@phdthesis{Ruppert2017,
  author    = {Ruppert, Markus},
  title     = {Wege der Analogiebildung - Eine qualitative Studie {\"u}ber den Prozess der Analogiebildung beim L{\"o}sen von Aufgaben},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-155910},
  school      = {Universit{\"a}t W{\"u}rzburg},
  pages     = {311},
  year      = {2017},
  abstract  = {{\"U}ber die besondere Bedeutung von Analogiebildungsprozessen beim Lernen im Allgemeinen und beim Lernen von Mathematik im Speziellen besteht ein breiter wissenschaftlicher Konsens. Es liegt deshalb nahe, von einem lernf{\"o}rderlichen Mathematikunterricht zu verlangen, dass er im Bewusstsein dieser Bedeutung entwickelt ist - dass er also einerseits Analogien aufzeigt und sich diese beim Lehren von Mathematik zunutze macht, dass er andererseits aber auch dem Lernenden Gelegenheiten bietet, Analogien zu erkennen und zu entwickeln. Kurz: Die F{\"a}higkeit zum Bilden von Analogien soll durch den Unterricht gezielt gef{\"o}rdert werden. Um diesem Anspruch gerecht werden zu k{\"o}nnen, m{\"u}ssen ausreichende Kenntnisse dar{\"u}ber vorliegen, wie Analogiebildungsprozesse beim Lernen von Mathematik und beim L{\"o}sen mathematischer Aufgaben ablaufen, wodurch sich erfolgreiche Analogiebildungsprozesse auszeichnen und an welchen Stellen m{\"o}glicherweise Schwierigkeiten bestehen. Der Autor zeigt auf, wie Prozesse der Analogiebildung beim L{\"o}sen mathematischer Aufgaben initiiert, beobachtet, beschrieben und interpretiert werden k{\"o}nnen, um auf dieser Grundlage Ansatzpunkte f{\"u}r geeignete F{\"o}rdermaßnahmen zu identifizieren, bestehende Ideen zur F{\"o}rderung der Analogiebildungsf{\"a}higkeit zu beurteilen und neue Ideen zu entwickeln. Es werden dabei Wege der Analogiebildung nachgezeichnet und untersucht, die auf der Verschr{\"a}nkung zweier Dimensionen der Analogiebildung im Rahmen des zugrundeliegenden theoretischen Modells beruhen. So k{\"o}nnen verschiedene Vorgehensweisen ebenso kontrastiert werden, wie kritische Punkte im Verlauf eines Analogiebildungsprozesses. Es ergeben sich daraus Unterrichtsvorschl{\"a}ge, die auf den Ideen zum beispielbasierten Lernen aufbauen.},
  subject      = {Analogie},
  language  = {de}
}
@phdthesis{Gaviraghi2017,
  author    = {Gaviraghi, Beatrice},
  title     = {Theoretical and numerical analysis of Fokker-Planck optimal control problems for jump-diffusion processes},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145645},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2017},
  abstract  = {The topic of this thesis is the theoretical and numerical analysis of optimal control problems, whose differential constraints are given by Fokker-Planck models related to jump-diffusion processes. We tackle the issue of controlling a stochastic process by formulating a deterministic optimization problem. The key idea of our approach is to focus on the probability density function of the process, whose time evolution is modeled by the Fokker-Planck equation. Our control framework is advantageous since it allows to model the action of the control over the entire range of the process, whose statistics are characterized by the shape of its probability density function. We first investigate jump-diffusion processes, illustrating their main properties. We define stochastic initial-value problems and present results on the existence and uniqueness of their solutions. We then discuss how numerical solutions of stochastic problems are computed, focusing on the Euler-Maruyama method. We put our attention to jump-diffusion models with time- and space-dependent coefficients and jumps given by a compound Poisson process. We derive the related Fokker-Planck equations, which take the form of partial integro-differential equations. Their differential term is governed by a parabolic operator, while the nonlocal integral operator is due to the presence of the jumps. The derivation is carried out in two cases. On the one hand, we consider a process with unbounded range. On the other hand, we confine the dynamic of the sample paths to a bounded domain, and thus the behavior of the process in proximity of the boundaries has to be specified. Throughout this thesis, we set the barriers of the domain to be reflecting. The Fokker-Planck equation, endowed with initial and boundary conditions, gives rise to Fokker-Planck problems. Their solvability is discussed in suitable functional spaces. The properties of their solutions are examined, namely their regularity, positivity and probability mass conservation. Since closed-form solutions to Fokker-Planck problems are usually not available, one has to resort to numerical methods. The first main achievement of this thesis is the definition and analysis of conservative and positive-preserving numerical methods for Fokker-Planck problems. Our SIMEX1 and SIMEX2 (Splitting-Implicit-Explicit) schemes are defined within the framework given by the method of lines. The differential operator is discretized by a finite volume scheme given by the Chang-Cooper method, while the integral operator is approximated by a mid-point rule. This leads to a large system of ordinary differential equations, that we approximate with the Strang-Marchuk splitting method. This technique decomposes the original problem in a sequence of different subproblems with simpler structure, which are separately solved and linked to each other through initial conditions and final solutions. After performing the splitting step, we carry out the time integration with first- and second-order time-differencing methods. These steps give rise to the SIMEX1 and SIMEX2 methods, respectively. A full convergence and stability analysis of our schemes is included. Moreover, we are able to prove that the positivity and the mass conservation of the solution to Fokker-Planck problems are satisfied at the discrete level by the numerical solutions computed with the SIMEX schemes. The second main achievement of this thesis is the theoretical analysis and the numerical solution of optimal control problems governed by Fokker-Planck models. The field of optimal control deals with finding control functions in such a way that given cost functionals are minimized. Our framework aims at the minimization of the difference between a known sequence of values and the first moment of a jump-diffusion process; therefore, this formulation can also be considered as a parameter estimation problem for stochastic processes. Two cases are discussed, in which the form of the cost functional is continuous-in-time and discrete-in-time, respectively. The control variable enters the state equation as a coefficient of the Fokker-Planck partial integro-differential operator. We also include in the cost functional a \$L^1\$-penalization term, which enhances the sparsity of the solution. Therefore, the resulting optimization problem is nonconvex and nonsmooth. We derive the first-order optimality systems satisfied by the optimal solution. The computation of the optimal solution is carried out by means of proximal iterative schemes in an infinite-dimensional framework.},
  subject      = {Fokker-Planck-Gleichung},
  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}
}
@article{RoyBorziHabbal2017,
  author    = {Roy, S. and Borz{\`i}, A. and Habbal, A.},
  title     = {Pedestrian motion modelled by Fokker-Planck Nash games},
  series = {Royal Society Open Science},
  volume    = {4},
  journal   = {Royal Society Open Science},
  number    = {9},
  doi       = {10.1098/rsos.170648},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-170395},
  pages     = {170648},
  year      = {2017},
  abstract  = {A new approach to modelling pedestrians' avoidance dynamics based on a Fokker-Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases.},
  language  = {en}
}