@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} }