Bilevel problems are optimization problems for which parts of the variables are constrained to be an optimal solution to another nested optimization problem. This structure renders bilevel problems particularly well-suited for modeling hierarchical decision-making processes. They are widely applicable in areas such as energy markets, transportation systems, security planning, and pricing. However, the hierarchical nature of these problems also makes them inherently challenging to solve, both in theory and in practice. In this thesis, we study different nonlinear problem settings for the nested optimization problem. First, we focus on nonlinear but convex bilevel problems with purely integer variables. We propose a solution algorithm that uses a branch-and-cut framework with tailored cutting planes. We prove correctness and finite termination of the method under suitable assumptions and put it into context of existing literature. Moreover, we provide an extensive numerical study to showcase the applicability of our method and we compare it to the state-of-the-art approach for a less general setting on suitable instances from the literature. Furthermore, we discuss challenges that arise when we try to generalize our approach to the mixed-integer setting. Next, we study mixed-integer bilevel problems for which the nested problem has a nonconvex and quadratic objective function, linear constraints, and continuous variables. We state and prove a complexity-theoretical hardness result for this problem class and develop a lower and upper bounding scheme to solve these problems. We prove correctness and finite termination of the proposed method under suitable assumptions and test its applicability in a numerical study. Finally, we consider bilevel problems with continuous variables, where the nested problem has a convex-quadratic objective function and linear constraints. We reformulate them as single-level optimization problems using necessary and sufficient optimality conditions for the nested problem. Then, we explore the family of so-called P-split reformulations for this single-level problem and test their applicability in a preliminary numerical study.

Extracellular enzymes in microbial communities play a central role in nutrient cycling and the degradation of (pollutant) substances in various natural and anthropogenic systems. Bound in aquatic biofilms, sludge aggregates, or even unbound at their interfaces, they are of great importance for both the environment and human health. In particular, in wastewater treatment plants and inland waters, hydrolytic activities influence the wide-reaching efficiency of nutrient removal and self-purification, thus contributing significantly to overall water quality. The main goal of this dissertation project was to investigate the factors that influence enzymatic activity and the health of microbial communities in activated sludge and river systems, particularly in relation to anthropogenic influences and natural environmental conditions. The aim was to contribute to a better understanding of the sensitivity of our freshwater ecosystems and to support the long-term preservation of water quality and ecological stability. The development and optimization of appropriate methods, as well as their testing and applicability, were the focal points. For this purpose, a fluorometric microplate assay was developed and adapted to determine both extracellular enzyme activities (EEAs) in activated sludge samples and in intact biofilms. Its suitability for field studies was subsequently tested. Inhibition and activity of selected hydrolases under different conditions were investigated to better understand the mechanisms and potential environmental risks posed by anthropogenic influences and seasonal fluctuations of hydrochemical and climatic parameters. The first phase of the doctoral thesis involved studies on the inhibition of alkaline phosphatase in activated sludge by oxyanions. Using the fluorometric microplate assay, the inhibitory effect was sensitively detected over a pH range of 7.0 to 8.5. IC50- and IC20-concentrations were calculated from modeled dose-response functions. It was found that vanadate and tungstate caused strong inhibitory effects, while molybdate moderately inhibited the enzyme. An increasing pH led to a reduction in the inhibitory effect of tungstate and molybdate. The inhibition effects of vanadate were not significantly affected by the pH. In municipal wastewater, the concentrations of such metal ions are usually low, but industrial wastewater may have pollutant loads that can significantly impact the removal of phosphorus-containing compounds, and thus the efficiency of treatment plants. In the second phase, an attempt was made to further adapt the developed methodology to investigate EEA and kinetics in intact freshwater biofilms. Four different types of bead materials (lava, glass, sintered quartz, and ceramics) fitting into a 96-well microplate were tested as carriers for biofilms on both the laboratory and field scale. The analysis included a total of seven hydrolases as representatives of key nutrient cycles such as phosphorus, carbon, and nitrogen: phosphatases, glucosidases, peptidases (two different types), and sulfatase. Experiments with increasing substrate concentrations led to classical kinetic profiles according to the Michaelis-Menten mechanism. This allowed for the prediction of the biofilm enzymes’ response to different substrate concentrations. Parameters such as Vmax and Km could be derived from the modeled curves. Ceramic beads are particularly suitable for long-term studies due to their high stability, while sintered quartz beads should be preferred for the use in stagnant media (material loss under turbulent conditions). Lava and glass beads, on the other and, proved suboptimal for uniform biofilm development due to their surface properties. The potential use of this fast and sensitive test for ecotoxicological or even human-toxicological studies was demonstrated by the effects of caffeine on the activity of PDE. The result of this part of the research represents a powerful tool for assessing environmental pollution and monitoring water quality. The high application potential was clearly highlighted in the final phase of the project. The goal here was to deepen the understanding of interactions between seasonal factors, anthropogenic influences, and biofilm processes in rivers by investigating EEA and biofilm parameters such as biomass and relating them to hydrochemical and climatic factors. Ceramic beads were exposed both upstream and downstream of a wastewater treatment plant discharge and sampled over a period of seven months. EEAs and biomass varied depending on the season and location, with higher microbial activity observed upstream in winter. Winter conditions led to the dilution of most nutrients as well as in an increse of dissolved oxygen. Nutrient concentrations analyzed downstream were significantly higher in the summer. Accumulation of nutrient or pollutants during the summer months cannot be excluded, which may have led to a general reduction in enzyme activities. Potential causes could be inhibitory effects on the enzymes, or a reduced enzyme activity due to a sufficiently high nutrient supply. In general, the sampling site upstream showed a more pronounced seasonal dynamics, with a significant proportion of the variance in biological parameters (activity and biomass) attributable to seasonal factors. A secondary component, likely reflecting the impact of the treatment plant discharge, explained another portion of the data variance. Regardless of the season, high correlations between biological parameters were observed upstream, while downstream the data were more decorrelated. This could be because the biofilms, under chronic stress, respond less dynamically to seasonal fluctuations. This dissertation illustrates that in addition to anthropogenic stress factors, seasonal fluctuations of hydrochemical and climatic parameters should also be considered in "stress downstream the pipe" studies. The selected methods are recommended for explaining and considering the data variance, as they highlight the complex interplay between microbial enzymatic activity, environmental factors, and pollutants in the activated sludge of wastewater treatment plants and also in aquatic systems. The novel bead assay could pave the way for the future standardization of effect-oriented studies on intact aquatic biofilms.

The application of machine learning and deep learning methods to hydrological modelling has advanced significantly in recent years, offering alternatives to traditional conceptual and physically based approaches. Within the numerous algorithms, long short-term memory (LSTM) networks have proven themselves particularly useful for the task of streamflow modelling. This thesis provides a collection of publications that investigate the capabilities, limitations and interpretability of LSTM for the purpose of streamflow modelling and climate change impact assessment within the lowland Ems catchment in Northwest Germany. Within a comparative performance evaluation, LSTM and its predecessor, the recurrent neural network, demonstrate superior accuracy compared to the conceptual HBV model across various statistical performance metrics. However, a decline in performance was observed during low-flow conditions in certain sub-catchments. The evaluation of the flow duration curve revealed that the ML models more effectively capture the water balance, while HBV better represents streamflow dynamics. To enhance the interpretability of LSTM, six explainable artificial intelligence techniques were applied. These methods consistently identified seasonal patterns in the temporal relevance of hydroclimatic input data. In combination with an observed correlation between the internal LSTM states and catchment-scale soil moisture dynamics, the findings suggest that LSTM models are capable of implicitly learning the relevant hydrological processes. Following, the capabilities of LSTM to model climate change impact scenarios, particularly when they extend beyond historically observed climate conditions, are addressed. An ensemble of climate change projections is provided as hydroclimatic input to evaluate the performance of LSTMs and conceptual models. While all models reveal heterogeneous alterations in streamflow under future climate conditions, significant differences emerge based on the model type. Results provide evidence that LSTMs, in combination with the temperature-based Haude formula for estimating potential evaporation, work inadequately under altered climatic regimes, raising concerns about their applicability in long-term projections. The study also indicates the potential need to incorporate physical constraints into LSTM architectures to ensure model robustness and hydrological plausibility beyond the historical training range. Collectively, this thesis contributes important insights into the applicability and interpretability of LSTM models in streamflow modelling. Despite the presence of a physically realistic representation of soil moisture dynamics of the Ems catchment, no robust change signals for streamflow under climate change can be derived. Those results underscore the potential of LSTM model approaches for accurate streamflow simulation, however, they require us to always critically question LSTM results, particularly when they are applied outside the training range.

Spatial microsimulation is an important tool for integrating geographical information into the evaluation of public policies and the analysis of social phenomena in urban regions. These models simulate the behavior and interaction between units of the region, such as individuals, households or firms, under specific conditions that may or not involve projections over time. This requires a representative base data set for their respective units. In this thesis, we focus on the geo-referencing step of the population in the construction of this data set, where we define the location of the individuals so that the allocation obtained is representative in relation to the population of the region. To do this, we consider the assignment of households to dwellings with specific coordinates by solving a maximum weight matching problem where side constraints are included so that the allocation obtained satisfies statistical structures intrinsic to the considered region. The model of this problem represents each feasible assignment of household to dwelling as a binary variable, which results in billions of variables for medium-sized municipalities such as the city of Trier, Germany. Therefore, standard solvers for mixed-integer linear optimization are not able to solve it due to their high time and memory consumption. Hence, we develop two approaches capable of producing high-quality allocations using a reasonable amount of computational resources, one based on specific decomposition algorithms, and the other characterized by the application of an approximation algorithm in the framework of Lagrangian relaxation of the side constraints. We theoretically explore the allocations obtained by both approaches and perform an extensive computational study using synthetic data sets and real-world data sets associated with the city of Trier. The results show that the developed methods are able to obtain near-optimal solutions using significantly less memory and time than the solver Gurobi, which enables them to tackle significantly larger instances, with approximately 100 000 households and dwellings. Furthermore, the allocations obtained for the real-world data sets correspond to a realistic population distribution, which strengthens the practical applicability of our methods.

The dissertation is devoted to the construction and justification of three-point difference schemes of high order of accuracy for solving the Sturm-Liouville problem. A new algorithmic realization of the exact three-point difference scheme on a non-uniform grid has been developed. We show that to compute the coefficients of the exact scheme in an arbitrary grid node, it is necessary to solve two auxiliary Cauchy problems for the system of three linear ordinary differential equations of the first order. The coefficient stability of the exact three-point difference scheme is proved. If the Cauchy problems are solved numerically using any one-step method, we obtain the truncated three-point difference scheme. The accuracy estimate of three-point difference schemes was obtained and the algorithm for finding their solution was developed. We also developed a new algorithmic realization of the exact three-point difference scheme for the Sturm-Liouville problem with singularities at the ends of the interval. As in the case of the classical Sturm-Liouville problem, to find the coefficients of the exact three-point difference scheme, it is necessary to solve two auxiliary Cauchy problems for each grid node. The coefficient stability of the exact three-point difference scheme is proved. Since the Cauchy problems for the first and last grid nodes are singular, the Taylor series method has been developed to solve them. The accuracy estimate of truncated three-point difference schemes was obtained. To solve the difference scheme, the Newton's iterative method is used. Numerical experiments are presented which confirm the efficiency of the proposed approach.