number = {1},
issn = {1573-2894},
doi = {10.1007/s10589-021-00298-z},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-269052},
pages = {185-211},
year = {2021},
abstract = {Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka-Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.},
language = {en}
}
@phdthesis{Moenius2021,
author = {M{\"o}nius, Katja},
title = {Algebraic and Arithmetic Properties of Graph Spectra},
doi = {10.25972/OPUS-23085},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-230850},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2021},
abstract = {In the present thesis we investigate algebraic and arithmetic properties of graph spectra. In particular, we study the algebraic degree of a graph, that is the dimension of the splitting field of the characteristic polynomial of the associated adjacency matrix over the rationals, and examine the question whether there is a relation between the algebraic degree of a graph and its structural properties. This generalizes the yet open question ``Which graphs have integral spectra?'' stated by Harary and Schwenk in 1974. We provide an overview of graph products since they are useful to study graph spectra and, in particular, to construct families of integral graphs. Moreover, we present a relation between the diameter, the maximum vertex degree and the algebraic degree of a graph, and construct a potential family of graphs of maximum algebraic degree. Furthermore, we determine precisely the algebraic degree of circulant graphs and find new criteria for isospectrality of circulant graphs. Moreover, we solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Those results generalize a theorem of So who characterized all integral circulant graphs. For our proofs we exploit the theory of Schur rings which was already used in order to solve the isomorphism problem for circulant graphs. Besides that, we study spectra of zero-divisor graphs over finite commutative rings. Given a ring \(R\), the zero-divisor graph over \(R\) is defined as the graph with vertex set being the set of non-zero zero-divisors of \(R\) where two vertices \(x,y\) are adjacent if and only if \(xy=0\). We investigate relations between the eigenvalues of a zero-divisor graph, its structural properties and the algebraic properties of the respective ring.},
subject = {Algebraische Zahlentheorie},
language = {en}
}
@phdthesis{Herrmann2021,
author = {Herrmann, Marc},
title = {The Total Variation on Surfaces and of Surfaces},
doi = {10.25972/OPUS-24073},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-240736},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2021},
abstract = {This thesis is concerned with applying the total variation (TV) regularizer to surfaces and different types of shape optimization problems. The resulting problems are challenging since they suffer from the non-differentiability of the TV-seminorm, but unlike most other priors it favors piecewise constant solutions, which results in piecewise flat geometries for shape optimization problems.The first part of this thesis deals with an analogue of the TV image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. A rigorous analytical framework is developed for this model and its Fenchel predual, which is a quadratic optimization problem with pointwise inequality constraints on the surface. A function space interior point method is proposed to solve it. Afterwards, a discrete variant (DTV) based on a nodal quadrature formula is defined for piecewise polynomial, globally discontinuous and continuous finite element functions on triangulated surface meshes. DTV has favorable properties, which include a convenient dual representation. Next, an analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. Its analysis is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. Shape calculus is used to characterize the relevant derivatives and an variant of the split Bregman method for manifold valued functions is proposed. This is followed by an extension of the total variation prior for the normal vector field for piecewise flat surfaces and the previous variant of split Bregman method is adapted. Numerical experiments confirm that the new prior favours polyhedral shapes.},
subject = {Gestaltoptimierung},
language = {en}
}
@phdthesis{Schmeller2022,
author = {Schmeller, Christof},
title = {Uniform distribution of zero ordinates of Epstein zeta-functions},
doi = {10.25972/OPUS-25199},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-251999},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2022},
abstract = {The dissertation investigates the wide class of Epstein zeta-functions in terms of uniform distribution modulo one of the ordinates of their nontrivial zeros. Main results are a proof of a Landau type theorem for all Epstein zeta-functions as well as uniform distribution modulo one for the zero ordinates of all Epstein zeta-functions asscoiated with binary quadratic forms.},
subject = {Zetafunktion},
language = {en}
}
@phdthesis{Suttner2020,
author = {Suttner, Raik},
title = {Output Optimization by Lie Bracket Approximations},
doi = {10.25972/OPUS-21177},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-211776},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2020},
abstract = {In this dissertation, we develop and analyze novel optimizing feedback laws for control-affine systems with real-valued state-dependent output (or objective) functions. Given a control-affine system, our goal is to derive an output-feedback law that asymptotically stabilizes the closed-loop system around states at which the output function attains a minimum value. The control strategy has to be designed in such a way that an implementation only requires real-time measurements of the output value. Additional information, like the current system state or the gradient vector of the output function, is not assumed to be known. A method that meets all these criteria is called an extremum seeking control law. We follow a recently established approach to extremum seeking control, which is based on approximations of Lie brackets. For this purpose, the measured output is modulated by suitable highly oscillatory signals and is then fed back into the system. Averaging techniques for control-affine systems with highly oscillatory inputs reveal that the closed-loop system is driven, at least approximately, into the directions of certain Lie brackets. A suitable design of the control law ensures that these Lie brackets point into descent directions of the output function. Under suitable assumptions, this method leads to the effect that minima of the output function are practically uniformly asymptotically stable for the closed-loop system. The present document extends and improves this approach in various ways. One of the novelties is a control strategy that does not only lead to practical asymptotic stability, but in fact to asymptotic and even exponential stability. In this context, we focus on the application of distance-based formation control in autonomous multi-agent system in which only distance measurements are available. This means that the target formations as well as the sensed variables are determined by distances. We propose a fully distributed control law, which only involves distance measurements for each individual agent to stabilize a desired formation shape, while a storage of measured data is not required. The approach is applicable to point agents in the Euclidean space of arbitrary (but finite) dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic (and even exponential) stability. A similar statement is also derived for nonholonomic unicycle agents with all-to-all communication. We also show how the findings can be used to solve extremum seeking control problems. Another contribution is an extremum seeking control law with an adaptive dither signal. We present an output-feedback law that steers a fully actuated control-affine system with general drift vector field to a minimum of the output function. A key novelty of the approach is an adaptive choice of the frequency parameter. In this way, the task of determining a sufficiently large frequency parameter becomes obsolete. The adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift. The proposed control law does not only lead to convergence into a neighborhood of a minimum, but leads to exact convergence. For the case of an output function with a global minimum and no other critical point, we prove global convergence. Finally, we present an extremum seeking control law for a class of nonholonomic systems. A detailed averaging analysis reveals that the closed-loop system is driven approximately into descent directions of the output function along Lie brackets of the control vector fields. Those descent directions also originate from an approximation of suitably chosen Lie brackets. This requires a two-fold approximation of Lie brackets on different time scales. The proposed method can lead to practical asymptotic stability even if the control vector fields do not span the entire tangent space. It suffices instead that the tangent space is spanned by the elements in the Lie algebra generated by the control vector fields. This novel feature extends extremum seeking by Lie bracket approximations from the class of fully actuated systems to a larger class of nonholonomic systems.},
subject = {Extremwertregelung},
language = {en}
}
@misc{Breitenbach2019,
author = {Breitenbach, Tim},
title = {Codes of examples for SQH method},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-178588},
year = {2019},
abstract = {Code examples for the paper "On the SQH Scheme to Solve Nonsmooth PDE Optimal Control Problems" by Tim Breitenbach and Alfio Borz{\`i} published in the journal "Numerical Functional Analysis and Optimization", in 2019, DOI: 10.1080/01630563.2019.1599911},
language = {en}
}
@phdthesis{Rehberg2020,
author = {Rehberg, Martin},
title = {Weighted uniform distribution related to primes and the Selberg Class},
doi = {10.25972/OPUS-20925},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-209252},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2020},
abstract = {In the thesis at hand, several sequences of number theoretic interest will be studied in the context of uniform distribution modulo one.
In the first part we deduce for positive and real \(z\not=1\) a discrepancy estimate for the sequence \( \left((2\pi )^{-1}(\log z)\gamma_a\right) \), where \(\gamma_a\) runs through the positive imaginary parts of the nontrivial \(a\)-points of the Riemann zeta-function. If the considered imaginary parts are bounded by \(T\), the discrepancy of the sequence \( \left((2\pi )^{-1}(\log z)\gamma_a\right) \) tends to zero like \( (\log\log\log T)^{-1} \) as \(T\rightarrow \infty\). The proof is related to the proof of Hlawka, who determined a discrepancy estimate for the sequence containing the positive imaginary parts of the nontrivial zeros of the Riemann zeta-function.
The second part of this thesis is about a sequence whose asymptotic behaviour is motivated by the sequence of primes. If \( \alpha\not=0\) is real and \(f\) is a function of logarithmic growth, we specify several conditions such that the sequence \( (\alpha f(q_n)) \) is uniformly distributed modulo one. The corresponding discrepancy estimates will be stated. The sequence \( (q_n)\) of real numbers is strictly increasing and the conditions on its counting function \( Q(x)=\\#\lbrace q_n \leq x \rbrace \) are satisfied by primes and primes in arithmetic progessions. As an application we obtain that the sequence \( \left( (\log q_n)^K\right)\) is uniformly distributed modulo one for arbitrary \(K>1\), if the \(q_n\) are primes or primes in arithmetic progessions. The special case that \(q_n\) equals the \(\textit{n}\)th prime number \(p_n\) was studied by Too, Goto and Kano.
In the last part of this thesis we study for irrational \(\alpha\) the sequence \( (\alpha p_n)\) of irrational multiples of primes in the context of weighted uniform distribution modulo one. A result of Vinogradov concerning exponential sums states that this sequence is uniformly distributed modulo one. An alternative proof due to Vaaler uses L-functions. We extend this approach in the context of the Selberg class with polynomial Euler product. By doing so, we obtain two weighted versions of Vinogradov's result: The sequence \( (\alpha p_n)\) is \( (1+\chi_{D}(p_n))\log p_n\)-uniformly distributed modulo one, where \( \chi_D\) denotes the Legendre-Kronecker character. In the proof we use the Dedekind zeta-function of the quadratic number field \( \Bbb Q (\sqrt{D})\). As an application we obtain in case of \(D=-1\), that \( (\alpha p_n)\) is uniformly distributed modulo one, if the considered primes are congruent to one modulo four. Assuming additional conditions on the functions from the Selberg class we prove that the sequence \( (\alpha p_n) \) is also \( (\sum_{j=1}^{\nu_F}{\alpha_j(p_n)})\log p_n\)-uniformly distributed modulo one, where the weights are related to the Euler product of the function.},
subject = {Zahlentheorie},
language = {en}
}
@phdthesis{Bartsch2021,
author = {Bartsch, Jan},
title = {Theoretical and numerical investigation of optimal control problems governed by kinetic models},
doi = {10.25972/OPUS-24906},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-249066},
school = {Universit{\"a}t W{\"u}rzburg},
year = {2021},
abstract = {This thesis is devoted to the numerical and theoretical analysis of ensemble optimal control problems governed by kinetic models. The formulation and study of these problems have been put forward in recent years by R.W. Brockett with the motivation that ensemble control may provide a more general and robust control framework for dynamical systems. Following this formulation, a Liouville (or continuity) equation with an unbounded drift function is considered together with a class of cost functionals that include tracking of ensembles of trajectories of dynamical systems and different control costs. Specifically, \$L^2\$, \$H^1\$ and \$L^1\$ control costs are taken into account which leads to non--smooth optimization problems. For the theoretical investigation of the resulting optimal control problems, a well--posedness theory in weighted Sobolev spaces is presented for Liouville and related transport equations. Specifically, existence and uniqueness results for these equations and energy estimates in suitable norms are provided; in particular norms in weighted Sobolev spaces. Then, non--smooth optimal control problems governed by the Liouville equation are formulated with a control mechanism in the drift function. Further, box--constraints on the control are imposed. The control--to--state map is introduced, that associates to any control the unique solution of the corresponding Liouville equation. Important properties of this map are investigated, specifically, that it is well--defined, continuous and Frechet differentiable. Using the first two properties, the existence of solutions to the optimal control problems is shown. While proving the differentiability, a loss of regularity is encountered, that is natural to hyperbolic equations. This leads to the need of the investigation of the control--to--state map in the topology of weighted Sobolev spaces. Exploiting the Frechet differentiability, it is possible to characterize solutions to the optimal control problem as solutions to an optimality system. This system consists of the Liouville equation, its optimization adjoint in the form of a transport equation, and a gradient inequality. Numerical methodologies for solving Liouville and transport equations are presented that are based on a non--smooth Lagrange optimization framework. For this purpose, approximation and solution schemes for such equations are developed and analyzed. For the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov--Tadmor method, a Runge--Kutta scheme, and a Strang splitting method are discussed. Stability and second--order accuracy of these resulting schemes are proven in the discrete \$L^1\$ norm. In addition, conservation of mass and positivity preservation are confirmed for the solution method of the Liouville model. As numerical optimization strategy, an adapted Krylow--Newton method is applied. Since the control is considered to be an element of \$H^1\$ and to obey certain box--constraints, a method for calculating a \$H^1\$ projection is presented. Since the optimal control problem is non-smooth, a semi-smooth adaption of Newton's method is taken into account. Results of numerical experiments are presented that successfully validate the proposed deterministic framework. After the discussion of deterministic schemes, the linear space--homogeneous Keilson--Storer master equation is investigated. This equation was originally developed for the modelling of Brownian motion of particles immersed in a fluid and is a representative model of the class of linear Boltzmann equations. The well--posedness of the Keilson--Storer master equation is investigated and energy estimates in different topologies are derived. To solve this equation numerically, Monte Carlo methods are considered. Such methods take advantage of the kinetic formulation of the Liouville equation and directly implement the behaviour of the system of particles under consideration. This includes the probabilistic behaviour of the collisions between particles. Optimal control problems are formulated with an objective that is constituted of certain expected values in velocity space and the \$L^2\$ and \$H^1\$ costs of the control. The problems are governed by the Keilson--Storer master equation and the control mechanism is considered to be within the collision kernel. The objective of the optimal control of this model is to drive an ensemble of particles to acquire a desired mean velocity and to achieve a desired final velocity configuration. Existence of solutions of the optimal control problem is proven and a Keilson--Storer optimality system characterizing the solution of the proposed optimal control problem is obtained. The optimality system is used to construct a gradient--based optimization strategy in the framework of Monte--Carlo methods. This task requires to accommodate the resulting adjoint Keilson--Storer model in a form that is consistent with the kinetic formulation. For this reason, we derive an adjoint Keilson--Storer collision kernel and an additional source term. A similar approach is presented in the case of a linear space--inhomogeneous kinetic model with external forces and with Keilson--Storer collision term. In this framework, a control mechanism in the form of an external space--dependent force is investigated. The purpose of this control is to steer the multi--particle system to follow a desired mean velocity and position and to reach a desired final configuration in phase space. An optimal control problem using the formulation of ensemble controls is stated with an objective that is constituted of expected values in phase space and \$H^1\$ costs of the control. For solving the optimal control problems, a gradient--based computational strategy in the framework of Monte Carlo methods is developed. Part of this is the denoising of the distribution functions calculated by Monte Carlo algorithms using methods of the realm of partial differential equations. A standalone C++ code is presented that implements the developed non--linear conjugated gradient strategy. Results of numerical experiments confirm the ability of the designed probabilistic control framework to operate as desired. An outlook section about optimal control problems governed by non--linear space--inhomogeneous kinetic models completes this thesis.},
subject = {Optimale Kontrolle},
language = {en}
}
@article{BartschBorziFanellietal.2021,
author = {Bartsch, Jan and Borz{\`i}, Alfio and Fanelli, Francesco and Roy, Souvik},
title = {A numerical investigation of Brockett's ensemble optimal control problems},
series = {Numerische Mathematik},
volume = {149},
journal = {Numerische Mathematik},
number = {1},
doi = {10.1007/s00211-021-01223-6},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-265352},
pages = {1-42},
year = {2021},
abstract = {This paper is devoted to the numerical analysis of non-smooth ensemble optimal control problems governed by the Liouville (continuity) equation that have been originally proposed by R.W. Brockett with the purpose of determining an efficient and robust control strategy for dynamical systems. A numerical methodology for solving these problems is presented that is based on a non-smooth Lagrange optimization framework where the optimal controls are characterized as solutions to the related optimality systems. For this purpose, approximation and solution schemes are developed and analysed. Specifically, for the approximation of the Liouville model and its optimization adjoint, a combination of a Kurganov-Tadmor method, a Runge-Kutta scheme, and a Strang splitting method are discussed. The resulting optimality system is solved by a projected semi-smooth Krylov-Newton method. Results of numerical experiments are presented that successfully validate the proposed framework.},
language = {en}
}
@article{MatlachDhillonHainetal.2015,
author = {Matlach, Juliane and Dhillon, Christine and Hain, Johannes and Schlunck, G{\"u}nther and Grehn, Franz and Klink, Thomas},
title = {Trabeculectomy versus canaloplasty (TVC study) in the treatment of patients with open-angle glaucoma: a prospective randomized clinical trial},
series = {Acta Ophthalmologica},
volume = {93},
journal = {Acta Ophthalmologica},
doi = {10.1111/aos.12722},
url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-149263},
pages = {753-761},
year = {2015},
abstract = {Purpose: To compare the outcomes of canaloplasty and trabeculectomy in open-angle glaucoma. Methods: This prospective, randomized clinical trial included 62 patients who randomly received trabeculectomy (n = 32) or canaloplasty (n = 30) and were followed up prospectively for 2 years. Primary endpoint was complete (without medication) and qualified success (with or without medication) defined as an intraocular pressure (IOP) of ≤18 mmHg (definition 1) or IOP ≤21 mmHg and ≥20\% IOP reduction (definition 2), IOP ≥5 mmHg, no vision loss and no further glaucoma surgery. Secondary endpoints were the absolute IOP reduction, visual acuity, medication, complications and second surgeries. Results: Surgical treatment significantly reduced IOP in both groups (p < 0.001). Complete success was achieved in 74.2\% and 39.1\% (definition 1, p = 0.01), and 67.7\% and 39.1\% (definition 2, p = 0.04) after 2 years in the trabeculectomy and canaloplasty group, respectively. Mean absolute IOP reduction was 10.8 ± 6.9 mmHg in the trabeculectomy and 9.3 ± 5.7 mmHg in the canaloplasty group after 2 years (p = 0.47). Mean IOP was 11.5 ± 3.4 mmHg in the trabeculectomy and 14.4 ± 4.2 mmHg in the canaloplasty group after 2 years. Following trabeculectomy, complications were more frequent including hypotony (37.5\%), choroidal detachment (12.5\%) and elevated IOP (25.0\%). Conclusions: Trabeculectomy is associated with a stronger IOP reduction and less need for medication at the cost of a higher rate of complications. If target pressure is attainable by moderate IOP reduction, canaloplasty may be considered for its relative ease of postoperative care and lack of complications.},
language = {en}
}