@phdthesis{Klotzky2018, author = {Klotzky, Jens}, title = {Well-posedness of a fluid-particle interaction model}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-169009}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {This thesis considers a model of a scalar partial differential equation in the presence of a singular source term, modeling the interaction between an inviscid fluid represented by the Burgers equation and an arbitrary, finite amount of particles moving inside the fluid, each one acting as a point-wise drag force with a particle related friction constant. \begin{align*} \partial_t u + \partial_x (u^2/2) \&= \sum_{i \in N(t)} \lambda_i \Big(h_i'(t)-u(t,h_i(t)\Big)\delta(x-h_i(t)) \end{align*} The model was introduced for the case of a single particle by Lagouti{\`e}re, Seguin and Takahashi, is a first step towards a better understanding of interaction between fluids and solids on the level of partial differential equations and has the unique property of considering entropy admissible solutions and the interaction with shockwaves. The model is extended to an arbitrary, finite number of particles and interactions like merging, splitting and crossing of particle paths are considered. The theory of entropy admissibility is revisited for the cases of interfaces and discontinuous flux conservation laws, existing results are summarized and compared, and adapted for regions of particle interactions. To this goal, the theory of germs introduced by Andreianov, Karlsen and Risebro is extended to this case of non-conservative interface coupling. Exact solutions for the Riemann Problem of particles drifting apart are computed and analysis on the behavior of entropy solutions across the particle related interfaces is used to determine physically relevant and consistent behavior for merging and splitting of particles. Well-posedness of entropy solutions to the Cauchy problem is proven, using an explicit construction method, L-infinity bounds, an approximation of the particle paths and compactness arguments to obtain existence of entropy solutions. Uniqueness is shown in the class of weak entropy solutions using almost classical Kruzkov-type analysis and the notion of L1-dissipative germs. Necessary fundamentals of hyperbolic conservation laws, including weak solutions, shocks and rarefaction waves and the Rankine-Hugoniot condition are briefly recapitulated.}, subject = {Hyperbolische Differentialgleichung}, language = {en} } @phdthesis{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{Lieb2017, author = {Lieb, Julia}, title = {Counting Polynomial Matrices over Finite Fields : Matrices with Certain Primeness Properties and Applications to Linear Systems and Coding Theory}, edition = {1. Auflage}, publisher = {W{\"u}rzburg University Press}, address = {W{\"u}rzburg}, isbn = {978-3-95826-064-1 (print)}, doi = {10.25972/WUP-978-3-95826-065-8}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-151303}, school = {W{\"u}rzburg University Press}, pages = {164}, year = {2017}, abstract = {This dissertation is dealing with three mathematical areas, namely polynomial matrices over finite fields, linear systems and coding theory. Coprimeness properties of polynomial matrices provide criteria for the reachability and observability of interconnected linear systems. Since time-discrete linear systems over finite fields and convolutional codes are basically the same objects, these results could be transfered to criteria for non-catastrophicity of convolutional codes. We calculate the probability that specially structured polynomial matrices are right prime. In particular, formulas for the number of pairwise coprime polynomials and for the number of mutually left coprime polynomial matrices are calculated. This leads to the probability that a parallel connected linear system is reachable and that a parallel connected convolutional codes is non-catastrophic. Moreover, the corresponding probabilities are calculated for other networks of linear systems and convolutional codes, such as series connection. Furthermore, the probabilities that a convolutional codes is MDP and that a clock code is MDS are approximated. Finally, we consider the probability of finding a solution for a linear network coding problem.}, subject = {Lineares System}, language = {en} } @phdthesis{Steck2018, author = {Steck, Daniel}, title = {Lagrange Multiplier Methods for Constrained Optimization and Variational Problems in Banach Spaces}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-174444}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {This thesis is concerned with a class of general-purpose algorithms for constrained minimization problems, variational inequalities, and quasi-variational inequalities in Banach spaces. A substantial amount of background material from Banach space theory, convex analysis, variational analysis, and optimization theory is presented, including some results which are refinements of those existing in the literature. This basis is used to formulate an augmented Lagrangian algorithm with multiplier safeguarding for the solution of constrained optimization problems in Banach spaces. The method is analyzed in terms of local and global convergence, and many popular problem classes such as nonlinear programming, semidefinite programming, and function space optimization are shown to be included as special cases of the general setting. The algorithmic framework is then extended to variational and quasi-variational inequalities, which include, by extension, Nash and generalized Nash equilibrium problems. For these problem classes, the convergence is analyzed in detail. The thesis then presents a rich collection of application examples for all problem classes, including implementation details and numerical results.}, subject = {Optimierung}, language = {en} } @phdthesis{Forster2016, author = {Forster, Johannes}, title = {Variational Approach to the Modeling and Analysis of Magnetoelastic Materials}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-147226}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2016}, abstract = {This doctoral thesis is concerned with the mathematical modeling of magnetoelastic materials and the analysis of PDE systems describing these materials and obtained from a variational approach. The purpose is to capture the behavior of elastic particles that are not only magnetic but exhibit a magnetic domain structure which is well described by the micromagnetic energy and the Landau-Lifshitz-Gilbert equation of the magnetization. The equation of motion for the material's velocity is derived in a continuum mechanical setting from an energy ansatz. In the modeling process, the focus is on the interplay between Lagrangian and Eulerian coordinate systems to combine elasticity and magnetism in one model without the assumption of small deformations. The resulting general PDE system is simplified using special assumptions. Existence of weak solutions is proved for two variants of the PDE system, one including gradient flow dynamics on the magnetization, and the other featuring the Landau-Lifshitz-Gilbert equation. The proof is based on a Galerkin method and a fixed point argument. The analysis of the PDE system with the Landau-Lifshitz-Gilbert equation uses a more involved approach to obtain weak solutions based on G. Carbou and P. Fabrie 2001.}, subject = {Magnetoelastizit{\"a}t}, language = {en} } @article{KasangKalluvyaMajingeetal.2016, author = {Kasang, Christa and Kalluvya, Samuel and Majinge, Charles and Kongola, Gilbert and Mlewa, Mathias and Massawe, Irene and Kabyemera, Rogatus and Magambo, Kinanga and Ulmer, Albrecht and Klinker, Hartwig and Gschmack, Eva and Horn, Anne and Koutsilieri, Eleni and Preiser, Wolfgang and Hofmann, Daniela and Hain, Johannes and M{\"u}ller, Andreas and D{\"o}lken, Lars and Weissbrich, Benedikt and Rethwilm, Axel and Stich, August and Scheller, Carsten}, title = {Effects of Prednisolone on Disease Progression in Antiretroviral-Untreated HIV Infection: A 2-Year Randomized, Double-Blind Placebo-Controlled Clinical Trial}, series = {PLoS One}, volume = {11}, journal = {PLoS One}, number = {1}, doi = {10.1371/journal.pone.0146678}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-146479}, pages = {e0146678}, year = {2016}, abstract = {Background HIV-disease progression correlates with immune activation. Here we investigated whether corticosteroid treatment can attenuate HIV disease progression in antiretroviral-untreated patients. Methods Double-blind, placebo-controlled randomized clinical trial including 326 HIV-patients in a resource-limited setting in Tanzania (clinicaltrials.gov NCT01299948). Inclusion criteria were a CD4 count above 300 cells/μl, the absence of AIDS-defining symptoms and an ART-na{\"i}ve therapy status. Study participants received 5 mg prednisolone per day or placebo for 2 years. Primary endpoint was time to progression to an AIDS-defining condition or to a CD4-count below 200 cells/μl. Results No significant change in progression towards the primary endpoint was observed in the intent-to-treat (ITT) analysis (19 cases with prednisolone versus 28 cases with placebo, p = 0.1407). In a per-protocol (PP)-analysis, 13 versus 24 study participants progressed to the primary study endpoint (p = 0.0741). Secondary endpoints: Prednisolone-treatment decreased immune activation (sCD14, suPAR, CD38/HLA-DR/CD8+) and increased CD4-counts (+77.42 ± 5.70 cells/μl compared to -37.42 ± 10.77 cells/μl under placebo, p < 0.0001). Treatment with prednisolone was associated with a 3.2-fold increase in HIV viral load (p < 0.0001). In a post-hoc analysis stratifying for sex, females treated with prednisolone progressed significantly slower to the primary study endpoint than females treated with placebo (ITT-analysis: 11 versus 21 cases, p = 0.0567; PP-analysis: 5 versus 18 cases, p = 0.0051): No changes in disease progression were observed in men. Conclusions This study could not detect any significant effects of prednisolone on disease progression in antiretroviral-untreated HIV infection within the intent-to-treat population. However, significant effects were observed on CD4 counts, immune activation and HIV viral load. This study contributes to a better understanding of the role of immune activation in the pathogenesis of HIV infection.}, language = {en} } @phdthesis{Koch2016, author = {Koch, Julia Diana}, title = {Value Ranges for Schlicht Functions}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-144978}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2016}, abstract = {This thesis deals with value sets, i.e. the question of what the set of values that a set of functions can take in a prescribed point looks like. Interest in such problems has been around for a long time; a first answer was given by the Schwarz lemma in the 19th century, and soon various refinements were proven. Since the 1930s, a powerful method for solving such problems has been developed, namely Loewner theory. We make extensive use of this tool, as well as variation methods which go back to Schiffer to examine the following questions: We describe the set of values a schlicht normalised function on the unit disc with prescribed derivative at the origin can take by applying Pontryagin's maximum principle to the radial Loewner equation. We then determine the value ranges for the set of holomorphic, normalised, and bounded functions that have only real coefficients in their power series expansion around 0, and for the smaller set of functions which are additionally typically real. Furthermore, we describe the values a univalent self-mapping of the upper half-plane with hydrodynamical normalization which is symmetric with respect to the imaginary axis can take. Lastly, we give a necessary condition for a schlicht bounded function f on the unit disc to have extremal derivative in a point z where its value f(z) is fixed by using variation methods.}, subject = {Pontrjagin-Maximumprinzip}, language = {en} } @article{SchindeleBorzi2016, author = {Schindele, Andreas and Borz{\`i}, Alfio}, title = {Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional}, series = {Applied Mathematics}, volume = {7}, journal = {Applied Mathematics}, number = {9}, doi = {10.4236/am.2016.79086}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145850}, pages = {967-992}, year = {2016}, abstract = {First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.}, language = {en} } @phdthesis{GallegoValencia2017, author = {Gallego Valencia, Juan Pablo}, title = {On Runge-Kutta discontinuous Galerkin methods for compressible Euler equations and the ideal magneto-hydrodynamical model}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-148874}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2017}, abstract = {An explicit Runge-Kutta discontinuous Galerkin (RKDG) method is used to device numerical schemes for both the compressible Euler equations of gas dynamics and the ideal magneto- hydrodynamical (MHD) model. These systems of conservation laws are known to have discontinuous solutions. Discontinuities are the source of spurious oscillations in the solution profile of the numerical approximation, when a high order accurate numerical method is used. Different techniques are reviewed in order to control spurious oscillations. A shock detection technique is shown to be useful in order to determine the regions where the spurious oscillations appear such that a Limiter can be used to eliminate these numeric artifacts. To guarantee the positivity of specific variables like the density and the pressure, a positivity preserving limiter is used. Furthermore, a numerical flux, proven to preserve the entropy stability of the semi-discrete DG scheme for the MHD system is used. Finally, the numerical schemes are implemented using the deal.II C++ libraries in the dflo code. The solution of common test cases show the capability of the method.}, subject = {Eulersche Differentialgleichung}, language = {en} } @unpublished{BreitenbachBorzi2019, author = {Breitenbach, Tim and Borz{\`i}, Alfio}, title = {On the SQH scheme to solve non-smooth PDE optimal control problems}, series = {Numerical Functional Analysis and Optimization}, journal = {Numerical Functional Analysis and Optimization}, doi = {10.1080/01630563.2019.1599911}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-180936}, year = {2019}, abstract = {A sequential quadratic Hamiltonian (SQH) scheme for solving different classes of non-smooth and non-convex PDE optimal control problems is investigated considering seven different benchmark problems with increasing difficulty. These problems include linear and nonlinear PDEs with linear and bilinear control mechanisms, non-convex and discontinuous costs of the controls, L\(^1\) tracking terms, and the case of state constraints. The SQH method is based on the characterisation of optimality of PDE optimal control problems by the Pontryagin's maximum principle (PMP). For each problem, a theoretical discussion of the PMP optimality condition is given and results of numerical experiments are presented that demonstrate the large range of applicability of the SQH scheme.}, language = {en} } @article{GaviraghiSchindeleAnnunziatoetal.2016, author = {Gaviraghi, Beatrice and Schindele, Andreas and Annunziato, Mario and Borz{\`i}, Alfio}, title = {On Optimal Sparse-Control Problems Governed by Jump-Diffusion Processes}, series = {Applied Mathematics}, volume = {7}, journal = {Applied Mathematics}, number = {16}, doi = {10.4236/am.2016.716162}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-147819}, pages = {1978 -- 2004}, year = {2016}, abstract = {A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.}, language = {en} } @phdthesis{Sapozhnikova2018, author = {Sapozhnikova, Kateryna}, title = {Robust Stability of Differential Equations with Maximum}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-173945}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2018}, abstract = {In this thesis stability and robustness properties of systems of functional differential equations which dynamics depends on the maximum of a solution over a prehistory time interval is studied. Max-operator is analyzed and it is proved that due to its presence such kind of systems are particular case of state dependent delay differential equations with piecewise continuous delay function. They are nonlinear, infinite-dimensional and may reduce to one-dimensional along its solution. Stability analysis with respect to input is accomplished by trajectory estimate and via averaging method. Numerical method is proposed.}, subject = {Differentialgleichung}, language = {en} } @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{Tichy2011, author = {Tichy, Diana}, title = {On the Fragility Index}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-73610}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {The Fragility Index captures the amount of risk in a stochastic system of arbitrary dimension. Its main mathematical tool is the asymptotic distribution of exceedance counts within the system which can be derived by use of multivariate extreme value theory. Thereby the basic assumption is that data comes from a distribution which lies in the domain of attraction of a multivariate extreme value distribution. The Fragility Index itself and its extension can serve as a quantitative measure for tail dependence in arbitrary dimensions. It is linked to the well known extremal index for stochastic processes as well the extremal coefficient of an extreme value distribution.}, subject = {Extremwertstatistik}, language = {en} } @phdthesis{Hofmann2012, author = {Hofmann, Martin}, title = {Contributions to Extreme Value Theory in the Space C[0,1]}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-74405}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {We introduce some mathematical framework for extreme value theory in the space of continuous functions on compact intervals and provide basic definitions and tools. Continuous max-stable processes on [0,1] are characterized by their "distribution functions" G which can be represented via a norm on function space, called D-norm. The high conformity of this setup with the multivariate case leads to the introduction of a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. We also introduce the concept of "sojourn time transformation" and compare several types of convergence on function space. Again in complete accordance with the uni- or multivariate case it is now possible to get functional generalized Pareto distributions (GPD) W via W = 1 + log(G) in the upper tail. In particular, this enables us to derive characterizations of the functional domain of attraction condition for copula processes. Moreover, we investigate the sojourn time above a high threshold of a continuous stochastic process. It turns out that the limit, as the threshold increases, of the expected sojourn time given that it is positive, exists if the copula process corresponding to Y is in the functional domain of attraction of a max-stable process. If the process is in a certain neighborhood of a generalized Pareto process, then we can replace the constant threshold by a general threshold function and we can compute the asymptotic sojourn time distribution.}, subject = {Extremwertstatistik}, language = {en} } @techreport{Englert2009, author = {Englert, Stefan}, title = {Mathematica in 15 Minuten (Mathematica Version 6.0)}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-70275}, year = {2009}, abstract = {Mathematica ist ein hervorragendes Programm um mathematische Berechnungen - auch sehr komplexe - auf relativ einfache Art und Weise durchf{\"u}hren zu lassen. Dieses Skript soll eine wirklich kurze Einf{\"u}hrung in Mathematica geben und als Nachschlagewerk einiger g{\"a}ngiger Anwendungen von Mathematica dienen. Dabei wird folgende Grobgliederung verwendet: - Grundlagen: Graphische Oberfl{\"a}che, einfache Berechnungen, Formeleingabe - Bedienung: Vorstellung einiger Kommandos und Einblick in die Funktionsweise - Praxis: Beispielhafte Berechnung einiger Abitur- und {\"U}bungsaufgaben}, subject = {Anwendungssoftware}, language = {de} } @techreport{Englert2012, author = {Englert, Stefan}, title = {Mathematica in 15 Minuten (Mathematica Version 8.0)}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-70287}, year = {2012}, abstract = {Mathematica ist ein hervorragendes Programm um mathematische Berechnungen - auch sehr komplexe - auf relativ einfache Art und Weise durchf{\"u}hren zu lassen. Dieses Skript soll eine wirklich kurze Einf{\"u}hrung in Mathematica geben und als Nachschlagewerk einiger g{\"a}ngiger Anwendungen von Mathematica dienen. Dabei wird folgende Grobgliederung verwendet: - Grundlagen: Graphische Oberfl{\"a}che, einfache Berechnungen, Formeleingabe - Bedienung: Vorstellung einiger Kommandos und Einblick in die Funktionsweise - Praxis: Beispielhafte Berechnung einiger Abitur- und {\"U}bungsaufgaben}, subject = {Anwendungssoftware}, language = {de} } @phdthesis{Dreves2011, author = {Dreves, Axel}, title = {Globally Convergent Algorithms for the Solution of Generalized Nash Equilibrium Problems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-69822}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2011}, abstract = {Es werden verschiedene Verfahren zur L{\"o}sung verallgemeinerter Nash-Gleichgewichtsprobleme mit dem Schwerpunkt auf deren globaler Konvergenz entwickelt. Ein globalisiertes Newton-Verfahren zur Berechnung normalisierter L{\"o}sungen, ein nichtglattes Optimierungsverfahren basierend auf einer unrestringierten Umformulierung des spieltheoretischen Problems, und ein Minimierungsansatz sowei eine Innere-Punkte-Methode zur L{\"o}sung der gemeinsamen Karush-Kuhn-Tucker-Bedingungen der Spieler werden theoretisch untersucht und numerisch getestet. Insbesondere das Innere-Punkte Verfahren erweist sich als das zur Zeit wohl beste Verfahren zur L{\"o}sung verallgemeinerter Nash-Gleichgewichtsprobleme.}, subject = {Nash-Gleichgewicht}, language = {en} } @phdthesis{Mauder2012, author = {Mauder, Markus}, title = {Time-Optimal Control of the Bi-Steerable Robot: A Case Study in Optimal Control of Nonholonomic Systems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-75036}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {In this thesis, time-optimal control of the bi-steerable robot is addressed. The bi-steerable robot, a vehicle with two independently steerable axles, is a complex nonholonomic system with applications in many areas of land-based robotics. Motion planning and optimal control are challenging tasks for this system, since standard control schemes do not apply. The model of the bi-steerable robot considered here is a reduced kinematic model with the driving velocity and the steering angles of the front and rear axle as inputs. The steering angles of the two axles can be set independently from each other. The reduced kinematic model is a control system with affine and non-affine inputs, as the driving velocity enters the system linearly, whereas the steering angles enter nonlinearly. In this work, a new approach to solve the time-optimal control problem for the bi-steerable robot is presented. In contrast to most standard methods for time-optimal control, our approach does not exclusively rely on discretization and purely numerical methods. Instead, the Pontryagin Maximum Principle is used to characterize candidates for time-optimal solutions. The resultant boundary value problem is solved by optimization to obtain solutions to the path planning problem over a given time horizon. The time horizon is decreased and the path planning is iterated to approximate a time-optimal solution. An optimality condition is introduced which depends on the number of cusps, i.e., reversals of the driving direction of the robot. This optimality condition allows to single out non-optimal solutions with too many cusps. In general, our approach only gives approximations of time-optimal solutions, since only normal regular extremals are considered as solutions to the path planning problem, and the path planning is terminated when an extremal with minimal number of cusps is found. However, for most desired configurations, normal regular extremals with the minimal number of cusps provide time-optimal solutions for the bi-steerable robot. The convergence of the approach is analyzed and its probabilistic completeness is shown. Moreover, simulation results on time-optimal solutions for the bi-steerable robot are presented.}, subject = {Mobiler Roboter}, language = {en} } @phdthesis{Akindeinde2012, author = {Akindeinde, Saheed Ojo}, title = {Numerical Verification of Optimality Conditions in Optimal Control Problems}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-76065}, school = {Universit{\"a}t W{\"u}rzburg}, year = {2012}, abstract = {This thesis is devoted to numerical verification of optimality conditions for non-convex optimal control problems. In the first part, we are concerned with a-posteriori verification of sufficient optimality conditions. It is a common knowledge that verification of such conditions for general non-convex PDE-constrained optimization problems is very challenging. We propose a method to verify second-order sufficient conditions for a general class of optimal control problem. If the proposed verification method confirms the fulfillment of the sufficient condition then a-posteriori error estimates can be computed. A special ingredient of our method is an error analysis for the Hessian of the underlying optimization problem. We derive conditions under which positive definiteness of the Hessian of the discrete problem implies positive definiteness of the Hessian of the continuous problem. The results are complemented with numerical experiments. In the second part, we investigate adaptive methods for optimal control problems with finitely many control parameters. We analyze a-posteriori error estimates based on verification of second-order sufficient optimality conditions using the method developed in the first part. Reliability and efficiency of the error estimator are shown. We illustrate through numerical experiments, the use of the estimator in guiding adaptive mesh refinement.}, subject = {Optimale Kontrolle}, language = {en} }