The topic of this thesis is the theoretical and numerical research of optimal control problems for uncertain nonlinear systems, described by semilinear parabolic differential equations with additive noise, where the state is not completely available. Based on a paper by Kazufumi Ito and Karl Kunisch, which was published in 2006 with the title "Receding Horizon Control with Incomplete Observations", we analyze a Model Predictive Control (MPC) approach where the resulting linear problems on small intervals are solved with a Linear Quadratic Gaussian (LQG) design. Further we define a performance index for the MPC/LQG approach, find estimates for it and present bounds for the solutions of the underlying Riccati equations. Another large part of the thesis is devoted to extensive numerical studies for an 1+1- and 3+1-dimensional problem to show the robustness of the MPC/LQG strategy. The last part is a generalization of the MPC/LQG approach to infinite-dimensional problems.

In this thesis we present some results on cross-diffusion and nonlocal interaction. In the first part we study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result. The analysis is motivated by the formulation of the system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Key ingredients are entropy dissipation methods as well as the recently developed boundedness by entropy principle. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their asymptotics to a previously studied case as the diffusivity tends to zero. Finally we briefly discuss coarsening dynamics in the system, which can be observed in numerical results and is motivated by rewriting the PDEs as a system of nonlocal Cahn-Hilliard equations. Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and very few results exist in this direction. In the second part of this thesis, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in [60]. Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence and uniqueness of nonnegative strong solutions. The proof of the existence relies on the use of an appropriate linearized problem and a fixed-point argument. In addition, a weak-strong stability result is obtained for this system in dimension one which also implies uniqueness of weak solutions. In the third part we focus on a class of integral functionals known as nonlocal perimeters. Intuitively, these functionals express a weighted interaction between a set and its complement. The weight is provided by a positive kernel K which might be singular. We show that these functionals are indeed perimeters in a generalised sense and we establish existence of minimisers for the corresponding Plateau’s problem. Also, when K is radial and strictly decreasing, we prove that halfspaces are minimisers if we prescribe “flat” boundary conditions. Furthermore, a Γ-convergence result is discussed. We study the limiting behaviour of the nonlocal perimeters associated with certain rescalings of a given kernel which might be singular in the origin but that have faster-than-L 1 decay at infinity and we show that the Γ-limit is the classical perimeter, up to a multiplicative constant that we give explicitly.

This thesis deals with the optimal control of PDEs. After a brief introduction in the theory of elliptic and parabolic PDEs, we introduce a software that solves systems of PDEs by the finite elements method. In the second chapter we derive optimality conditions in terms of function spaces, i.e. a systems of PDEs coupled by some pointwise relations. Now we present algorithms to solve the optimality systems numerically and present some numerical test cases. A further chapter deals with the so called lack of adjointness, an issue of gradient methods applied on parabolic optimal control problems. However, since optimal control problems lead to large numerical schemes, model reduction becomes popular. We analyze the proper orthogonal decomposition method and apply it to our model problems. Finally, we apply all considered techniques to a real world problem.

This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case. / Die vorliegende Dissertation untersucht den Gradientenfluss von Dirac-harmonischen Abbildungen. Dirac-harmonische Abbildungen sind kritische Punkte eines Energiefunktionals, welches aus supersymmetrischen Feldtheorien motiviert ist. Die kritischen Punkte dieses Energiefunktionals koppeln die Gleichung für harmonische Abbildungen mit Spinorfeldern. Viele analytische Eigenschaften von Dirac-harmonischen Abbildungen sind bereits bekannt, ein allgemeines Existenzresultat wurde aber noch nicht erzielt. Diese Dissertation untersucht das Existenzproblem, indem der Gradientenfluss von einer regularisierten Version Dirac-harmonischer Abbildungen untersucht wird. Die Methode des Gradientenflusses kann nicht direkt angewendet werden, da das Energiefunktional für Dirac-harmonische Abbildungen nach unten unbeschränkt ist. Daher wird zunächst eine Regularisierungsvorschrift für Dirac-harmonische Abbildungen eingeführt und dann der Gradientenfluss betrachtet. Kapitel 1 stellt für die Arbeit wichtige Resultate über harmonische Abbildungen/harmonische Spinoren zusammen. Außerdem werden die zur Zeit bekannten Resultate über Dirac-harmonische Abbildungen zusammengefasst. In Kapitel 2 werden Dirac-harmonische Abbildungen im Detail eingeführt, außerdem wird eine Regularisierungsvorschrift präsentiert. Kapitel 3 führt die Evolutionsgleichungen für regularisierte Dirac-harmonische Abbildungen ein. Zusätzlich wird die Evolution von verschiedenen Energien diskutiert. Schließlich wird die Existenz einer Kurzzeitlösung bewiesen. In Kapitel 4 werden die Evolutionsgleichungen für den Fall analysiert, dass die Ursprungsmannigfaltigkeit eine geschlossene Kurve ist. Die Existenz einer Langzeitlösung der Evolutionsgleichungen wird bewiesen. Es wird außerdem gezeigt, dass die Evolutionsgleichungen konvergieren, falls die Regularisierung groß genug gewählt wurde. Schließlich wird diskutiert, ob die Regularisierung wieder entfernt werden kann. Kapitel 5 schlussendlich untersucht die Evolutionsgleichungen für den Fall, dass die Ursprungsmannigfaltigkeit eine geschlossene Riemannsche Spin Fläche ist. Es wird die Existenz einer global schwachen Lösung bewiesen, welche bis auf endlich viele Singularitäten glatt ist. Die Lösung konvergiert im schwachen Sinne gegen eine regularisierte Dirac-harmonische Abbildung. Auch hier wird schließlich untersucht, ob die Regularisierung wieder entfernt werden kann.

The discretization of the stationary diffusion equation with random parameters by the Stochastic Finite Element Method requires the solution of a highly structured but very large linear system of equations. Depending on the stochastic properties of the diffusion coefficient together with the stochastic discretization we consider three solver cases. If the diffusion coefficient is given by a stochastically linear expansion, e.g. a truncated Karhunen-Loeve expansion, and tensor product polynomial stochastic shape functions are employed, the Galerkin matrix can be transformed to a block-diagonal matrix. For the solution of the resulting sequence of linear systems we study Krylov subspace recycling methods whose success depends on the ordering and grouping of the linear systems as well as the preconditioner. If we use complete polynomials for the stochastic discretization instead, we show that decoupling of the Galerkin matrix with respect to the stochastic degrees of freedom is impossible. For a stochastically nonlinear diffusion coefficient, e.g. a lognormal random field, together with complete polynomials serving as stochastic shape functions, we introduce and test the performance of a new Kronecker product preconditioner, which is not exclusively based on the mean value of the diffusion coefficient.

Die Dissertation beschäftigt sich mit Parameteridentifikationsproblemen, wie sie häufig in Fragestellungen der Festkörpermechanik zu finden sind. Hierbei betrachten wir die Identifikation von Materialparametern -- die typischerweise die Eigenschaften der zugrundeliegenden Materialien repräsentieren -- aus gemessenen Verformungen oder Belastungen eines Testkörpers. In mathematischem Sinne entspricht dies der Lösung von Identifikationsproblemen, die eine spezielle Klasse von inversen Problemen bilden. Der Inhalt der Dissertation ist folgendermaßen gegliedert. Nach dem einführenden Abschnitt 1 wird in Abschnitt 2 ein Überblick von Optimierungs- und Regularisierungsverfahren zur stabilen Lösung nichtlinearer inverser Probleme diskutiert. In Abschnitt 3 betrachten wir die Identifikation von skalaren und stückweise konstanten Parametern in linearen elliptischen Differentialgleichungen. Hierbei werden zwei Testprobleme erörtert, die Identifikation von Diffusions- und Reaktionsparameter in einer allgemeinen elliptischen Differentialgleichung und die Identifikation der Lame-Konstanten in einem Modell der linearisierten Elastizität. Die zugrunde liegenden PDE-Modelle und Lösungszugänge werden erläutert. Insbesondere betrachten wir hier Newton-artige Algorithmen, Gradientenmethoden, Multi-Parameter Regularisierung and den evolutionären Algorithmus CMAES. Abschließend werden Ergebnisse einer numerischen Studie präsentiert. Im Abschnitt 4 konzentrieren wir uns auf die Identifikation von verteilten Parametern in hyperelastischen Materialmodellen. Das nichtlineare Elastizitätsproblem wird detailiert erläutert und verschiedene Materialmodelle werden diskutiert (linear elastisches St.-Venant-Kirchhoff Material und nichtlineare Neo-Hooke, Mooney-Rivlin und Modified-Fung Materialien. Zur Lösung des resultierenden Parameteridentifikationsproblems werden Lösungsansätze aus der optimalen Steuerung in Form eines Newton-Lagrange SQP Algorithmus verwendet. Die Resultate einer numerischen Studie werden präsentiert, basierend auf einem zweidimensionales Testproblem mit einer sogenannten Cook-Mebran. Abschließend wird im Abschnitt 5 die Verwendung adaptiver FEM für die Lösung von Parameteridentifikationsproblems kurz erörtert. / The dissertation is focussed on parameter identification problems arising in the context of structural mechanics. At this, we consider the identification of material parameters - which typically represent the properties of an underlying material - from given measured displacements and forces of a loaded test body. In mathematical terms such problems denote identification problems as a special case of general inverse problems. The dissertation is organized as follows. After the introductive section 1, section 2 is devoted to a survey of optimization and regularization methods for the stable solution of nonlinear inverse problems. In section 3 we consider the identification of scalar and piecewise constant parameters in linear elliptic differential equations and examine two test problems, namely the identification of diffusion and reaction parameters in a generalized linear elliptic differential equation of second order and the identification of the Lame constants in the linearized elasticity model. The underlying PDE models are introduced and solution approaches are discussed in detail. At this, we consider Newton-type algorithms, gradient methods, multi-parameter regularization, and the evolutionary algorithm CMAES. Consequently, numerical studies for a two-dimensional test problem are presented. In section 4 we point out the identification of distributed material parameters in hyperelastic deformation models. The nonlinear elasticity boundary value problem for large deformations is introduced. We discuss several material laws for linear elastic (St.-Venant-Kirchhoff) materials and nonlinear Neo-Hooke, Mooney-Rivlin, and Modified-Fung materials. For the solution of the corresponding parameter identification problem, we focus on an optimal control solution approach and introduce a regularized Newton-Lagrange SQP method. The Newton-Lagrange algorithm is demonstrated within a numerical study. Therefore, a simplified two-dimensional Cook membrane test problem is solved. Additionally, in section 5 the application of adaptive methods for the solution of parameter identification problems is discussed briefly.

The subject of the thesis is the investigation of asymptotic properties of solutions of the Cauchy problem for structurally damped sigma-evolution operators with time dependent, monotonous, dissipation term. An appropriate energy for solutions of the sigma-evolution equations is defined and some estimates for energies of higher order are proved. In the scale invariant case the optimality of these estimates is shown. Further, the influence of properties of the time dependent dissipation on L^p-L^q estimates for the energy with p and q bigger or equal to 2 and from the conjugate line is clarified. Also smoothing properties of the operators under consideration are investigated. The connection between the regularity of the data and the regularity of the solution in terms of L^2 based Gevrey spaces is considered. Finally, L^1-L^1-estimates in the special case delta = sigma/2 and decreasing dissipative coefficient. / Thema der vorliegenden Dissertation ist die Untersuchung asymptotischer Eigenschaften von Lösungen des Cauchy Problems für strukturell gedämpfte sigma-Evolutions-Operatoren mit zeitabhängigem, monotonen Dissipationskoeffizienten. Es wird eine geeignete Energie definiert und für diese Abschätzungen, auf für entsprechende Energien höherer Ordnung gezeigt. Darüber hinaus wird der Einfluss des Dissipationskoeffizienten auf L^p-L^q Abschätzungen auf und entfernt von der konjugierten Linie untersucht. Im skaleninvarianten Fall wird die Schärfe der Abschätzungen bewiesen. Weiterhin wird der Zusammenhang zwischen der Regularität der Daten und der der Lösung in Termen von L^2-basierten Gevrey-Räumen untersucht. Schließlich werden L^1-L^1-Abschätzungen für den Spezialfall delta = sigma/2 und monoton fallenden Dissipationskoeffizienten gezeigt.

The analytical solution of a given inhomogeneous boundary value problem of a linear, elliptic, inhomogeneous partial differential equation and a set of inhomogeneous mixed Dirichlet- and Neumann-type boundary conditions is derived in the present paper. In the context of elasticity theory, the problem arises for a non-conservative symmetric ansatz and an extended constitutive law shown earlier. For convenient user application, the scalar function expressed in cylindrical coordinates is primarily obtained for the general case before being expatiated on a special case of linear boundary conditions.

In addition to previous publications, the paper presents the analytical solution of a special boundary value problem which arises in the context of elasticity theory for an extended constitutive law and a non-conservative symmetric ansatz. Besides deriving the general analytical solution, a specific form for linear boundary conditions is given for user convenience.

Techn. University, Diss., 2004--Braunschweig.