@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}
}