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Sat, 07 Sep 2024 12:31:45 +0200Sat, 07 Sep 2024 12:31:45 +0200Computational Solution of Nonlocal Problems
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/2334
Differential equations yield solutions that necessarily contain a certain amount of regularity and are based on local interactions. There are various natural phenomena that are not well described by local models. An important class of models that describe long-range interactions are the so-called nonlocal models, which are the subject of this work.
The nonlocal operators considered here are integral operators with a finite range of interaction and the resulting models can be applied to anomalous diffusion, mechanics and multiscale problems.
While the range of applications is vast, the applicability of nonlocal models can face problems such as the high computational and algorithmic complexity of fundamental tasks. One of them is the assembly of finite element discretizations of truncated, nonlocal operators.
The first contribution of this thesis is therefore an openly accessible, documented Python code which allows to compute finite element approximations for nonlocal convection-diffusion problems with truncated interaction horizon.
Another difficulty in the solution of nonlocal problems is that the discrete systems may be ill-conditioned which complicates the application of iterative solvers. Thus, the second contribution of this work is the construction and study of a domain decomposition type solver that is inspired by substructuring methods for differential equations. The numerical results are based on the abstract framework of nonlocal subdivisions which is introduced here and which can serve as a guideline for general nonlocal domain decomposition methods.Manuel Klardoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/2334Tue, 09 Jul 2024 12:31:45 +0200An Integer Optimization Approach to k-Anonymity on Nominal Data
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/2153
The publication of statistical databases is subject to legal regulations, e.g. national statistical offices are only allowed to publish data if the data cannot be attributed to individuals. Achieving this privacy standard requires anonymizing the data prior to publication. However, data anonymization inevitably leads to a loss of information, which should be kept minimal. In this thesis, we analyze the anonymization method SAFE used in the German census in 2011 and we propose a novel integer programming-based anonymization method for nominal data.
In the first part of this thesis, we prove that a fundamental variant of the underlying SAFE optimization problem is NP-hard. This justifies the use of heuristic approaches for large data sets. In the second part, we propose a new anonymization method belonging to microaggregation methods, specifically designed for nominal data. This microaggregation method replaces rows in a microdata set with representative values to achieve k-anonymity, ensuring each data row is identical to at least k − 1 other rows. In addition to the overall dissimilarities of the data rows, the method accounts for errors in resulting frequency tables, which are of high interest for nominal data in practice. The method employs a typical two-step structure: initially partitioning the data set into clusters and subsequently replacing all cluster elements with representative values to achieve k-anonymity. For the partitioning step, we propose a column generation scheme followed by a heuristic to obtain an integer solution, which is based on the dual information. For the aggregation step, we present a mixed-integer problem formulation to find cluster representatives. To this end, we take errors in a subset of frequency tables into account. Furthermore, we show a reformulation of the problem to a minimum edge-weighted maximal clique problem in a multipartite graph, which allows for a different perspective on the problem. Moreover, we formulate a mixed-integer program, which combines the partitioning and the aggregation step and aims to minimize the sum of chi-squared errors in frequency tables.
Finally, an experimental study comparing the methods covered or developed in this work shows particularly strong results for the proposed method with respect to relative criteria, while SAFE shows its strength with respect to the maximum absolute error in frequency tables. We conclude that the inclusion of integer programming in the context of data anonymization is a promising direction to reduce the inevitable information loss inherent in anonymization, particularly for nominal data.Vera Charlotte Costdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/2153Tue, 16 Jan 2024 15:17:47 +0100A Penalty Branch-and-Bound Method for Piecewise Convex Objective Functions
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/2021
Lukas Winkeldoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/2021Thu, 25 May 2023 12:58:26 +0200On Lacunary Approximation of Mergelyan Type
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1913
Let K be a compact subset of the complex plane. Then the family of polynomials P is dense in A(K), the space of all continuous functions on K that are holomorphic on the interior of K, endowed with the uniform norm, if and only if the complement of K is connected. This is the statement of Mergelyan's celebrated theorem.
There are, however, situations where not all polynomials are required to approximate every f ϵ A(K) but where there are strict subspaces of P that are still dense in A(K). If, for example, K is a singleton, then the subspace of all constant polynomials is dense in A(K). On the other hand, if 0 is an interior point of K, then no strict subspace of P can be dense in A(K).
In between these extreme cases, the situation is much more complicated. It turns out that it is mostly determined by the geometry of K and its location in the complex plane which subspaces of P are dense in A(K). In Chapter 1, we give an overview of the known results.
Our first main theorem, which we will give in Chapter 3, deals with the case where the origin is not an interior point of K. We will show that if K is a compact set with connected complement and if 0 is not an interior point of K, then any subspace Q ⊂ P which contains the constant functions and all but finitely many monomials is dense in A(K).
There is a close connection between lacunary approximation and the theory of universality. At the end of Chapter 3, we will illustrate this connection by applying the above result to prove the existence of certain universal power series. To be specific, if K is a compact set with connected complement, if 0 is a boundary point of K and if A_0(K) denotes the subspace of A(K) of those functions that satisfy f(0) = 0, then there exists an A_0(K)-universal formal power series s, where A_0(K)-universal means that the family of partial sums of s forms a dense subset of A_0(K).
In addition, we will show that no formal power series is simultaneously universal for all such K.
The condition on the subspace Q in the main result of Chapter 3 is quite restrictive, but this should not be too surprising: The result applies to the largest possible class of compact sets.
In Chapter 4, we impose a further restriction on the compact sets under consideration, and this will allow us to weaken the condition on the subspace Q. The result that we are going to give is similar to one of those presented in the first chapter, namely the one due to Anderson. In his article “Müntz-Szasz type approximation and the angular growth of lacunary integral functions”, he gives a criterion for a subspace Q of P to be dense in A(K) where K is entirely contained in some closed sector with vertex at the origin.
We will consider compact sets with connected complement that are -- with the possible exception of the origin -- entirely contained in some open sector with vertex at the origin. What we are going to show is that if K\{0} is contained in an open sector of opening angle 2α and if Λ is some subset of the nonnegative integers, then the span of {z → z^λ : λ ϵ Λ} is dense in A(K) whenever 0 ϵ Λ and some Müntz-type condition is satisfied.
Conversely, we will show that if a similar condition is not satisfied, then we can always find a compact set K with connected complement such that K\{0} is contained in some open sector of opening angle 2α and such that the span of {z → z^λ : λ ϵ Λ} fails to be dense in A(K).Markus Borndoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1913Mon, 26 Sep 2022 12:34:14 +0200Pre-Shape Calculus - a Unified Framework for Mesh Quality and Shape Optimization
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1886
In common shape optimization routines, deformations of the computational mesh
usually suffer from decrease of mesh quality or even destruction of the mesh.
To mitigate this, we propose a theoretical framework using so-called pre-shape
spaces. This gives an opportunity for a unified theory of shape optimization, and of
problems related to parameterization and mesh quality. With this, we stay in the
free-form approach of shape optimization, in contrast to parameterized approaches
that limit possible shapes. The concept of pre-shape derivatives is defined, and
according structure and calculus theorems are derived, which generalize classical
shape optimization and its calculus. Tangential and normal directions are featured
in pre-shape derivatives, in contrast to classical shape derivatives featuring only
normal directions on shapes. Techniques from classical shape optimization and
calculus are shown to carry over to this framework, and are collected in generality
for future reference.
A pre-shape parameterization tracking problem class for mesh quality is in-
troduced, which is solvable by use of pre-shape derivatives. This class allows for
non-uniform user prescribed adaptations of the shape and hold-all domain meshes.
It acts as a regularizer for classical shape objectives. Existence of regularized solu-
tions is guaranteed, and corresponding optimal pre-shapes are shown to correspond
to optimal shapes of the original problem, which additionally achieve the user pre-
scribed parameterization.
We present shape gradient system modifications, which allow simultaneous nu-
merical shape optimization with mesh quality improvement. Further, consistency
of modified pre-shape gradient systems is established. The computational burden
of our approach is limited, since additional solution of possibly larger (non-)linear
systems for regularized shape gradients is not necessary. We implement and com-
pare these pre-shape gradient regularization approaches for a 2D problem, which
is prone to mesh degeneration. As our approach does not depend on the choice of
forms to represent shape gradients, we employ and compare weak linear elasticity
and weak quasilinear p-Laplacian pre-shape gradient representations.
We also introduce a Quasi-Newton-ADM inspired algorithm for mesh quality,
which guarantees sufficient adaption of meshes to user specification during the rou-
tines. It is applicable in addition to simultaneous mesh regularization techniques.
Unrelated to mesh regularization techniques, we consider shape optimization
problems constrained by elliptic variational inequalities of the first kind, so-called
obstacle-type problems. In general, standard necessary optimality conditions cannot
be formulated in a straightforward manner for such semi-smooth shape optimization
problems. Under appropriate assumptions, we prove existence and convergence of
adjoints for smooth regularizations of the VI-constraint. Moreover, we derive shape
derivatives for the regularized problem and prove convergence to a limit object.
Based on this analysis, an efficient optimization algorithm is devised and tested
numerically.
All previous pre-shape regularization techniques are applied to a variational
inequality constrained shape optimization problem, where we also create customized
targets for increased mesh adaptation of changing embedded shapes and active set
boundaries of the constraining variational inequality.Daniel Luftdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1886Thu, 21 Jul 2022 17:24:14 +0200Hybrid Modelling of Dynamical Systems in Mechanics
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1899
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.Jan Sokolowskidoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1899Wed, 20 Jul 2022 15:37:24 +0200Strictly and -robust counterparts of electricity market models
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1826
This paper mainly studies two topics: linear complementarity problems for modeling electricity market equilibria and optimization under uncertainty. We consider both perfectly competitive and Nash–Cournot models of electricity markets and study their robustifications using strict robustness and the -approach. For three out of the four combinations of economic competition and robustification, we derive algorithmically tractable convex optimization counterparts that have a clear-cut economic interpretation. In the case of perfect competition, this result corresponds to the two classic welfare theorems, which also apply in both considered robust cases that again yield convex robustified problems. Using the mentioned counterparts, we can also prove the existence and, in some cases, uniqueness of robust equilibria. Surprisingly, it turns out that there is no such economic sensible counterpart for the case of -robustifications of Nash–Cournot models. Thus, an analog of the welfare theorems does not hold in this case. Finally, we provide a computational case study that illustrates the different effects of the combination of economic competition and uncertainty modeling.Anja Kramer; Vanessa Krebs; Martin Schmidtarticlehttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1826Tue, 15 Mar 2022 10:19:04 +0100De Rham and Cech-de Rham Cohomologies of Smooth Foliated Manifolds
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1635
In order to classify smooth foliated manifolds, which are smooth maifolds equipped with a smooth foliation, we introduce the de Rham cohomologies of smooth foliated manifolds. These cohomologies are build in a similar way as the de Rham cohomologies of smooth manifolds. We develop some tools to compute these cohomologies. For example we proof a Mayer Vietoris theorem for foliated de Rham cohomology and show that these cohomologys are invariant under integrable homotopy. A generalization of a known Künneth formula, which relates the cohomologies of a product foliation with its factors, is discussed. In particular, this envolves a splitting theory of sequences between Frechet spaces and a theory of projective spectrums. We also prove, that the foliated de Rham cohomology is isomorphic to the Cech-de Rham cohomology and the Cech cohomology of leafwise constant functions of an underlying so called good cover.Thorben Schlierkampdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1635Tue, 03 Aug 2021 09:45:11 +0200On particles, fibers and suspension flows
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1506
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.Alexander Vibedoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1506Tue, 12 Jan 2021 16:53:39 +0100Stochastic Particle Systems and Optimization - Branching Processes, Mean Field Games and Impulse Control
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1491
This thesis addresses three different topics from the fields of mathematical finance, applied probability and stochastic optimal control. Correspondingly, it is subdivided into three independent main chapters each of which approaches a mathematical problem with a suitable notion of a stochastic particle system.
In Chapter 1, we extend the branching diffusion Monte Carlo method of Henry-Labordère et. al. (2019) to the case of parabolic PDEs with mixed local-nonlocal analytic nonlinearities. We investigate branching diffusion representations of classical solutions, and we provide sufficient conditions under which the branching diffusion representation solves the PDE in the viscosity sense. Our theoretical setup directly leads to a Monte Carlo algorithm, whose applicability is showcased in two stylized high-dimensional examples. As our main application, we demonstrate how our methodology can be used to value financial positions with defaultable, systemically important counterparties.
In Chapter 2, we formulate and analyze a mathematical framework for continuous-time mean field games with finitely many states and common noise, including a rigorous probabilistic construction of the state process. The key insight is that we can circumvent the master equation and reduce the mean field equilibrium to a system of forward-backward systems of (random) ordinary differential equations by conditioning on common noise events. We state and prove a corresponding existence theorem, and we illustrate our results in three stylized application examples. In the absence of common noise, our setup reduces to that of Gomes, Mohr and Souza (2013) and Cecchin and Fischer (2020).
In Chapter 3, we present a heuristic approach to tackle stochastic impulse control problems in discrete time. Based on the work of Bensoussan (2008) we reformulate the classical Bellman equation of stochastic optimal control in terms of a discrete-time QVI, and we prove a corresponding verification theorem. Taking the resulting optimal impulse control as a starting point, we devise a self-learning algorithm that estimates the continuation and intervention region of such a problem. Its key features are that it explores the state space of the underlying problem by itself and successively learns the behavior of the optimally controlled state process. For illustration, we apply our algorithm to a classical example problem, and we give an outlook on open questions to be addressed in future research.Daniel Hoffmanndoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1491Tue, 17 Nov 2020 13:25:51 +0100On robust small area estimation
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1480
Tobias Schochdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1480Tue, 17 Nov 2020 13:24:55 +0100New Concise Extended Formulations for Circular Structures in Optimization Problems
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1478
Many NP-hard optimization problems that originate from classical graph theory, such as the maximum stable set problem and the maximum clique problem, have been extensively studied over the past decades and involve the choice of a subset of edges or vertices. There usually exist combinatorial methods that can be applied to solve them directly in the graph.
The most simple method is to enumerate feasible solutions and select the best. It is not surprising that this method is very slow oftentimes, so the task is to cleverly discard fruitless search space during the search. An alternative method to solve graph problems is to formulate integer linear programs, such that their solution yields an optimal solution to the original optimization problem in the graph. In order to solve integer linear programs, one can start with relaxing the integer constraints and then try to find inequalities for cutting off fractional extreme points. In the best case, it would be possible to describe the convex hull of the feasible region of the integer linear program with a set of inequalities. In general, giving a complete description of this convex hull is out of reach, even if it has a polynomial number of facets. Thus, one tries to strengthen the (weak) relaxation of the integer linear program best possible via strong inequalities that are valid for the convex hull of feasible integer points.
Many classes of valid inequalities are of exponential size. For instance, a graph can have exponentially many odd cycles in general and therefore the number of odd cycle inequalities for the maximum stable set problem is exponential. It is sometimes possible to check in polynomial time if some given point violates any of the exponentially many inequalities. This is indeed the case for the odd cycle inequalities for the maximum stable set problem. If a polynomial time separation algorithm is known, there exists a formulation of polynomial size that contains a given point if and only if it does not violate one of the (potentially exponentially many) inequalities. This thesis can be divided into two parts. The first part is the main part and it contains various new results. We present new extended formulations for several optimization problems, i.e. the maximum stable set problem, the nonconvex quadratic program with box
constraints and the p-median problem. In the second part we modify a very fast algorithm for finding a maximum clique in very large sparse graphs. We suggest and compare three alternative versions of this algorithm to the original version and compare their strengths and weaknesses.Bernd Perscheiddoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1478Tue, 13 Oct 2020 10:38:38 +0200Robust Training of Artificial Neural Networks via p-Quasinorms
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1435
Data used for the purpose of machine learning are often erroneous. In this thesis, p-quasinorms (p<1) are employed as loss functions in order to increase the robustness of training algorithms for artificial neural networks. Numerical issues arising from these loss functions are addressed via enhanced optimization algorithms (proximal point methods; Frank-Wolfe methods) based on the (non-monotonic) Armijo-rule. Numerical experiments comprising 1100 test problems confirm the effectiveness of the approach. Depending on the parametrization, an average reduction of the absolute residuals of up to 64.6% is achieved (aggregated over 100 test problems).Stefan Geisendoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1435Thu, 06 Aug 2020 08:45:16 +0200On port-Hamiltonian modeling and structure-preserving model reduction
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1449
In this thesis we study structure-preserving model reduction methods for the efficient and reliable approximation of dynamical systems. A major focus is the approximation of a nonlinear flow problem on networks, which can, e.g., be used to describe gas network systems. Our proposed approximation framework guarantees so-called port-Hamiltonian structure and is general enough to be realizable by projection-based model order reduction combined with complexity reduction. We divide the discussion of the flow problem into two parts, one concerned with the linear damped wave equation and the other one with the general nonlinear flow problem on networks.
The study around the linear damped wave equation relies on a Galerkin framework, which allows for convenient network generalizations. Notable contributions of this part are the profound analysis of the algebraic setting after space-discretization in relation to the infinite dimensional setting and its implications for model reduction. In particular, this includes the discussion of differential-algebraic structures associated to the network-character of our problem and the derivation of compatibility conditions related to fundamental physical properties. Amongst the different model reduction techniques, we consider the moment matching method to be a particularly well-suited choice in our framework.
The Galerkin framework is then appropriately extended to our general nonlinear flow problem. Crucial supplementary concepts are required for the analysis, such as the partial Legendre transform and a more careful discussion of the underlying energy-based modeling. The preservation of the port-Hamiltonian structure after the model-order- and complexity-reduction-step represents a major focus of this work. Similar as in the analysis of the model order reduction, compatibility conditions play a crucial role in the analysis of our complexity reduction, which relies on a quadrature-type ansatz. Furthermore, energy-stable time-discretization schemes are derived for our port-Hamiltonian approximations, as structure-preserving methods from literature are not applicable due to our rather unconventional parametrization of the solution.
Apart from the port-Hamiltonian approximation of the flow problem, another topic of this thesis is the derivation of a new extension of moment matching methods from linear systems to quadratic-bilinear systems. Most system-theoretic reduction methods for nonlinear systems rely on multivariate frequency representations. Our approach instead uses univariate frequency representations tailored towards user-defined families of inputs. Then moment matching corresponds to a one-dimensional interpolation problem rather than to a multi-dimensional interpolation as for the multivariate approaches, i.e., it involves fewer interpolation frequencies to be chosen. The notion of signal-generator-driven systems, variational expansions of the resulting autonomous systems as well as the derivation of convenient tensor-structured approximation conditions are the main ingredients of this part. Notably, our approach allows for the incorporation of general input relations in the state equations, not only affine-linear ones as in existing system-theoretic methods.Björn Liljegren-Sailerdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1449Wed, 05 Aug 2020 14:37:38 +0200Nonlocal models with truncated interaction kernels - analysis, finite element methods and shape optimization
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1225
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.Christian Vollmanndoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1225Tue, 27 Aug 2019 14:06:24 +0200Cardinality-Constrained Discrete Optimization for Regression
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1209
We consider a linear regression model for which we assume that some of the observed variables are irrelevant for the prediction. Including the wrong variables in the statistical model can either lead to the problem of having too little information to properly estimate the statistic of interest, or having too much information and consequently describing fictitious connections. This thesis considers discrete optimization to conduct a variable selection. In light of this, the subset selection regression method is analyzed. The approach gained a lot of interest in recent years due to its promising predictive performance. A major challenge associated with the subset selection regression is the computational difficulty. In this thesis, we propose several improvements for the efficiency of the method. Novel bounds on the coefficients of the subset selection regression are developed, which help to tighten the relaxation of the associated mixed-integer program, which relies on a Big-M formulation. Moreover, a novel mixed-integer linear formulation for the subset selection regression based on a bilevel optimization reformulation is proposed. Finally, it is shown that the perspective formulation of the subset selection regression is equivalent to a state-of-the-art binary formulation. We use this insight to develop novel bounds for the subset selection regression problem, which show to be highly effective in combination with the proposed linear formulation.
In the second part of this thesis, we examine the statistical conception of the subset selection regression and conclude that it is misaligned with its intention. The subset selection regression uses the training error to decide on which variables to select. The approach conducts the validation on the training data, which oftentimes is not a good estimate of the prediction error. Hence, it requires a predetermined cardinality bound. Instead, we propose to select variables with respect to the cross-validation value. The process is formulated as a mixed-integer program with the sparsity becoming subject of the optimization. Usually, a cross-validation is used to select the best model out of a few options. With the proposed program the best model out of all possible models is selected. Since the cross-validation is a much better estimate of the prediction error, the model can select the best sparsity itself.
The thesis is concluded with an extensive simulation study which provides evidence that discrete optimization can be used to produce highly valuable predictive models with the cross-validation subset selection regression almost always producing the best results.Dennis Kreberdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1209Tue, 20 Aug 2019 09:02:53 +0200Optimization in Tensor Spaces for Data Science and Scientific Computing
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1107
In this thesis, we consider the solution of high-dimensional optimization problems with an underlying low-rank tensor structure. Due to the exponentially increasing computational complexity in the number of dimensions—the so-called curse of dimensionality—they present a considerable computational challenge and become infeasible even for moderate problem sizes.
Multilinear algebra and tensor numerical methods have a wide range of applications in the fields of data science and scientific computing. Due to the typically large problem sizes in practical settings, efficient methods, which exploit low-rank structures, are essential. In this thesis, we consider an application each in both of these fields.
Tensor completion, or imputation of unknown values in partially known multiway data is an important problem, which appears in statistics, mathematical imaging science and data science. Under the assumption of redundancy in the underlying data, this is a well-defined problem and methods of mathematical optimization can be applied to it.
Due to the fact that tensors of fixed rank form a Riemannian submanifold of the ambient high-dimensional tensor space, Riemannian optimization is a natural framework for these problems, which is both mathematically rigorous and computationally efficient.
We present a novel Riemannian trust-region scheme, which compares favourably with the state of the art on selected application cases and outperforms known methods on some test problems.
Optimization problems governed by partial differential equations form an area of scientific computing which has applications in a variety of areas, ranging from physics to financial mathematics. Due to the inherent high dimensionality of optimization problems arising from discretized differential equations, these problems present computational challenges, especially in the case of three or more dimensions. An even more challenging class of optimization problems has operators of integral instead of differential type in the constraint. These operators are nonlocal, and therefore lead to large, dense discrete systems of equations. We present a novel solution method, based on separation of spatial dimensions and provably low-rank approximation of the nonlocal operator. Our approach allows the solution of multidimensional problems with a complexity which is only slightly larger than linear in the univariate grid size; this improves the state of the art for a particular test problem problem by at least two orders of magnitude.Gennadij Heideldoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1107Tue, 16 Apr 2019 08:27:22 +0200Copositivity in Infinite Dimension
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1074
Many combinatorial optimization problems on finite graphs can be formulated as conic convex programs, e.g. the stable set problem, the maximum clique problem or the maximum cut problem. Especially NP-hard problems can be written as copositive programs. In this case the complexity is moved entirely into the copositivity constraint.
Copositive programming is a quite new topic in optimization. It deals with optimization over the so-called copositive cone, a superset of the positive semidefinite cone, where the quadratic form x^T Ax has to be nonnegative for only the nonnegative vectors x. Its dual cone is the cone of completely positive matrices, which includes all matrices that can be decomposed as a sum of nonnegative symmetric vector-vector-products.
The related optimization problems are linear programs with matrix variables and cone constraints.
However, some optimization problems can be formulated as combinatorial problems on infinite graphs. For example, the kissing number problem can be formulated as a stable set problem on a circle.
In this thesis we will discuss how the theory of copositive optimization can be lifted up to infinite dimension. For some special cases we will give applications in combinatorial optimization.Claudia Adamsdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/1074Thu, 14 Mar 2019 11:52:05 +0100Optimal Control of Partial Integro-Differential Equations and Analysis of the Gaussian Kernel
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/914
An important field of applied mathematics is the simulation of complex financial, mechanical, chemical, physical or medical processes with mathematical models. In addition to the pure modeling of the processes, the simultaneous optimization of an objective function by changing the model parameters is often the actual goal. Models in fields such as finance, biology or medicine benefit from this optimization step.
While many processes can be modeled using an ordinary differential equation (ODE), partial differential equations (PDEs) are needed to optimize heat conduction and flow characteristics, spreading of tumor cells in tissue as well as option prices. A partial integro-differential equation (PIDE) is a parital differential equation involving an integral operator, e.g., the convolution of the unknown function with a given kernel function. PIDEs occur for example in models that simulate adhesive forces between cells or option prices with jumps.
In each of the two parts of this thesis, a certain PIDE is the main object of interest. In the first part, we study a semilinear PIDE-constrained optimal control problem with the aim to derive necessary optimality conditions. In the second, we analyze a linear PIDE that includes the convolution of the unknown function with the Gaussian kernel.Lukas Aaron Zimmerdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/914Tue, 27 Nov 2018 09:01:28 +0100The Nonlocal Spatial Ramsey Model with Endogenous Productivity Growth
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/918
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.Laura Viktoria Somorowskydoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/918Tue, 25 Sep 2018 09:44:03 +0200A Method for Completely Positive and Nonnegative Matrix Factorization
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/879
A matrix A is called completely positive if there exists an entrywise nonnegative matrix B such that A = BB^T. These matrices can be used to obtain convex reformulations of for example nonconvex quadratic or combinatorial problems. One of the main problems with completely positive matrices is checking whether a given matrix is completely positive. This is known to be NP-hard in general. rnrnFor a given matrix completely positive matrix A, it is nontrivial to find a cp-factorization A=BB^T with nonnegative B since this factorization would provide a certificate for the matrix to be completely positive. But this factorization is not only important for the membership to the completely positive cone, it can also be used to recover the solution of the underlying quadratic or combinatorial problem. In addition, it is not a priori known how many columns are necessary to generate a cp-factorization for the given matrix. The minimal possible number of columns is called the cp-rank of A and so far it is still an open question how to derive the cp-rank for a given matrix. Some facts on completely positive matrices and the cp-rank will be given in Chapter 2. Moreover, in Chapter 6, we will see a factorization algorithm, which, for a given completely positive matrix A and a suitable starting point, computes the nonnegative factorization A=BB^T. The algorithm therefore returns a certificate for the matrix to be completely positive. As introduced in Chapter 3, the fundamental idea of the factorization algorithm is to start from an initial square factorization which is not necessarily entrywise nonnegative, and extend this factorization to a matrix for which the number of columns is greater than or equal to the cp-rank of A. Then it is the goal to transform this generated factorization into a cp-factorization. This problem can be formulated as a nonconvex feasibility problem, as shown in Section 4.1, and solved by a method which is based on alternating projections, as proven in Chapter 6. On the topic of alternating projections, a survey will be given in Chapter 5. Here we will see how to apply this technique to several types of sets like subspaces, convex sets, manifolds and semialgebraic sets. Furthermore, we will see some known facts on the convergence rate for alternating projections between these types of sets. Considering more than two sets yields the so called cyclic projections approach. Here some known facts for subspaces and convex sets will be shown. Moreover, we will see a new convergence result on cyclic projections among a sequence of manifolds in Section 5.4. In the context of cp-factorizations, a local convergence result for the introduced algorithm will be given. This result is based on the known convergence for alternating projections between semialgebraic sets. To obtain cp-facrorizations with this first method, it is necessary to solve a second order cone problem in every projection step, which is very costly. Therefore, in Section 6.2, we will see an additional heuristic extension, which improves the numerical performance of the algorithm. Extensive numerical tests in Chapter 7 will show that the factorization method is very fast in most instances. In addition, we will see how to derive a certificate for the matrix to be an element of the interior of the completely positive cone. As a further application, this method can be extended to find a symmetric nonnegative matrix factorization, where we consider an additional low-rank constraint. Here again, the method to derive factorizations for completely positive matrices can be used, albeit with some further adjustments, introduced in Section 8.1. Moreover, we will see that even for the general case of deriving a nonnegative matrix factorization for a given rectangular matrix A, the key aspects of the completely positive factorization approach can be used. To this end, it becomes necessary to extend the idea of finding a completely positive factorization such that it can be used for rectangular matrices. This yields an applicable algorithm for nonnegative matrix factorization in Section 8.2. Numerical results for this approach will suggest that the presented algorithms and techniques to obtain completely positive matrix factorizations can be extended to general nonnegative factorization problems.Patrick Hermann Groetznerdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/879Tue, 04 Sep 2018 09:59:32 +0200Dynamics of the Taylor Shift on Spaces of Holomorphic Functions
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/867
We will consider discrete dynamical systems (X,T) which consist of a state space X and a linear operator T acting on X. Given a state x in X at time zero, its state at time n is determined by the n-th iteration T^n(x). We are interested in the long-term behaviour of this system, that means we want to know how the sequence (T^n (x))_(n in N) behaves for increasing n and x in X. In the first chapter, we will sum up the relevant definitions and results of linear dynamics. In particular, in topological dynamics the notions of hypercyclic, frequently hypercyclic and mixing operators will be presented. In the setting of measurable dynamics, the most important definitions will be those of weakly and strongly mixing operators. If U is an open set in the (extended) complex plane containing 0, we can define the Taylor shift operator on the space H(U) of functions f holomorphic in U as Tf(z) = (f(z)- f(0))/z if z is not equal to 0 and otherwise Tf(0) = f'(0). In the second chapter, we will start examining the Taylor shift on H(U) endowed with the topology of locally uniform convergence. Depending on the choice of U, we will study whether or not the Taylor shift is weakly or strongly mixing in the Gaussian sense. Next, we will consider Banach spaces of functions holomorphic on the unit disc D. The first section of this chapter will sum up the basic properties of Bergman and Hardy spaces in order to analyse the dynamical behaviour of the Taylor shift on these Banach spaces in the next part. In the third section, we study the space of Cauchy transforms of complex Borel measures on the unit circle first endowed with the quotient norm of the total variation and then with a weak-* topology. While the Taylor shift is not even hypercyclic in the first case, we show that it is mixing for the latter case. In Chapter 4, we will first introduce Bergman spaces A^p(U) for general open sets and provide approximation results which will be needed in the next chapter where we examine the Taylor shift on these spaces on its dynamical properties. In particular, for 1<=p<2 we will find sufficient conditions for the Taylor shift to be weakly mixing or strongly mixing in the Gaussian sense. For p>=2, we consider specific Cauchy transforms in order to determine open sets U such that the Taylor shift is mixing on A^p(U). In both sections, we will illustrate the results with appropriate examples. Finally, we apply our results to universal Taylor series. The results of Chapter 5 about the Taylor shift allow us to consider the behaviour of the partial sums of the Taylor expansion of functions in general Bergman spaces outside its disc of convergence.Maike Thelendoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/867Tue, 28 Aug 2018 09:03:26 +0200Extension Operators with Optimal Continuity Estimates
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/866
Given a compact set K in R^d, the theory of extension operators examines the question, under which conditions on K, the linear and continuous restriction operators r_n:E^n(R^d)→E^n(K),f↦(∂^α f|_K)_{|α|≤n}, n in N_0 and r:E(R^d)→E(K),f↦(∂^α f|_K)_{α in N_0^d}, have a linear and continuous right inverse. This inverse is called extension operator and this problem is known as Whitney's extension problem, named after Hassler Whitney. In this context, E^n(K) respectively E(K) denote spaces of Whitney jets of order n respectively of infinite order. With E^n(R^d) and E(R^d), we denote the spaces of n-times respectively infinitely often continuously partially differentiable functions on R^d. Whitney already solved the question for finite order completely. He showed that it is always possible to construct a linear and continuous right inverse E_n for r_n. This work is concerned with the question of how the existence of a linear and continuous right inverse of r, fulfilling certain continuity estimates, can be characterized by properties of K. On E(K), we introduce a full real scale of generalized Whitney seminorms (|·|_{s,K})_{s≥0}, where |·|_{s,K} coincides with the classical Whitney seminorms for s in N_0. We equip also E(R^d) with a family (|·|_{s,L})_{s≥0} of those seminorms, where L shall be a a compact set with K in L-°. This family of seminorms on E(R^d) suffices to characterize the continuity properties of an extension operator E, since we can without loss of generality assume that E(E(K)) in D^s(L).
In Chapter 2, we introduce basic concepts and summarize the classical results of Whitney and Stein.
In Chapter 3, we modify the classical construction of Whitney's operators E_n and show that |E_n(·)|_{s,L}≤C|·|_{s,K} for s in[n,n+1).
In Chapter 4, we generalize a result of Frerick, Jordá and Wengenroth and show that LMI(1) for K implies the existence of an extension operator E without loss of derivatives, i.e. we have it fulfils |E(·)|_{s,L}≤C|·|_{s,K} for all s≥0. We show that a large class of self similar sets, which includes the Cantor set and the Sierpinski triangle, admits an extensions operator without loss of derivatives.
In Chapter 5 we generalize a result of Frerick, Jordá and Wengenroth and show that WLMI(r) for r≥1 implies the existence of a tame linear extension operator E having a homogeneous loss of derivatives, such that |E(·)|_{s,L}≤C|·|_{(r+ε)s,K} for all s≥0 and all ε>0.
In the last chapter we characterize the existence of an extension operator having an arbitrary loss of derivatives by the existence of measures on K.Arne Jakobsdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/866Tue, 28 Aug 2018 09:02:20 +0200Modeling, Simulation and Optimization of Wine Fermentation
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/860
Industrial companies mainly aim for increasing their profit. That is why they intend to reduce production costs without sacrificing the quality. Furthermore, in the context of the 2020 energy targets, energy efficiency plays a crucial role. Mathematical modeling, simulation and optimization tools can contribute to the achievement of these industrial and environmental goals. For the process of white wine fermentation, there exists a huge potential for saving energy. In this thesis mathematical modeling, simulation and optimization tools are customized to the needs of this biochemical process and applied to it. Two different models are derived that represent the process as it can be observed in real experiments. One model takes the growth, division and death behavior of the single yeast cell into account. This is modeled by a partial integro-differential equation and additional multiple ordinary integro-differential equations showing the development of the other substrates involved. The other model, described by ordinary differential equations, represents the growth and death behavior of the yeast concentration and development of the other substrates involved. The more detailed model is investigated analytically and numerically. Thereby existence and uniqueness of solutions are studied and the process is simulated. These investigations initiate a discussion regarding the value of the additional benefit of this model compared to the simpler one. For optimization, the process is described by the less detailed model. The process is identified by a parameter and state estimation problem. The energy and quality targets are formulated in the objective function of an optimal control or model predictive control problem controlling the fermentation temperature. This means that cooling during the process of wine fermentation is controlled. Parameter and state estimation with nonlinear economic model predictive control is applied in two experiments. For the first experiment, the optimization problems are solved by multiple shooting with a backward differentiation formula method for the discretization of the problem and a sequential quadratic programming method with a line search strategy and a Broyden-Fletcher-Goldfarb-Shanno update for the solution of the constrained nonlinear optimization problems. Different rounding strategies are applied to the resulting post-fermentation control profile. Furthermore, a quality assurance test is performed. The outcomes of this experiment are remarkable energy savings and tasty wine. For the next experiment, some modifications are made, and the optimization problems are solved by using direct transcription via orthogonal collocation on finite elements for the discretization and an interior-point filter line-search method for the solution of the constrained nonlinear optimization problems. The second experiment verifies the results of the first experiment. This means that by the use of this novel control strategy energy conservation is ensured and production costs are reduced. From now on tasty white wine can be produced at a lower price and with a clearer conscience at the same time.Christina Schenkdoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/860Wed, 08 Aug 2018 06:59:20 +0200Quadratic Optimization: Copositive Modelling, Algorithms and Aspects of Duality
https://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/852
Quadratische Optimierungsprobleme (QP) haben ein breites Anwendungsgebiet, wie beispielsweise kombinatorische Probleme einschließlich des maximalen Cliquenroblems. Motzkin und Straus [25] zeigten die Äquivalenz zwischen dem maximalen Cliquenproblem und dem standard quadratischen Problem. Auch mathematische Statistik ist ein weiteres Anwendungsgebiet von (QP), sowie eine Vielzahl von ökonomischen Modellen basieren auf (QP), z.B. das quadratische Rucksackproblem. In [5] Bomze et al. haben das standard quadratische Optimierungsproblem (StQP) in ein Copositive-Problem umformuliert. Im Folgenden wurden Algorithmen zur Lösung dieses copositiviten Problems von Bomze und de Klerk in [6] und Dür und Bundfuss in [9] entwickelt. Während die Implementierung dieser Algorithmen einige vielversprechende numerische Ergebnisse hervorbrachten, konnten die Autoren nur die copositive Neuformulierung des (StQP)s lösen. In [11] präsentierte Burer eine vollständig positive Umformulierung für allgemeine (QP)s, sogar mit binären Nebenbedingungen. Leider konnte er keine Methode zur Lösung für ein solches vollständig positives Problem präsentieren, noch wurde eine copositive Formulierung vorgeschlagen, auf die man die oben erwähnten Algorithmen modifizieren und anwenden könnte, um diese zu lösen. Diese Arbeit wird einen neuen endlichen Algorithmus zur Lösung eines standard quadratischen Optimierungsproblems aufstellen. Desweiteren werden in dieser Thesis copositve Darstellungen für ungleichungsbeschränkte sowie gleichungsbeschränkte quadratische Optimierungsprobleme vorgestellt. Für den ersten Ansatz wurde eine vollständig positive Umformulierung des (QP) entwickelt. Die copositive Umformulierung konnte durch Betrachtung des dualen Problems des vollständig positiven Problems erhalten werden. Ein direkterer Ansatz wurde gemacht, indem das Lagrange-Duale eines äquivalenten quadratischen Optimierungsproblems betrachtet wurde, das durch eine semidefinite quadratische Nebenbedingung beschränkt wurde. In diesem Zusammenhang werden Bedingungen für starke Dualität vorgeschlagen.Duy Van Nguyendoctoralthesishttps://ubt.opus.hbz-nrw.de/frontdoor/index/index/docId/852Tue, 12 Jun 2018 14:01:30 +0200