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#### Institute

- Fachbereich 4 (57) (remove)

Survey data can be viewed as incomplete or partially missing from a variety of perspectives and there are different ways of dealing with this kind of data in the prediction and the estimation of economic quantities. In this thesis, we present two selected research contexts in which the prediction or estimation of economic quantities is examined under incomplete survey data.
These contexts are first the investigation of composite estimators in the German Microcensus (Chapters 3 and 4) and second extensions of multivariate Fay-Herriot (MFH) models (Chapters 5 and 6), which are applied to small area problems.
Composite estimators are estimation methods that take into account the sample overlap in rotating panel surveys such as the German Microcensus in order to stabilise the estimation of the statistics of interest (e.g. employment statistics). Due to the partial sample overlaps, information from previous samples is only available for some of the respondents, so the data are partially missing.
MFH models are model-based estimation methods that work with aggregated survey data in order to obtain more precise estimation results for small area problems compared to classical estimation methods. In these models, several variables of interest are modelled simultaneously. The survey estimates of these variables, which are used as input in the MFH models, are often partially missing. If the domains of interest are not explicitly accounted for in a sampling design, the sizes of the samples allocated to them can, by chance, be small. As a result, it can happen that either no estimates can be calculated at all or that the estimated values are not published by statistical offices because their variances are too large.

Die Dissertation beschäftigt sich mit einer neuartigen Art von Branch-and-Bound Algorithmen, deren Unterschied zu klassischen Branch-and-Bound Algorithmen darin besteht, dass
das Branching durch die Addition von nicht-negativen Straftermen zur Zielfunktion erfolgt
anstatt durch das Hinzufügen weiterer Nebenbedingungen. Die Arbeit zeigt die theoretische Korrektheit des Algorithmusprinzips für verschiedene allgemeine Klassen von Problemen und evaluiert die Methode für verschiedene konkrete Problemklassen. Für diese Problemklassen, genauer Monotone und Nicht-Monotone Gemischtganzzahlige Lineare Komplementaritätsprobleme und Gemischtganzzahlige Lineare Probleme, präsentiert die Arbeit
verschiedene problemspezifische Verbesserungsmöglichkeiten und evaluiert diese numerisch.
Weiterhin vergleicht die Arbeit die neue Methode mit verschiedenen Benchmark-Methoden
mit größtenteils guten Ergebnissen und gibt einen Ausblick auf weitere Anwendungsgebiete
und zu beantwortende Forschungsfragen.

Sample surveys are a widely used and cost effective tool to gain information about a population under consideration. Nowadays, there is an increasing demand not only for information on the population level but also on the level of subpopulations. For some of these subpopulations of interest, however, very small subsample sizes might occur such that the application of traditional estimation methods is not expedient. In order to provide reliable information also for those so called small areas, small area estimation (SAE) methods combine auxiliary information and the sample data via a statistical model.
The present thesis deals, among other aspects, with the development of highly flexible and close to reality small area models. For this purpose, the penalized spline method is adequately modified which allows to determine the model parameters via the solution of an unconstrained optimization problem. Due to this optimization framework, the incorporation of shape constraints into the modeling process is achieved in terms of additional linear inequality constraints on the optimization problem. This results in small area estimators that allow for both the utilization of the penalized spline method as a highly flexible modeling technique and the incorporation of arbitrary shape constraints on the underlying P-spline function.
In order to incorporate multiple covariates, a tensor product approach is employed to extend the penalized spline method to multiple input variables. This leads to high-dimensional optimization problems for which naive solution algorithms yield an unjustifiable complexity in terms of runtime and in terms of memory requirements. By exploiting the underlying tensor nature, the present thesis provides adequate computationally efficient solution algorithms for the considered optimization problems and the related memory efficient, i.e. matrix-free, implementations. The crucial point thereby is the (repetitive) application of a matrix-free conjugated gradient method, whose runtime is drastically reduced by a matrx-free multigrid preconditioner.

Nonlocal operators are used in a wide variety of models and applications due to many natural phenomena being driven by nonlocal dynamics. Nonlocal operators are integral operators allowing for interactions between two distinct points in space. The nonlocal models investigated in this thesis involve kernels that are assumed to have a finite range of nonlocal interactions. Kernels of this type are used in nonlocal elasticity and convection-diffusion models as well as finance and image analysis. Also within the mathematical theory they arouse great interest, as they are asymptotically related to fractional and classical differential equations.
The results in this thesis can be grouped according to the following three aspects: modeling and analysis, discretization and optimization.
Mathematical models demonstrate their true usefulness when put into numerical practice. For computational purposes, it is important that the support of the kernel is clearly determined. Therefore nonlocal interactions are typically assumed to occur within an Euclidean ball of finite radius. In this thesis we consider more general interaction sets including norm induced balls as special cases and extend established results about well-posedness and asymptotic limits.
The discretization of integral equations is a challenging endeavor. Especially kernels which are truncated by Euclidean balls require carefully designed quadrature rules for the implementation of efficient finite element codes. In this thesis we investigate the computational benefits of polyhedral interaction sets as well as geometrically approximated interaction sets. In addition to that we outline the computational advantages of sufficiently structured problem settings.
Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface-dependent kernels. We derive the shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques.

This work studies typical mathematical challenges occurring in the modeling and simulation of manufacturing processes of paper or industrial textiles. In particular, we consider three topics: approximate models for the motion of small inertial particles in an incompressible Newtonian fluid, effective macroscopic approximations for a dilute particle suspension contained in a bounded domain accounting for a non-uniform particle distribution and particle inertia, and possibilities for a reduction of computational cost in the simulations of slender elastic fibers moving in a turbulent fluid flow.
We consider the full particle-fluid interface problem given in terms of the Navier-Stokes equations coupled to momentum equations of a small rigid body. By choosing an appropriate asymptotic scaling for the particle-fluid density ratio and using an asymptotic expansion for the solution components, we derive approximations of the original interface problem. The approximate systems differ according to the chosen scaling of the density ratio in their physical behavior allowing the characterization of different inertial regimes.
We extend the asymptotic approach to the case of many particles suspended in a Newtonian fluid. Under specific assumptions for the combination of particle size and particle number, we derive asymptotic approximations of this system. The approximate systems describe the particle motion which allows to use a mean field approach in order to formulate the continuity equation for the particle probability density function. The coupling of the latter with the approximation for the fluid momentum equation then reveals a macroscopic suspension description which accounts for non-uniform particle distributions in space and for small particle inertia.
A slender fiber in a turbulent air flow can be modeled as a stochastic inextensible one-dimensionally parametrized Kirchhoff beam, i.e., by a stochastic partial differential algebraic equation. Its simulations involve the solution of large non-linear systems of equations by Newton's method. In order to decrease the computational time, we explore different methods for the estimation of the solution. Additionally, we apply smoothing techniques to the Wiener Process in order to regularize the stochastic force driving the fiber, exploring their respective impact on the solution and performance. We also explore the applicability of the Wiener chaos expansion as a solution technique for the simulation of the fiber dynamics.

For decades, academics and practitioners aim to understand whether and how (economic) events affect firm value. Optimally, these events occur exogenously, i.e. suddenly and unexpectedly, so that an accurate evaluation of the effects on firm value can be conducted. However, recent studies show that even the evaluation of exogenous events is often prone to many challenges that can lead to diverse interpretations, resulting in heated debates. Recently, there have been intense debates in particular on the impact of takeover defenses and of Covid-19 on firm value. The announcements of takeover defenses and the propagation of Covid-19 are exogenous events that occur worldwide and are economically important, but have been insufficiently examined. By answering open research questions, this dissertation aims to provide a greater understanding about the heterogeneous effects that exogenous events such as the announcements of takeover defenses and the propagation of Covid-19 have on firm value. In addition, this dissertation analyzes the influence of certain firm characteristics on the effects of these two exogenous events and identifies influencing factors that explain contradictory results in the existing literature and thus can reconcile different views.

Even though proper research on Cauchy transforms has been done, there are still a lot of open questions. For example, in the case of representation theorems, i.e. the question when a function can be represented as a Cauchy transform, there is 'still no completely satisfactory answer' ([9], p. 84). There are characterizations for measures on the circle as presented in the monograph [7] and for general compactly supported measures on the complex plane as presented in [27]. However, there seems to exist no systematic treatise of the Cauchy transform as an operator on $L_p$ spaces and weighted $L_p$ spaces on the real axis.
This is the point where this thesis draws on and we are interested in developing several characterizations for the representability of a function by Cauchy transforms of $L_p$ functions. Moreover, we will attack the issue of integrability of Cauchy transforms of functions and measures, a topic which is only partly explored (see [43]). We will develop different approaches involving Fourier transforms and potential theory and investigate into sufficient conditions and characterizations.
For our purposes, we shall need some notation and the concept of Hardy spaces which will be part of the preliminary Chapter 1. Moreover, we introduce Fourier transforms and their complex analogue, namely Fourier-Laplace transforms. This will be of extraordinary usage due to the close connection of Cauchy and Fourier(-Laplace) transforms.
In the second chapter we shall begin our research with a discussion of the Cauchy transformation on the classical (unweighted) $L_p$ spaces. Therefore, we start with the boundary behavior of Cauchy transforms including an adapted version of the Sokhotski-Plemelj formula. This result will turn out helpful for the determination of the image of the Cauchy transformation under $L_p(\R)$ for $p\in(1,\infty).$ The cases $p=1$ and $p=\infty$ are playing special roles here which justifies a treatise in separate sections. For $p=1$ we will involve the real Hardy space $H_{1}(\R)$ whereas the case $p=\infty$ shall be attacked by an approach incorporating intersections of Hardy spaces and certain subspaces of $L_{\infty}(\R).$
The third chapter prepares ourselves for the study of the Cauchy transformation on subspaces of $L_{p}(\R).$ We shall give a short overview of the basic facts about Cauchy transforms of measures and then proceed to Cauchy transforms of functions with support in a closed set $X\subset\R.$ Our goal is to build up the main theory on which we can fall back in the subsequent chapters.
The fourth chapter deals with Cauchy transforms of functions and measures supported by an unbounded interval which is not the entire real axis. For convenience we restrict ourselves to the interval $[0,\infty).$ Bringing once again the Fourier-Laplace transform into play, we deduce complex characterizations for the Cauchy transforms of functions in $L_{2}(0,\infty).$ Moreover, we analyze the behavior of Cauchy transform on several half-planes and shall use these results for a fairly general geometric characterization. In the second section of this chapter, we focus on Cauchy transforms of measures with support in $[0,\infty).$ In this context, we shall derive a reconstruction formula for these Cauchy transforms holding under pretty general conditions as well as results on the behaviur on the left half-plane. We close this chapter by rather technical real-type conditions and characterizations for Cauchy transforms of functions in $L_p(0,\infty)$ basing on an approach in [82].
The most common case of Cauchy transforms, those of compactly supported functions or measures, is the subject of Chapter 5. After complex and geometric characterizations originating from similar ideas as in the fourth chapter, we adapt a functional-analytic approach in [27] to special measures, namely those with densities to a given complex measure $\mu.$ The chapter is closed with a study of the Cauchy transformation on weighted $L_p$ spaces. Here, we choose an ansatz through the finite Hilbert transform on $(-1,1).$
The sixth chapter is devoted to the issue of integrability of Cauchy transforms. Since this topic has no comprehensive treatise in literature yet, we start with an introduction of weighted Bergman spaces and general results on the interaction of the Cauchy transformation in these spaces. Afterwards, we combine the theory of Zen spaces with Cauchy transforms by using once again their connection with Fourier transforms. Here, we shall encounter general Paley-Wiener theorems of the recent past. Lastly, we attack the issue of integrability of Cauchy transforms by means of potential theory. Therefore, we derive a Fourier integral formula for the logarithmic energy in one and multiple dimensions and give applications to Fourier and hence Cauchy transforms.
Two appendices are annexed to this thesis. The first one covers important definitions and results from measure theory with a special focus on complex measures. The second appendix contains Cauchy transforms of frequently used measures and functions with detailed calculations.

The economic growth theory analyses which factors affect economic growth and tries to analyze how it can last. A popular neoclassical growth model is the Ramsey-Cass-Koopmans model, which aims to determine how much of its income a nation or an economy should save in order to maximize its welfare. In this thesis, we present and analyze an extended capital accumulation equation of a spatial version of the Ramsey model, balancing diffusive and agglomerative effects. We model the capital mobility in space via a nonlocal diffusion operator which allows for jumps of the capital stock from one location to an other. Moreover, this operator smooths out heterogeneities in the factor distributions slower, which generated a more realistic behavior of capital flows. In addition to that, we introduce an endogenous productivity-production operator which depends on time and on the capital distribution in space. This operator models the technological progress of the economy. The resulting mathematical model is an optimal control problem under a semilinear parabolic integro-differential equation with initial and volume constraints, which are a nonlocal analog to local boundary conditions, and box-constraints on the state and the control variables. In this thesis, we consider this problem on a bounded and unbounded spatial domain, in both cases with a finite time horizon. We derive existence results of weak solutions for the capital accumulation equations in both settings and we proof the existence of a Ramsey equilibrium in the unbounded case. Moreover, we solve the optimal control problem numerically and discuss the results in the economic context.

Hybrid Modelling in general, describes the combination of at least two different methods to solve one specific task. As far as this work is concerned, Hybrid Models describe an approach to combine sophisticated, well-studied mathematical methods with Deep Neural Networks to solve parameter estimation tasks. To combine these two methods, the data structure of artifi- cially generated acceleration data of an approximate vehicle model, the Quarter-Car-Model, is exploited. Acceleration of individual components within a coupled dynamical system, can be described as a second order ordinary differential equation, including velocity and dis- placement of coupled states, scaled by spring - and damping-coefficient of the system. An appropriate numerical integration scheme can then be used to simulate discrete acceleration profiles of the Quarter-Car-Model with a random variation of the parameters of the system. Given explicit knowledge about the data structure, one can then investigate under which con- ditions it is possible to estimate the parameters of the dynamical system for a set of randomly generated data samples. We test, if Neural Networks are capable to solve parameter estima- tion problems in general, or if they can be used to solve several sub-tasks, which support a state-of-the-art parameter estimation method. Hybrid Models are presented for parameter estimation under uncertainties, including for instance measurement noise or incompleteness of measurements, which combine knowledge about the data structure and several Neural Networks for robust parameter estimation within a dynamical system.

Our goal is to approximate energy forms on suitable fractals by discrete graph energies and certain metric measure spaces, using the notion of quasi-unitary equivalence. Quasi-unitary equivalence generalises the two concepts of unitary equivalence and norm resolvent convergence to the case of operators and energy forms defined in varying Hilbert spaces.
More precisely, we prove that the canonical sequence of discrete graph energies (associated with the fractal energy form) converges to the energy form (induced by a resistance form) on a finitely ramified fractal in the sense of quasi-unitary equivalence. Moreover, we allow a perturbation by magnetic potentials and we specify the corresponding errors.
This aforementioned approach is an approximation of the fractal from within (by an increasing sequence of finitely many points). The natural step that follows this realisation is the question whether one can also approximate fractals from outside, i.e., by a suitable sequence of shrinking supersets. We partly answer this question by restricting ourselves to a very specific structure of the approximating sets, namely so-called graph-like manifolds that respect the structure of the fractals resp. the underlying discrete graphs. Again, we show that the canonical (properly rescaled) energy forms on such a sequence of graph-like manifolds converge to the fractal energy form (in the sense of quasi-unitary equivalence).
From the quasi-unitary equivalence of energy forms, we conclude the convergence of the associated linear operators, convergence of the spectra and convergence of functions of the operators – thus essentially the same as in the case of the usual norm resolvent convergence.