## Department Mathematik

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In applications, often not all parameters of optimization problems are known precisely, for example, due to measurement inaccuracies or imprecise statistical predictions.
Optimization under uncertainty, therefore, plays an increasingly important role in current research.
Robust optimization is one approach to dealing with such uncertainties.
Here, we require that a robust solution is feasible for all parameters from a previously chosen so-called uncertainty set.
In the classical concept of robustness, one then searches for the robust solution that has the best worst-case objective function value.
We consider three relevant problems under this and other robustness concepts in the following. In each case, we focus on a different aspect of robust optimization.
In addition, for all problems, we investigate whether and how problem-specific properties extend to the robust case as well.
After a short introduction to robust optimization, we summarize the considered problems and present the respective results in the first part of this cumulative dissertation.
The second part contains reprints of the publications and manuscripts on which the first part is based.
First, we consider flow problems under uncertainty:
The well-known maximum flow problem is extended by the assumption that up to Γ many edges may fail.
Several robust models already exist for this problem. We introduce a unifying model and analyze its complexity. Further, we compare the models with each other and extend our analysis to a dynamic setting.
Next, we address robust market equilibria under the assumption of uncertain production costs: We model robust producers and compare the resulting equilibria with the so-called robust central planner.
In the case that the uncertainty set contains correlations, there is a gap between the welfare of the equilibrium and the optimal value of the robust central planner, which is in contrast to the nominal problem.
We quantify this gap and provide bounds.
Since the market model we chose involves investments, the established concept of adjustable robustness is also suitable for the problem. This means that the investments are fixed in the beginning, while the production decisions may be made after the production costs are known. Among other things, we show how to use subsidies to induce an equilibrium that corresponds to the optimal solution of the robust central planner.
Third, we consider the Linear Complementarity Problem (LCP): Again, we discuss adjustable robustness but restrict the solution set to affine functions. This restriction allows us to derive compact characterizations of affinely adjustable robust solutions. With these, a mixed-integer linear problem can be set up to compute solutions.
Under further assumptions, particularly on the LCP matrix M, we show solvability in polynomial time and the uniqueness of the solution.
In the last content chapter, we study the concept of Pareto optimality for robust optimization.
In many problems, the robust optimal solution is not necessarily unique.
For this case, the concept of Pareto optimality has already been applied to linear problems in the literature.
We extend this to problems with a nonlinear objective function and demonstrate how Pareto robust optimal solutions can be computed.
Finally, we discuss two approaches to extend the concept of Pareto optimality to robust combinatorial problems.

A collection of mathematical optimization models: Formulations, solution methods and applications
(2023)

This is a cumulative habilitation thesis that includes a summary of ten previously published articles in mathematical optimization. Stochastic optimization is the unifying theme of these works, although there are a few topics dealing with deterministic optimization as well. Chance-constrained optimization is presented in particular.

This doctoral thesis develops mathematical aspects of the theory of topological insulators and topological semimetals in an operator-algebraic framework. Among the main results are index theorems for so-called weak Chern numbers, which are associated to the Fermi projections of non-interacting free Fermions, under Sobolev-type regularity conditions which make them applicable to systems with only a Mobility gap or Pseudogap, namely strongly disordered topological insulators and Dirac-/Weyl-semimetals respectively. Another novelty is a systematic treatment of topological phases which are defined with respect to a reference system with a focus on continuous models for topological insulators. A K-theoretic bulk-boundary and bulk-interface correspondence for such relative topological invariants is developed and illustrated at the hand of continuum models like massive Dirac Hamiltonians.
The final chapter treats two approaches to bulk-boundary correspondence of strongly disordered insulators and semimetals: The first is an index-theoretic approach to flat band eigenstates of certain chirally symmetric systems, which applies for example to some models of graphene. The second presents an approach to the interface conductivitiy between (weak) Quantum Hall systems where bulk-interface correspondence is derived using regularization and explicit calculations.

Deterministic linear complementarity systems (LCPs) are often used to model
equilibrium problems where all parameters are certain. However, in real-world
applications as, e.g., in game-theoretic settings, traffic modeling, or energy
markets, many parameters are uncertain. One reason for that are estimates and
prognoses of future events and data. There exist different approaches to include
such uncertainties in the modeling. Thus, the question arises how “efficient” the
chosen approach for the given application is. How should one formulate and
integrate the uncertainties? Is the modeled uncertain problem tractable, i.e.,
easy to solve? Does a solution exist in general and, if so, is the solution unique?
In the past, this topic was mainly considered using a stochastic approach.
This cumulative PhD thesis is concerned with uncertain LCPs using the
concept of Γ-robust optimization to integrate uncertainties. As there exist
no solution in general, we consider “worst-case” minima, i.e., we study the
minimization of the worst-case gap function formulation of Γ-robust counterparts
of LCPs.
Part I of this dissertation is split in two topics. The first is a theoretical
consideration of such problems. We investigate uncertainties in the vector and
in the matrix defining the LCP. For box, ` 1 -norm, and ellipsoidal uncertainty
sets we derive conditions for tractable convex counterparts of the gap function
formulation and study their feasibility as well as the existence and uniqueness of
solutions. To illustrate the effects of the Γ-robust concept applied to LCPs, we
present case studies containing the uncertain traffic equilibrium problem and
market equilibrium modeling. The second topic is the study of different robust
electricity market equilibrium problems. These models are considered using
lossless DC networks. We state deterministic market formulations and after-
ward derive Γ-robust models. In one model we have possible transmission and
generation investments. Here, we show that the Γ-robust welfare maximization
problem is the counterpart of a market equilibrium problem with robustified
player problems. We see in a case study for this scenario that the transmission
system operator acts more risk-neutral whereas the generating firms behave more
risk-averse in the robust setting. Afterward, we consider models of electricity
markets using the economic assumptions of perfect competition or Nash–Cournot
and without transmission and generation investments. We study their robustifi-
cation using the Γ-approach and consider strict robustness separately. Except
the combination of Nash–Cournot competition and Γ-robustification, we ob-
tain the same systems if we robustify the deterministic welfare problem and if
we robustify the separate optimization problems of the players with uncertain
data and consider the complementarity problem which contains their first-order
optimality conditions with added equilibrium conditions. This means that in the case of Nash–Cournot competition and Γ-robustification no analogy of the
welfare theorems hold. The different effects of the combination of economic
competition and uncertainty modeling is illustrated in a case study.
The second part of this thesis contains reprints of our original articles under-
lying the first part and comprise all details.

This dissertation connects different topics and shows new results in the research
field of applied analysis.
We investigate the regularity of solutions of partial differential
equations (=PDEs) of elliptic and parabolic type on non-smooth domains. Special attention is paid to the regularity of the solution in certain weighted Sobolev spaces as well as in a specific scale of Besov spaces, the so called adaptivity scale. Our studies are motivated by some fundamental questions arising in the context of the numerical treatment when solving such PDEs: The smoothness of the solutions in this specific scale of Besov spaces determines the order of convergence that can be achieved when using adaptive algorithms. On the other hand, it is the (fractional) Sobolev regularity, which encodes information on the convergence order for linear approximation of non-adaptive (uniform) methods. Since we want to justify the use of adaptive schemes for solving PDEs on non-smooth domains, an analysis of the regularity of the solution in the Besov scale and a comparison with its Sobolev regularity is needed.

This thesis is concerned with the investigation of a toxicological research question with the help of a mathematical model which is parameterized from experimental data. This is an interdisciplinary and fundamental but also intricate task.
The beginning of this thesis deals with the modeling of a metabolic network to investigate the cellular response dynamics to xenobiotic-induced oxidative stress, especially in view of the contributions of the pentose phosphate pathway (PPP) and the gluconate shunt. As specific application, the exposure to the toxin 3-nitrobenzanthrone (3-NBA) which is contained for instance in diesel engine exhaust is considered. Its bioactivation in cells which can be monitored via fluorescence goes hand in hand with the generation of reactive oxygen species (ROS). We identify the metabolic network which is mainly responsible for the antioxidative cellular response to oxidative stress and corresponding regulatory mechanisms adapting the metabolism to counteract detrimental oxidative stress. Such regulatory mechanisms have an important impact on the dose-effect relationship. We derive a mathematical model in form of ordinary differential equations based on enzyme kinetics for multisubstrate reactions and mass action. The model is parameterized employing literature data and data from diverse in vitro experiments such as relative fluorescence intensity measurements. Simulations of the mathematical model give insight into the cellular response dynamics to oxidative stress and reveal the importance of the gluconate shunt in the low-dose region of exposure. A local sensitivity analysis in Matlab underlines that the solution is particularly sensitive to parameters from the bioactivation of 3-NBA, the regulatory mechanisms and the rate determining reactions.
Therefore, we improve the parameter estimation for the bioactivation of 3-NBA from fluorescence intensity measurements. From statistical analysis of the raw experimental data, the conjecture arises that samples might influence each other during the measurement process. This phenomenon is known as crosstalk. We derive two mathematical models for fluorescence intensity measurements (one with crosstalk and one without crosstalk) from the experimental setup to describe the connection between fluorophore concentration and raw experimental data. For parameter estimation from raw fluorescence data it is required to identify the more plausible model first.
For this task, we focus on the Bayesian approach for model comparison and parameter inference and employ the algorithm nested sampling which has been created to approximate the evidence of a model and to generate simultaneously samples from the posterior distribution. The centerpiece of nested sampling is an integral transformation and the subsequent approximation of the transformed integral by a Monte Carlo approximation based on the uniformity assumption. The proof of the integral transformation has been sketched previously for bounded and nonnegative likelihood functions exhibiting no plateau with positive prior measure. We prove that the integral transformation is valid in more general settings allowing particularly plateaus in the likelihood function. Furthermore, we show that the uniformity assumption is violated in such settings (even though it is fulfilled in settings without plateaus). A preprocessing splitting approach is derived to overcome this difficulty and, thus, to adapt vanilla nested sampling to more general settings. It is shown that this preprocessing approach is more efficient than the established randomization strategy which simply avoids likelihood plateaus by perturbing the likelihood function.
The thesis concludes by formulating a Bayesian framework for parameter estimation from fluorescence data. Within the framework nested sampling is applied to approximate the evidences of both fluorescence models (with and without crosstalk) and to estimate simultaneously the unknown parameters. The performance of the Bayesian framework is demonstrated using artificial data corresponding to reduced (bio)chemical models.

Feasibility for Maximal Uncertainty Sets in Robust Optimization with Application to Gas Networks
(2022)

Robust optimization is a popular approach to protect an optimization problem against uncertain data within a user-specified set of scenarios, the so-called uncertainty set. In many cases, the choice of the uncertainty set is driven by the application. In general, it can be elusive to assume that the exact “size” of the uncertainty set can be specified prior to the optimization process. Overly large sized uncertainty sets can lead to infeasible robust optimization problems. To avoid robust infeasibility due to the choice of the uncertainty set, it is useful to know the maximal “size” of a given uncertainty set such that feasibility of the robust optimization problem is still guaranteed. We study maximal uncertainty sets that guarantee robust feasibility for general mixed-integer linear problems (MIPs) and in the context of gas networks in this cumulative dissertation.
In the first part, we summarize and discuss our results developed over the last years. The second part of this cumulative dissertation contains reprints of our original articles and preprints, which contain all details of the presented results. We also refer to these articles throughout the first part of this dissertation.
For general MIPs, we consider a specific notion for the maximal size of a given uncertainty set: the radius of robust feasibility (RRF). We introduce and study the RRF for MIPs under common assumptions from the literature and then extend the RRF to include “safe” variables and constraint, i.e., variables and constraints that are not affected by uncertainties. We further develop methods for computing the RRF of linear and mixed-integer linear problems with safe variables and constraints and successfully apply them to instances of the MIPLIB 2017 library. Based on our results, we can control the price of robustness by adjusting the size of the uncertainty set.
Moreover, we study the two-stage robust problems of deciding the feasibility of a booking as well as of computing maximal technical capacities within the European entry-exit gas market system. A booking is a capacity-right contract for which the transmission system operator has to guarantee that every balanced load flow below the booking can be transported through the network. Maximal technical capacities bound these bookings and, thus, describe maximal bookable capacities. Except for some technical subtleties, these robust problems lead to deciding the feasibility as well as solving a specific two-stage robust nonlinear optimization problem. The main goal of this problem consists of computing a maximal uncertainty set of balanced load flows so that for each of these load flows there is a feasible transport through the network. We study this problem algorithmically with focus on nonlinear models of gas transport. We analyze structural properties such as (non-)convexity of the set of feasible bookings and of the set of feasible balanced load flows for different models of gas transport. For deciding the feasibility of a booking, we develop a polynomial-time algorithm for single-cycle networks consisting of pipes. We also characterize feasible bookings in networks with compressors and control valves. Based on the results for bookings, we provide results for computing maximal technical capacities in tree-shaped networks. We note that our results can also contribute to other potential-based network problems such as computing a robust diameter selection for tree-shaped hydrogen networks with demand uncertainties.

In this thesis, we study well-posedness, stabilization and control problems involving freely vibrating beams that may undergo motions of large magnitude -- i.e. large displacements of the reference line and large rotations of the cross sections. Such beams, shearable and very flexible, are often called geometrically exact beams and are especially needed in modern highly flexible light-weight structures, where one cannot neglect these large motions. The mathematical model is represented by a nonlinear governing system to account for such motions, and one thus also speaks of geometric nonlinearities. The constitutive law however is linear, and allows for possibly composite, anisotropic materials.
We view these beams from two perspectives. The first perspective is one in which the beam is described in terms of the position of its reference line and the orientation of its cross sections (expressed in some fixed coordinate system). This is the generally encountered model, due to Eric Reissner and Juan C. Simo. Of second order in time and space, it is a quasilinear system of six equations. The second perspective is one in which the beam is rather described by intrinsic variables -- velocities and strains or internal forces and moments -- which are moreover expressed in a moving coordinate system attached to the beam. This system, proposed in its most general form by Dewey H. Hodges, consists of twice as many equations, but is of first order in time and space, hyperbolic and only semilinear (quadratic). While the first model has a wave-like form, the intrinsic beam model may be seen as part of the Hamiltonian framework in continuum mechanics.
Looking at the definition of the state of the latter model, we can see that both perspectives are linked by a nonlinear transformation. The questions of well-posedness, stabilization and control are addressed for geometrically exact beams (and networks of such beams) governed by the intrinsic model, while by using the transformation we also prove that the existence and uniqueness of a classical solution to the intrinsic model implies that of a classical solution to the model written in terms of positions and rotations. In particular, this enables us to deduce corresponding well-posedness, stabilization and control results for the latter model. We also also address these questions for networks of beams attached to each other by means of rigid joints.
The stabilization is realised by means of velocity feedback controls applied at the boundary. The first step consists in proving local in time existence and uniqueness of solutions in $C_t^0H_x^1$ (and $C_t^0H_x^2$ for more regular initial data) for general networks. Then, by means of quadratic Lyapunov functionals we show that, first for a single beam controlled at one end, and then for a star-shaped network controlled at all simple nodes, one can achieve local exponential stability of the zero steady state for the $H^1$ and $H^2$ norms. Other than the quadratic nonlinearity, the main difficulty in finding such a functional lies in the fact that the linearized system is not homogeneous and thus one has to take into account not only the boundary conditions but also the governing system in order to find the functional.
The control problem we address herein is often called nodal profile control. It consists in steering the network states to given profiles at a prescribed node, over a prescribed time interval, by means of sufficiently many controls actuating along the remainder of the nodes.
This notion is not hindered by the presence of cycles in the network. We first prove the semi-global in time existence and uniqueness of $C_{x,t}^1$ solutions -- i.e., for arbitrarily large time intervals, provided
that the initial and boundary data are small enough -- and then use the so-called constructive method developed by Tatsien Li and collaborators to prove local exact controllability of nodal profiles for an A-shaped network.
We conclude with perspectives of future research and interesting open problems.

Based on Onsager's variational principle, the motion of superparamagnetic nanoparticles – which are suspended in an incompressible carrier fluid and subjected to an external magnetic field – is described by systems of partial differential quations. Various modeling assumptions lead to three different systems denoted by "model GW", "model W" and "model B". When proposed in our joint work (2019), "model GW" was – to the best of our knowledge – the first one to include evolution equations for the magnetization field and for the magnetic particle density – all of them are nonlinearly coupled to the magnetostatic and the Navier-Stokes equations. The other two models use algebraic equations to determine the magnetization based on the linearized Langevin formula and are derived by us for the purpose of comparison. In case of "model W", we assume that the suspension yields a single-phase flow with negligible mass of magnetic nanoparticles. The other model is derived under the assumption that fluid particles and magnetic particles set up a two-phase fluid. It shows similarities to the model from Himmelsbach et al. (2017). All three models follow a two-domain approach – considering the magnetic field on a possibly larger domain compared to the fluid domain – of which we expect higher accuracy in determining the total magnetic field. The case when the two domains coincide requires some slight changes which are discussed when needed.
In contrast to other works in the mathematical literature, the boundary conditions of the magnetization equation in "model GW" are motivated by physical arguments only, not by practical aspects with respect to mathematical analysis. They entail H(div,curl)-regularity of the magnetic quantities – magnetization and (total) magnetic field – but possibly not H1-regularity. To establish existence of solutions to this model, the absence of H1-regularity is compensated by an intricate approximation procedure. For this, we give a meaning to the Kelvin force (m*nabla)h in the distributional sense. Existence of distributional global-in-time solutions is guaranteed under appropriate assumptions. The latter include nonlinear diffusion to be used in the evolution equation for the magnetic particles' density and the restriction to the two-dimensional setting. An existence result of global-in-time weak solutions to a regularized model is presented also in the three-dimensional setting.
To each of the three models, we propose an unconditionally energy stable finite element scheme. The discretization of "model GW" has already been published in our joint work (2019). Here, non-conforming finite elements are used to approximate the magnetization – similar as in a paper of Nochetto et al. (2015). For the first time, however, the second order differential operators nabla div and curl curl in the magnetization equation have been discretized by introducing the operators divh and curlh – discrete versions of div and curl. The latter are defined by duality and yield H1-conforming approximations of the divergence and curl of a vector field. By means of Schaefer's fixed point theorem, existence of discrete solutions is guaranteed for all three schemes. The existence results are independent of the discretization parameters.
The thesis concludes with simulations that serve as a proof of concept for the models and their numerical schemes. The three models are compared in the case of linear diffusion and the case of nonlinear diffusion which has been used in the analysis part of this thesis. For simplicity, most simulations are performed under the assumption that the domain of the magnetic field coincides with the fluid domain. Using "model W" – for practical reasons – the effects of multiple different external magnetic fields are examined as well as the impact of using a strictly larger domain for the magnetic field.

This dissertation "Defects and symmetries in Hopf algebra lattice models" is concerned with generalizations of Kitaev models.
In first part of the thesis constructs a Kitaev model with defects and boundaries, based on finite-dimensional semisimple Hopf algebras. It shows that this model is a counterpart to a topological quantum field theory of Turaev-Viro type with defects and boundaries.
Furthermore it gives a description of the generation, transport, fusion and braiding of excitations.
The second part of the thesis constructs a generalized lattice model based on pivotal Hopf monoids in symmetric monoidal categories. It is shown that mapping class group of the surface of genus g>0 with n>0 boundary components acts on the model for this surface. The action is given in terms of generating Dehn twists of a presentation by Gervais. When the symmetric monoidal category is finitely complete and cocomplete, then the action for n=1 induces an action of the mapping class group of the closed surface on the biinvariants of a Yetter-Drinfeld-module structure on this model.

Let \(\mathcal{H}\) be a complex Hilbert space.
A closed real subspace \(\mathtt{V} \subset \mathcal{H}\) is called a standard subspace if \(\mathtt{V} + i\mathtt{V}\) is dense in \(\mathcal{H}\) and \(\mathtt{V} \cap i\mathtt{V} = \{0\}\).
We denote the set of standard subspaces of \(\mathcal{H}\) by \(\mathrm{Stand}(\mathcal{H})\).
Let \((G,\tau_G)\) be a finite-dimensional symmetric Lie group, i.e. \(G\) is a Lie group and \(\tau_G\) is an involutive automorphism of \(G\).
Let \((\mathfrak{g},\tau)\) be the symmetric Lie algebra of \((G,\tau_G)\) and let \((\rho,\mathcal{H})\) be a strongly continuous antiunitary representation of \(G_\tau := G \rtimes \{\mathbf{1},\tau_G\}\), i.e. \(\rho(G)\) consists of unitary operators and \(\rho(\tau_G)\) is an antilinear surjective isometry.
Let \(h \in \mathfrak{g}^\tau := \mathrm{ker}(\tau - \mathrm{id}_\mathfrak{g})\) and
\[\Delta := e^{2i \pi \partial\rho(h)} \quad \text{and} \quad J := \rho(\tau_G).\]
Then \(\mathtt{V}_{(h,\tau_G)} := \mathrm{Fix}(J\Delta^{1/2})\) is a standard subspace.
This construction is called the Brunetti-Guido-Longo construction (or BGL construction).
This thesis investigates, for standard subspaces of the form \(\mathtt{V} := \mathtt{V}_{(h,\tau_G)} \subset \mathcal{H}\), the structure of the endomorphism semigroup
\[S_\mathtt{V} := \{g \in G : \rho(g)\mathtt{V} \subset \mathtt{V}\}\]
using the representation theory of finite dimensional Lie groups and the structure theory of Lie algebras.
The semigroup \(S_\mathtt{V}\) encodes the inclusion order on the set \(\rho(G)\mathtt{V} \subset \mathrm{Stand}(\mathcal{H})\), which plays a decisive role in the construction of nets of standard subspaces and of von Neumann algebras in the context of Algebraic Quantum Field Theory.

This thesis is based on three articles written in the course of the author's doctoral studies. An overview of important results on Anderson localization and former approaches to the open problem of Anderson delocalization is followed by the discussion of an idea for a proof of Anderson delocalization by means of the transfer matrix technique in finite volume approximations. A particular reference is made to a certain random dynamics on the complex Grassmannian that is induced by randomly perturbed hyperbolic matrices. A comparable random dynamics on the real projective space was studied in the first article, namely with a specific focus on the invariant measure of the system. This first article contains a proof that builds a bridge towards the second article, in which pseudo-gaps of the density of states in random hopping models are proven. Further aspects of the idea for a proof of Anderson delocalization motivate the third article, in which an approximate formula for the Lyapunov exponent of certain two parameter random perturbations of elliptic random matrices is proven by analyzing the invariant measure of the associated Möbius dynamics.

Atherosclerosis is a disease of the arteries which can cause a reduction or complete blockage of blood flow and may thus lead to heart attacks or strokes, two of the most common causes of death worldwide. This motivates mathematical modelling and simulation of the disease. The timescale of disease progression, driven by the growth of plaque in the artery wall over years, differs greatly from other processes which are assumed to be relevant, e.g. the shear stresses exerted by the blood flow on the artery wall, which oscillates with the heart beat every second. This prevents a direct numerical simulation of such models, since long timescales would have to be resolved with a very fine step size. The goal of this thesis is to construct approximations which are simpler to solve numerically, through the use of multiscale methods, and to prove quantitative convergence results. Motivated by a model by Yang et. al. (2015), we will investigate two simplified submodels of atherosclerosis for which a rigorous multiscale analysis is possible. The first model studies slow plaque growth coupled to fast oscillating shear stresses caused by the blood flow. Mathematically this is realized through a slow ordinary differential equation coupled to a fluid equation with rapidly oscillating boundary conditions and growth-dependent, non-cylindrical space-time domain. The second model investigates substances quickly advected through the artery but only slowly diffusing into the semi-permeable wall. It consists of a system of coupled advection-diffusion and diffusion-reaction equations. With a small parameter epsilon, which expresses the timescale separation, the behavior of the solutions to these models in the limit epsilon to 0 is investigated. Both models are singularly perturbed, meaning that their solutions converge to functions which solve a differential equation of different type. For the first model it will be shown that the solution converges with order O(epsilon) to the solution of a limit equation which averages the effect of a time-periodic fluid equation. The second model yields a limit consisting of a coupled advection and diffusion-reaction equation. The order of convergence depends on the solution regularity and the behavior of the advection field. For e.g. the stationary problem and Poiseuille flow it will be shown that the spatial L2- and H1-errors are of order O(epsilon^(1/2)), respectively O(epsilon^(1/6)), in the advection domain. Inside the wall the H1-error will be of order O(epsilon^(1/3)). The derivation of this result combines qualitative convergence theory for advection-diffusion equations in the vanishing diffusion limit with a specific trace estimate for the coupling through the permeable wall. Numerical calculations are carried out for both models. For the plaque growth the focus lies on the solution of the time-periodic Navier–Stokes equation which will be required for the limit system, an existing algorithm from the literature is improved here. Furthermore, the error of the time-discrete equation is analyzed, which quantifies and emphasizes how the errors made in the different solution steps must be balanced for efficiency. For the second model a discontinuous Galerkin discretization is proposed and the agreement between theoretical and numerical results shown.

Life Cycle Assessment (LCA) deals with factors that impact the environment, such
as the greenhouse gas potential. It considers the complete life cycle of a product or
service. In the first case, this means the analysis of raw material extraction, through
production and usage to end of life. The ISO norms 14040 and 14044 recommend a
precisely defined procedure: Goal and Scope Definition, Life Cycle Inventory Analysis,
Life Cycle Impact Assessment, and Interpretation.
So far, this instrument has mostly been applied only as a balancing method, in order to
determine the ecological footprint of an available product. In our further development of
this approach, we use the techniques of the mathematical optimization to influence the
design of the life cycle. This allows the improvement of the environmental properties for
both during development and existing products while at the same time including social
and financial considerations.
For Vitesco Technologies, this resulting optimized Integrated Life Cycle Sustainability
Assessment (ILCSA) is an important aspect of sustainability which goes beyond the
scope of LCA in the ordinary sense. We implemented a highly customizable mixed-integer
optimization model which provides the product manager guidance to choose,
for example, the locations for gaining the raw materials, production sites, and material
composition.
For its solution, we developed a software with a user-friendly interface and show the
benefit in a detailed case study at the example of a lithium-ion battery. It is a main part
of an electric vehicle and, therefore, it is of great significance in the current focus topic
of electromobility. Our method can be transferred to all other automotive components,
up to the optimization of the life cycle for the complete vehicle.

In this thesis, problems in material and topology optimization of a linear elastic continuum
motivated by applications in additive manufacturing are studied. More precisely,
specific material characteristics arising in layered additive manufacturing are considered,
but the analysis is done in great generality to also incorporate other applications.
First, simultaneous optimization of topology and parametrized material is discussed.
Anisotropic material properties and a material with a graded, oriented microstructure
are considered as applications from additive manufacturing, as well as more abstract
parametrizations for theoretical comparisons with global lower bounds. To be able to
constrain the local change of the material grading, pointwise bounds on design variable
gradients are imposed ("slope constraints"). Furthermore, a filtering technique based on
a convolution product ("density filtering") is applied and a new estimate of the maximal
gradient of filtered design variables is derived. For the combination of these two known
regularization methods and the suggested general formulation, a novel proof for the existence
of solutions and convergence of the finite element discretization is conducted.
Second, the problem of nonlinear material parametrizations potentially leading to suboptimal
local solutions is studied. For this, an asymptotic expansion of the compliance
is used. This approximation is separable between different finite element patches, which
makes it possible to find the globally optimal material for the model. While the formula
in principle only holds for elliptic inclusions within a matrix material, numerical tests
suggest a good performance of the proposed method. Subsequently, an extension for the
simultaneous optimization of a topology is introduced and tested numerically.
Finally, a method for topology optimization with uncertainties in the material properties
such as in additive manufacturing is developed. The uncertainties are handled using a
worst-case approach, i. e. they are always distributed such that the compliance is maximally
weakened. The resulting optimization problem is a minimax problem with a large
number of variables. A relaxation of the inner maximization problem is suggested, which
allows the solution of the minimax problem through the minimization of an optimal value
function. A Tikhonov type and a barrier regularization scheme are suggested, which
render the resulting minimization problem continuously differentiable. The barrier regularization
scheme is studied in more detail, as it can be closely linked to a highly efficient
interior point approach for a precise evaluation of the optimal value function and its gradient.
Examples from additive manufacturing as well as material degradation are examined.
Lastly, the method is extended to solve the original problem without relaxation by using
a RAMP-type continuation approach from the concave to the original model.

The present thesis is concerned with certain doubly nonlinear evolutionary partial differential equations, whose study has been motivated by a number of applications including fluid dynamics and filtration. Since they generalize both the parabolic p-Laplace equation and the porous medium equation, these equations are also interesting from a mathematical point of view.
The thesis is divided into three parts. In the first part, the existence of variational solutions to the Cauchy-Dirichlet problem associated with singular doubly nonlinear systems and obstacle problems for doubly nonlinear equations is discussed. The novelty of the results is the inclusion of quite general time-dependent boundary values and obstacle functions. For each result, the overall proof strategy is the same. First, a variant of the method of minimizing movements yields a preliminary existence result for more regular boundary data and – in case of the obstacle problems – more regular obstacle functions. In a second step, the general existence result is obtained by means of approximation.
The second part is devoted to stability results for parabolic p-Laplace type systems as p tends to 1, which can also be interpreted as existence results for the limit case p=1. As in the first part, the notion of variational solution is used. After an introduction to functions of bounded variation, related parabolic function spaces and relaxation of linear growth functionals, the Cauchy-Dirichlet problem with time-dependent boundary values is considered. Subsequently, motivated by an application in image processing, a lower order term is included in the system and a Cauchy-Neumann boundary condition is imposed. While the proof of the existence result is close to the one for Cauchy-Dirichlet data, additional results concerning the continuity of variational solutions with respect to the time variable and their uniqueness are achieved.
Finally, the third part contains a result on an intrinsic Harnack inequality for non-negative weak solutions of doubly nonlinear equations of slow diffusion type as supplementary material to the thesis. The proof relies on the method of expansion of positivity.

Problems with free surfaces are ubiquitous in nature. The propagation of those surfaces is affected by various parameters, some of which are uncertain. Such effects can be interpreted as random noise and may lead to probabilistic terms in modeling [13, 15, 27, 34]. The scope of this thesis is to analyze the impact of stochastic terms in the Ito-sense on the propagation of solutions. In particular, we concentrate on porous-medium equations and parabolic p-Laplace equations.
Concerning stochastic porous-medium equations, we derive upper bounds on average waiting times and criteria for instantaneous propagation. Moreover, we present numerical experiments on average propagation.
Regarding stochastic degenerate-parabolic p-Laplace equations, we prove finite speed of propagation as well as sufficient conditions for the pathwise occurrence of waiting times. Afterwards, we derive a finite-element scheme for stochastic degenerate-parabolic p-Laplace equations that is nonnegativity preserving and also convergent. Finally, the quantitative propagation of solutions subjected to noise is investigated by means of numerical simulations.

The discontinuous Galerkin method for free surface and subsurface flows in geophysical applications
(2020)

Free surface flows and subsurface flows appear in a broad range of geophysical applications and in many environmental settings situations arise which even require the coupling of free surface and subsurface flows. Many of these application scenarios are characterized by large domain sizes and long simulation times. Hence, they need considerable amounts of computational work to achieve accurate solutions and the use of efficient algorithms and high performance computing resources to obtain results within a reasonable time frame is mandatory.
Discontinuous Galerkin methods are a class of numerical methods for solving differential equations that share characteristics with methods from the finite volume and finite element frameworks. They feature high approximation orders, offer a large degree of flexibility, and are well-suited for parallel computing.
This thesis consists of eight articles and an extended summary that describe the application of discontinuous Galerkin methods to mathematical models including free surface and subsurface flow scenarios with a strong focus on computational aspects. It covers discretization and implementation aspects, the parallelization of the method, and discrete stability analysis of the coupled model.

Transparent and conductive thin films find broad application in optoelectronic devices such as touchscreens and solar panels, and are intended to satisfy two opposite properties.
On the one hand, these films should appear transparent in the visible light, which means that a large amount of light can pass through such a film.
On the other hand, electric energy induced by an applied voltage should be transported with a low electric resistance.
Besides the transparent and conductive thin films, we want to consider particle monolayers as another class of photonic nanostructures.
The particle monolayers are utilized, for instance, to control the diffuse scattering behavior of photodetectors used in solar cells.
This optical property is quantified by the haze factor.
Experiments show that the design, which includes both the material composition and the overall shape of the photonic nanostructures, has a noteworthy influence on the performance with respect to the intended purpose.
The main objective of this thesis is to optimize the design of such photonic nanostructures with respect to transmission, conductivity and haze factor by changing the material, the shape and
the geometry using gradient-based algorithms.
Before the individual optimization problems are specified, analytical and numerical solution methods for the involved partial differential equations to determine the optical and electrical properties are discussed.
The electromagnetic scattering of a single spherical particle and assemblies of spherical particles is formulated in terms of fundamental solutions of Maxwell's equations, i. e. the vector spherical wave functions.
In this context, the order of convergence of dedicated errors is numerically studied with respect to various parameters.
In particular, the numerical evaluation of the haze factor for particle monolayers consisting of non-spherical particle is challenging and
a suitable numerical solution scheme has been developed.
For this purpose, the Finite Element Method and a spectral method based on vector spherical wave functions are combined to a two-stage hybrid simulation scheme in which computationally expensive tasks can be computed in a so-called offline stage.
Hence, sophisticated algorithms for the optimization of material, shape and geometry accomplish the gradient-based design optimization of photonic nanostructures.

We consider the gas dynamics on networks modeled by the isothermal Euler equations.
The demands of the customers at the exits of the network
represent a mass flow boundary condition. The true demands are not
exactly known and are modeled by probability distributions.
First, the stationary solutions of the quasilinear isothermal Euler
equations are derived on a single pipe. For this purpose, a real gas model is used.
Important monotonicity properties of the solution, which are of crucial
importance for the extension to networks, are shown. The existence theory
is developed in a more general framework for flow problems on networks.
This makes it possible to treat different coupling conditions at the nodes
and to use various models that are not necessarily restricted to gas, or
active elements along the edges.
For the optimization, the uncertain boundary conditions are represented
by means of chance constraints. These guarantee that the system is kept
within technical pressure limits with a certain probability. The calculation
of the probability is realized by a combination of quasi-Monte Carlo methods
and the spherical radial decomposition of Gaussian random vectors. For
the gradient-based optimization, a result for a gradient representation is
extended to include the case of convex sets of feasible realizations. It is
shown that, for the case of tree networks, the required assumptions are
fulfilled. For the numerical implementation, a multilevel method is proposed
that uses the solution for low sample numbers as a warm start for the more
complex model. This leads to a significant reduction in computational
time.
In the transient case, the linear wave equation is considered on a single
edge. The initial and boundary data at the end of an interval are uncertain
and given by stochastic processes. The other end of the interval is controlled
by a Neumann feedback law. The stochastic processes are approximated
using the Karhunen-Loève theorem. This enables the representation of
the initial-boundary data by a finite number of random variables. As a
stability measure, the probability to stay below a given threshold in the
L∞-norm, is considered. The system can be stabilized by adjusting the
feedback parameter.