## 510 Mathematik

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This thesis is concerned with the efficient and accurate simulation and optimization of linear Timoshenko beam networks subjected to external loads.
For this we develop a solution scheme, which is based on so-called analytic ansatz-functions known to provide analytic solutions
for the unknowns of a single beam subjected to given boundary data. In the present work we prove that this concept can be extended to a network of beams
which enables the analytic calculation of the unique deformation of the whole network.
Moreover, we show that this approach is equivalent to a finite element method where only one finite element with a particular shape function per
beam is required. With this we can provide the analytic solution on each node of the network, from which the state of the whole network can be uniquely determined - no further spatial discretization is needed.
Based on this analytic solution scheme for the simulation problem, we investigate a number of problems from structural optimization, more precisely
topology optimization, material and multi-material optimization, cross section optimization, geometry optimization and simultaneous versions of the before mentioned.
Hereby we provide the problem formulation, analytic formulae for first order derivatives, an implementation and numerical examples for each type of optimization model.
In the next part of this thesis we develop a novel approach for geometry, cross section and material optimization for a system of high-pressure steam pipes in a power plant.
This industrial application is efficiently modeled by including a special parametrization for the geometry of pipes.
Then we formulate
an optimization problem with an objective function considering monetary costs and a large number of constraints.
Moreover, we derive analytic expressions for the first order derivatives of the problem and provide an implementation. We demonstrate the
capabilities of our program with numerical tests for a real-world instance.

This paper presents a new approach for the determination of the wind speed distribution based on wind speed data. This approach is based on the fact that, in general, wind speed distributions restricted to seasons of year or months are different. Therefore, instead of one Weibull density function, a convex combination of Weibull density functions is considered for a calendar year. This model improves the maximum likelihood of the estimated wind speed distribution. Numerical results including a Kolmogorov–Smirnov test are given for a site at Jamaica. Numerical comparisons are carried out for different sites and various known methods for the estimation of the wind speed distribution.

In an example for the determination of publicly registered land values in urban building conflict situations an interval value duality and regression are applied. The result is incumbent upon a new interval value optimum – beginning with non-ordered point sets of publicly registered German land values for residential land. These two mathematical approaches with set-valued order relations describe the relation to the socio-economy in an urban building conflict situation.

This thesis deals with properties of singular orbits in multi-body-systems.
Here, systems are examined, in which the dynamics defining potential solely depends on the relative positions of the particles. Celestial mechanics, which is the motion of any number of particles due to Newton's law of universal gravitation, can be regarded as a standard example. With this in mind, the class of so-called long-ranged and moderated potentials are introduced, which among others also includes the Coulomb field of electrostatics. In these systems, the flow which is induced by the set of differential equations of motion, is generally not complete, and the non-global solutions are called singular. The goal is to show their improbability in due situations, that is to show that the set of initial conditions leading to non-global solution is a null-set in the sense of Lebesgue.
An important distinction of singular orbits are collision and non-collision orbits: the former are defined by the property that all particles have definitive limit points as time approaches the singularity. Previous results in the field of celestial mechanics show the improbability of collisions in general, and the improbability of non-collision singularities in the four-body-problem. Some generalizations to a wider class of potentials is possible with the techniques imposed there, however, certain restrictions like on the homogeneity of the force field cannot be dropped, and even within these restrictions optimal bounds cannot be shown.
Hence, the provision of a new technique seems to be necessary, in order to generalize the results to a wider class of systems, which is done within this work...

In this paper we present a chain of mathematical models that enables the numerical simulation of the airlay process and the investigation of the resulting nonwoven material by means of virtual tensile strength tests. The models range from a highly turbulent dilute fiber suspension flow to stochastic surrogates for fiber lay-down and web formation and further to Cosserat networks with effective material laws. Crucial is the consistent mathematical mapping between the parameters of the process and the material. We illustrate the applicability of the model chain for an industrial scenario, regarding data from computer tomography and experiments. By this proof of concept we show the feasibility of future simulation-based process design and material optimization which are long-term objectives in the technical textile industry.

Mathematical modeling of biochemical pathways is an important resource in Synthetic Biology, as the predictive power of simulating synthetic pathways represents an important step in the design of synthetic metabolons. In this paper, we are concerned with the mathematical modeling, simulation, and optimization of metabolic processes in biochemical microreactors able to carry out enzymatic reactions and to exchange metabolites with their surrounding medium. The results of the reported modeling approach are incorporated in the design of the first microreactor prototypes that are under construction. These microreactors consist of compartments separated by membranes carrying specific transporters for the input of substrates and export of products. Inside the compartments of the reactor multienzyme complexes assembled on nano-beads by peptide adapters are used to carry out metabolic reactions. The spatially resolved mathematical model describing the ongoing processes consists of a system of diffusion equations together with boundary and initial conditions. The boundary conditions model the exchange of metabolites with the neighboring compartments and the reactions at the surface of the nano-beads carrying the multienzyme complexes. Efficient and accurate approaches for numerical simulation of the mathematical model and for optimal design of the microreactor are developed. As a proof-of-concept scenario, a synthetic pathway for the conversion of sucrose to glucose-6-phosphate (G6P) was chosen. In this context, the mathematical model is employed to compute the spatio-temporal distributions of the metabolite concentrations, as well as application relevant quantities like the outflow rate of G6P. These computations are performed for different scenarios, where the number of beads as well as their loading capacity are varied. The computed metabolite distributions show spatial patterns, which differ for different experimental arrangements. Furthermore, the total output of G6P increases for scenarios where microcompartimentation of enzymes occurs. These results show that spatially resolved models are needed in the description of the conversion processes. Finally, the enzyme stoichiometry on the nano-beads is determined, which maximizes the production of glucose-6-phosphate.

We use the example of hot rolling to develop a comprehensive optimization approach that is based on a mathematical model of the underlying manufacturing process. More precisely, we study an optimal control problem that is designed to minimize cutting scrap while taking industrial specifications and technical limitations into account.
It is well known that the associated control-to-observation map is non-differentiable due to changes of state resulting from elasto-viscoplastic material behavior and frictional contact. However, we still want to apply gradient-based methods to solve the optimal control problem and therefore have to compute derivatives of cost functional and constraints. To resolve this issue, we first regularize all non-differentiabilities before computing sensitivity information via direct differentiation.
We moreover present solution techniques for the regularized problem and discuss numerical results for real-world examples.

Let K→X be a smooth Lie algebra bundle over a σ-compact manifold X whose typical fiber is the compact Lie algebra k. We give a complete description of the irreducible bounded (i.e., norm continuous) unitary representations of the Fréchet–Lie algebra Γ(K) of all smooth sections of K, and of the LF-Lie algebra Γc(K) of compactly supported smooth sections. For Γ(K), irreducible bounded unitary representations are finite tensor products of so-called evaluation representations, hence in particular finite dimensional. For Γc(K), bounded unitary irreducible (factor) representations are possibly infinite tensor products of evaluation representations, which reduces the classification problem to results of Glimm and Powers on irreducible (factor) representations of UHF C∗-algebras. The key part in our proof is the result that every irreducible bounded unitary representation of a Lie algebra of the form k⊗RAR, where AR is a unital real complete continuous inverse algebra, is a finite product of evaluation representations. On the group level, our results cover in particular the bounded unitary representations of the identity component Gau(P)0 of the group of smooth gauge transformations of a principal fiber bundle P→X with compact base and structure group, and the groups SUn(A)0 with A a complete involutive commutative continuous inverse algebra.

We consider the heat flow associated with the H-surface system given by
∂tu−Δuu(⋅,0)=−2(H∘u)D1u×D2uon B×(0,∞),=uo on B,u=g on ∂B×(0,∞),
for a prescribed bounded and continuous function H:R3→R satisfying an isoperimetric condition of type c for 0<c<1 and time-independent Dirichlet data g of class C1,γ. Here, B⊂R2 denotes the unit disk. For the global solutions u:B→R3 constructed in a preceding work, we prove short-time regularity in the sense that u and Du are locally Hölder continuous on B¯×(0,T) up to a singular time T>0. From this, we deduce the existence of a global solution u~:B×(0,∞)→R3 that is regular except from finitely many singular times.

A priori error analysis for finite element approximations of the Stokes problem on dynamic meshes
(2014)

In this article we study finite element approximations of the time-dependent Stokes system on dynamically changing meshes. Applying the backward Euler method for time discretization we use the discrete Helmholtz or Stokes projection to evaluate the solution at time tn−1 on the new spatial mesh at time tn. The theoretical results consist of a priori error estimates that show a dependence on the time step size not better than O(1/Δt). These surprisingly pessimistic upper bounds are complemented by numerical examples giving evidence for a negative convergence rate, at least for a large range of time step sizes, and in this sense backing our theory. These observations imply that using adaptive meshes for incompressible flow problems is delicate and requires further investigation.

We develop a numerical framework for efficiently
simulating partially miscible multiphase multicomponent flow in porous media with general chemical reactions,
modeled by a system of coupled and strongly nonlinear partial differential equations, ordinary differential equations, and algebraic equations.
The equations are transformed with the help of a reduction
scheme which preserves the model and allows the unknown equilibrium reaction rates to be eliminated. In the transformed system, the algebraic equations resulting from the chemical
equilibria are eliminated in terms of a nonlinear, implicitly defined resolution
function. Hereby, the equilibrium conditions consisting of equations and inequality constraints are formulated as a complementarity problem and rewritten as
algebraic equations using the minimum function. The existence of the resolution
function is established by using the connection of the nonlinear system of algebraic equations to a constrained minimization problem based on a modified Gibbs
functional, and numerical strategies to evaluate it numerically are presented. Using a global implicit approach, the nonlinear systems that remain to be solved
in each time step are treated with the semismooth Newton method. With the help of the
resolution function, we are ready to define a persistent set of primary variables that
are valid in either phase state and for an arbitrary mineral assemblage. Different numerical tests related to hydrogen migration in deep geological repositories of radioactive waste and to CO2 sequestration show that our solver provides accurate numerical results, and that it is capable of
handling the strongly nonlinear coupling of flow, transport, chemical reactions,
and mass transfer across the phases.
The second part of this work is concerned with the analysis of robust mixed hybrid finite
element methods of lowest order for advection–diffusion problems. In the approach studied
here, the classical scheme is extended with the help of the
Lagrange multipliers associated with the hybrid problem formulation. More precisely, it relies
on the fact that the Lagrange multipliers represent approximations of the scalar
unknown on the interelement boundaries and uses them in the approximation of
the advective fluxes. Using techniques from the a posteriori error analysis, we are
able to establish optimal order convergence for a new class of methods including
the standard method, a partial upwind method, and a full upwind method. The
advantage of the specific choice of the upwind weights is the fact that the method
remains local. In this case, static condensation can be employed, whereas the standard upwind-mixed method requires information from neighbor cells such that
static condensation is not applicable.
Concerning approximations with the BDM1 mixed finite element, we show that
the use of Lagrange multipliers in the discretization of the advective term provides optimal second order convergence for the total flux variable in the L2 norm, whereas the standard mixed method
is known to be of suboptimal first order accuracy only.

In dieser Arbeit wird eine neue Strategie zur Lösung von Optimierungsproblemen mit lokal Lipschitz-stetiger Zielfunktion vorgestellt. Wir werden insbesondere nicht voraussetzen, dass die Zielfunktion semismooth ist.
Die Grundlage dieser Strategie bilden die sogenannten stetigen äußeren Subdifferentiale, die den mangelnden Informationsgehalt eines Subdifferentials beheben. Es wird ein Abstiegsverfahren basierend auf diesen äußeren Subdifferentialen entwickelt und dessen Konvergenz bewiesen. In diesem Zusammenhang wird untersucht, welche Eigenschaften eines Subdifferentials essentiell zur Optimierung nichtdifferenzierbarer Funktionen sind, und damit wird die Frage nach einer geeigneten Wahl eines Subdifferentials beantwortet.
Ein weiterer Abschnitt der Arbeit widmet sich der Konstruktion stetiger äußerer Subdifferentiale. Dabei nimmt die Klasse der Optimalwertfunktionen einen besonderen Stellenwert ein. Die bei der Konstruktion zum Vorschein kommenden Schwierigkeiten werden anhand akademischer Beispiele diskutiert.
Im letzten Abschnitt der Arbeit wird ein neues Verfahren zur Minimierung lokal Lipschitz-stetiger Optimalwertfunktionen vorgestellt (BTO). Die Grundlage bilden Bundle-Trust-Region-Ideen der nichtglatten, konvexen Optimierung und die Armijo-Regel zur Bestimmung einer geeigneten Schrittweite in der glatten Optimierung. Von Bundle- Trust-Region-Verfahren wird die Idee zur Bildung einer Modellfunktion adaptiert, die auf den in dieser Arbeit zuvor eingeführten approximativen stetigen äußeren Subdifferentialen basiert.
Wir verallgemeinern weiterhin die Strategie zur Anpassung des Trust-Region-Radius und präsentieren ein neues Verfahren zur sukzessiven Verbesserung der Modellfunktion. Abschließend wird die globale Konvergenz des BTO-Verfahrens bewiesen.

In this note we characterize those unitary one-parameter groups (Utc)t∈R which admit euclidean realizations in the sense that they are obtained by the analytic continuation process corresponding to reflection positivity from a unitary representation U of the circle group. These are precisely the ones for which there exists an anti-unitary involution J commuting with Uc. This provides an interesting link with the modular data arising in Tomita-Takesaki theory. Introducing the concept of a positive definite function with values in the space of sesquilinear forms, we further establish a link between KMS states and reflection positivity on the circle.

Let G and T be topological groups, α : T → Aut(G) a homomorphism defining a continuous action of T on G and G ♯ := G ⋊α T the corresponding semidirect product group. In this paper, we address several issues concerning irreducible continuous unitary representations (π♯, ) of G ♯ whose restriction to G remains irreducible. First, we prove that, for T = , this is the case for any irreducible positive energy representation of G ♯, i.e. for which the one-parameter group Ut := π♯(1,t) has non-negative spectrum. The passage from irreducible unitary representations of G to representations of G ♯ requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results of Borchers for W*-dynamical systems, we also derive a characterization of the continuous positive definite functions on G that extend to G ♯.

This thesis is about the information-theoretic difference between hypergraphs and
undirected graphs, which arises in spectral bisections of these objects. We also
investigate the effect of general deletion and contraction operations of hyperedges
in a hypergraph from different points of view. We restrict ourselves to a wellstructured
class of hypergraphs, namely, simplicial complexes. This allows us to
define higher-dimensional Laplace operators, which generalize graph Laplacians in
a natural way. Similarly, isoperimetric numbers for graphs - combinatorial Cheeger
constants - can be extended to simplicial complexes. We suggest a new isoperimetric
quantity for simplicial complexes, based on minimal interaction loss, and show
an isoperimetric inequality for them. The Kullback-Leibler divergence, also known
as relative entropy, is used to consider the difference between simplicial complexes
and undirected graphs from an information-theoretic point of view. By means of
hierarchical exponential families, which are naturally induced by simplicial complexes,
we define a relative entropy between simplicial complexes. For a systematic
investigation of this relative entropy we consider its effect under certain operations,
like deletion and contraction of simplices in a simplicial complex. With a detailed
analysis of marginalized exponential families and with already known decomposition
formulas for the Kullback-Leibler divergence, we succeed to identify a class of
operations on simplicial complexes for which we can deduce many exact expressions
and inequalities for the relative entropy. For more general operations on simplicial
complexes we can show an upper bound for the relative entropy. Through an
explicit construction of probability distributions we are able to state lower bounds
of the relative entropy between separated and non separated simplicial complexes.
Based on the observation from spectral graph theory, in which the smallest nontrivial
eigenvalue of the Laplacian is related to a combinatorial Cheeger constant,
we show a connection of this eigenvalue to scaled relative entropies. For this purpose,
an information-theoretic Cheeger constant is introduced, which is related to
a combinatorial Cheeger constant. This allows us to derive information-theoretic
Cheeger inequalities for undirected graphs and simplicial complexes. In particular,
it is possible to see the information-theoretic difference between undirected graphs
and simplicial complexes by means of an entropy gap.

Vector Optimization and Control with Partial Differential Equations and Pointwise State Constraints
(2014)

This thesis is dedicated to the generalization of state-constrained optimal control problems with PDE-constraints from problems with scalar valued cost functions to problems with vector valued cost functions. Apart from being a real Banach space, no further restrictions on the objective space are made.
The presence of pointwise constraints on the state function raises difficulties when it comes to checking admissibility or formulating optimality conditions. Moreover, Lagrange multipliers corresponding to pointwise state constraints are typically very irregular such that regularization is necessary before solving state-constrained problems numerically. In order to cover a wide range of problems, we introduce an abstract class of possibly nonconvex extremal problems,
and we analyze this class of problems in terms of admissibility, solvability, scalarization, regularization and optimality conditions.
All results are verified by applying them to linear and nonlinear, elliptic and parabolic examples.
Furthermore, solutions to certain test problems are computed by using efficient numerical algorithms.