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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.
The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, explicit and independent on the initial time bounds of the output are provided. Uniform BIBO stability and uniform BIBO stability with input threshold are studied. A numerical simulation is carried out to show the system's dynamic response, and demonstrate the effectiveness of our theoretical results.
We give a collection of 16 examples which show that compositions \(g\) \(\circ\) \(f\) of well-behaved functions \(f\) and \(g\) can be badly behaved. Remarkably, in 10 of the 16 examples it suffices to take as outer function \(g\) simply a power-type or characteristic function. Such a collection of examples may serve as a source of exercises for a calculus course.
Extreme value theory aims at modeling extreme but rare events from a probabilistic point of view. It is well-known that so-called generalized Pareto distributions, which are briefly reviewed in Chapter 1, are the only reasonable probability distributions suited for modeling observations above a high threshold, such as waves exceeding the height of a certain dike, earthquakes having at least a certain intensity, and, after applying a simple transformation, share prices falling below some low threshold. However, there are cases for which a generalized Pareto model might fail. Therefore, Chapter 2 derives certain neighborhoods of a generalized Pareto distribution and provides several statistical tests for these neighborhoods, where the cases of observing finite dimensional data and of observing continuous functions on [0,1] are considered. By using a notation based on so-called D-norms it is shown that these tests consistently link both frameworks, the finite dimensional and the functional one. Since the derivation of the asymptotic distributions of the test statistics requires certain technical restrictions, Chapter 3 analyzes these assumptions in more detail. It provides in particular some examples of distributions that satisfy the null hypothesis and of those that do not. Since continuous copula processes are crucial tools for the functional versions of the proposed tests, it is also discussed whether those copula processes actually exist for a given set of data. Moreover, some practical advice is given how to choose the free parameters incorporated in the test statistics. Finally, a simulation study in Chapter 4 compares the in total three different test statistics with another test found in the literature that has a similar null hypothesis. This thesis ends with a short summary of the results and an outlook to further open questions.
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.
We present a technique for computing multi-branch-point covers with prescribed ramification and demonstrate the applicability of our method in relatively large degrees by computing several families of polynomials with symplectic and linear Galois groups.
As a first application, we present polynomials over \(\mathbb{Q}(\alpha,t)\) for the primitive rank-3 groups \(PSp_4(3)\) and \(PSp_4(3).C_2\) of degree 27 and for the 2-transitive group \(PSp_6(2)\) in its actions on 28 and 36 points, respectively. Moreover, the degree-28 polynomial for \(PSp_6(2)\) admits infinitely many totally real specializations.
Next, we present the first (to the best of our knowledge) explicit polynomials for the 2-transitive linear groups \(PSL_4(3)\) and \(PGL_4(3)\) of degree 40, and the imprimitive group \(Aut(PGL_4(3))\) of degree 80.
Additionally, we negatively answer a question by König whether there exists a degree-63 rational function with rational coefficients and monodromy group \(PSL_6(2)\) ramified over at least four points. This is achieved due to the explicit computation of the corresponding hyperelliptic genus-3 Hurwitz curve parameterizing this family, followed by a search for rational points on it. As a byproduct of our calculations we obtain the first explicit \(Aut(PSL_6(2))\)-realizations over \(\mathbb{Q}(t)\).
At last, we present a technique by Elkies for bounding the transitivity degree of Galois groups. This provides an alternative way to verify the Galois groups from the previous chapters and also yields a proof that the monodromy group of a degree-276 cover computed by Monien is isomorphic to the sporadic 2-transitive Conway group \(Co_3\).
Theoretical and numerical investigation of optimal control problems governed by kinetic models
(2021)
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.
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.
Der Einzug des Rechners in den Mathematikunterricht hat eine Vielzahl neuer Möglichkeiten der Darstellung mit sich gebracht, darunter auch multiple, dynamisch verbundene Repräsentationen mathematischer Probleme. Die Arbeit beantwortet die Frage, ob und wie diese Repräsentationsarten von Schülerinnen und Schüler in Argumentationen genutzt werden. In der empirischen Untersuchung wurde dabei einerseits quantitativ erforscht, wie groß der Einfluss der in der Aufgabenstellung gegebenen Repräsentationsform auf die schriftliche Argumentationen der Schülerinnen und Schüler ist. Andererseits wurden durch eine qualitative Analyse spezifische Nutzungsweisen identifiziert und mittels Toulmins Argumentationsmodell beschrieben. Diese Erkenntnisse wurden genutzt, um Konsequenzen bezüglich der Verwendung von multiplen und/oder dynamischen Repräsentationen im Mathematikunterricht der Sekundarstufe zu formulieren.