Seit 1993 veranstaltet der Fachbereich Mathematik der TU Kaiserslautern jährlich die mathematischen Modellierungswochen. Die Veranstaltung erwuchs parallel zu der steigenden Relevanz angewandter mathematischer Forschungsgebiete, wie der Technomathematik und der Wirtschaftsmathematik. Sie soll dazu dienen, Schülerinnen und Schülern die Bedeutung mathematischer Arbeitsweisen in der heutigen Berufswelt, insbesondere in Industrie und Wirtschaft, begreifbar zu machen. Darüber hinaus bietet die Modellierungswoche den teilnehmenden Lehrkräften einen Einblick in die Projektarbeit mit offenen Fragestellungen im Rahmen der mathematischen Modellierung. In diesem Report beschreiben wir die Projekte, die während der Modellierungswoche im Dezember 2021 durchgeführt wurden. Der Themenschwerpunkt der Veranstaltung lautete "Wetter und Katastrophenschutz".

An increasing number of nowadays tasks, such as speech recognition, image generation, translation, classification or prediction are performed with the help of machine learning. Especially artificial neural networks (ANNs) provide convincing results for these tasks. The reasons for this success story are the drastic increase of available data sources in our more and more digitalized world as well as the development of remarkable ANN architectures. This development has led to an increasing number of model parameters together with more and more complex models. Unfortunately, this yields a loss in the interpretability of deployed models. However, there exists a natural desire to explain the deployed models, not just by empirical observations but also by analytical calculations. In this thesis, we focus on variational autoencoders (VAEs) and foster the understanding of these models. As the name suggests, VAEs are based on standard autoencoders (AEs) and therefore used to perform dimensionality reduction of data. This is achieved by a bottleneck structure within the hidden layers of the ANN. From a data input the encoder, that is the part up to the bottleneck, produces a low dimensional representation. The decoder, the part from the bottleneck to the output, uses this representation to reconstruct the input. The model is learned by minimizing the error from the reconstruction. In our point of view, the most remarkable property and, hence, also a central topic in this thesis is the auto-pruning property of VAEs. Simply speaking, the auto-pruning is preventing the VAE with thousands of parameters from overfitting. However, such a desirable property comes with the risk for the model of learning nothing at all. In this thesis, we look at VAEs and the auto-pruning from two different angles and our main contributions to research are the following: (i) We find an analytic explanation of the auto-pruning. We do so, by leveraging the framework of generalized linear models (GLMs). As a result, we are able to explain training results of VAEs before conducting the actual training. (ii) We construct a time dependent VAE and show the effects of the auto-pruning in this model. As a result, we are able to model financial data sequences and estimate the value-at-risk (VaR) of associated portfolios. Our results show that we surpass the standard benchmarks for VaR estimation.

Risk management is an indispensable component of the financial system. In this context, capital requirements are built by financial institutions to avoid future bankruptcy. Their calculation is based on a specific kind of maps, so-called risk measures. There exist several forms and definitions of them. Multi-asset risk measures are the starting point of this dissertation. They determine the capital requirements as the minimal amount of money invested into multiple eligible assets to secure future payoffs. The dissertation consists of three main contributions: First, multi-asset risk measures are used to calculate pricing bounds for European type options. Second, multi-asset risk measures are combined with recently proposed intrinsic risk measures to obtain a new kind of a risk measure which we call a multi-asset intrinsic (MAI) risk measure. Third, the preferences of an agent are included in the calculation of the capital requirements. This leads to another new risk measure which we call a scalarized utility-based multi-asset (SUBMA) risk measure. In the introductory chapter, we recall the definition and properties of multi-asset risk measures. Then, each of the aforementioned contributions covers a separate chapter. In the following, the content of these three chapters is explained in more detail: Risk measures can be used to calculate pricing bounds for financial derivatives. In Chapter 2, we deal with the pricing of European options in an incomplete financial market model. We use the common risk measures Value-at-Risk and Expected Shortfall to define good deals on a financial market with log-normally distributed rates of return. We show that the pricing bounds obtained from Value-at-Risk may have a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller's bound for a call option is smaller than the buyer's bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between the obtained pricing bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of multi-asset risk measures. In Chapter 3, we construct the MAI risk measure. Therefore, recall that a multi-asset risk measure describes the minimal external capital that has to be raised into multiple eligible assets to make a future financial position acceptable, i.e., that it passes a capital adequacy test. Recently, the alternative methodology of intrinsic risk measures was introduced in the literature. These ask for the minimal proportion of the financial position that has to be reallocated to pass the capital adequacy test, i.e., only internal capital is used. We combine these two concepts and call this new type of risk measure an MAI risk measure. It allows to secure the financial position by external capital as well as reallocating parts of the portfolio as an internal rebooking. We investigate several properties to demonstrate similarities and differences to the two aforementioned classical types of risk measures. We find out that diversification reduces the capital requirement only in special situations depending on the financial positions. With the help of Sion's minimax theorem we also prove a dual representation for MAI risk measures. Finally, we determine capital requirements in a model motivated by the Solvency II methodology. In the final Chapter 4, we construct the SUBMA risk measure. In doing so, we consider the situation in which a financial institution has to satisfy a capital adequacy test, e.g., by the Basel Accords for banks or by Solvency II for insurers. If the financial situation of this institution is tight, then it can happen that no reallocation of the initial endowment would pass the capital adequacy test. The classical portfolio optimization approach breaks down and a capital increase is needed. We introduce the SUBMA risk measure which optimizes the hedging costs and the expected utility of the institution simultaneously subject to the capital adequacy test. We find out that the SUBMA risk measure is coherent if the utility function has constant relative risk aversion and the capital adequacy test leads to a coherent acceptance set. In a one-period financial market model we present a sufficient condition for the SUBMA risk measure to be finite-valued and continuous. Finally, we calculate the SUBMA risk measure in a continuous-time financial market model for two benchmark capital adequacy tests.

Die Felix-Klein-Modellierungswoche ist eine Projektwoche für Schülerinnen und Schüler der gymnasialen Oberstufe. Wir beschreiben hier die Projekte, die während der Modellierungswoche im Februar 2022 durchgeführt wurden. Der Themenschwerpunkt der Veranstaltung lautete "Winter".

The great interest in robust covering problems is manifold, especially due to the plenitude of real world applications and the additional incorporation of uncertainties which are inherent in many practical issues. In this thesis, for a fixed positive integer \(q\), we introduce and elaborate on a new robust covering problem, called Robust Min-\(q\)-Multiset-Multicover, and related problems. The common idea of these problems is, given a collection of subsets of a ground set, to decide on the frequency of choosing each subset so as to satisfy the uncertain demand of each overall occurring element. Yet, in contrast to general covering problems, the subsets may only cover at most \(q\) of their elements. Varying the properties of the occurring elements leads to a selection of four interesting robust covering problems which are investigated. We extensively analyze the complexity of the arising problems, also for various restrictions to particular classes of uncertainty sets. For a given problem, we either provide a polynomial time algorithm or show that, unless \(\text{P}=\text{NP}\), such an algorithm cannot exist. Furthermore, in the majority of cases, we even give evidence that a polynomial time approximation scheme is most likely not possible for the hard problem variants. Moreover, we aim for approximations and approximation algorithms for these hard variants, where we focus on Robust Min-\(q\)-Multiset-Multicover. For a wide class of uncertainty sets, we present the first known polynomial time approximation algorithm for Robust Min-\(q\)-Multiset-Multicover having a provable worst-case performance guarantee.

Im Projekt MAFoaM - Modular Algorithms for closed Foam Mechanics - des Fraunhofer ITWM in Zusammenarbeit mit dem Fraunhofer IMWS wurde eine Methode zur Analyse und Simulation geschlossenzelliger PMI-Hartschäume entwickelt. Die Zellstruktur der Hartschäume wurde auf Basis von CT-Aufnahmen modelliert, um ihr Verformungs- und Versagensverhalten zu simulieren, d.h. wie sich die Schäume unter Belastungen bis hin zum totalen Defekt verhalten. In der Diplomarbeit wird die bildanalytische Zellrekonstruktion für PMI-Hartschäume automatisiert. Die Zellrekonstruktion dient der Bestimmung von Mikrostrukturgrößen, also geometrischer Eigenschaften der Schaumzellen, wie z.B. Mittelwert und Varianz des Zellvolumens oder der Zelloberfläche.

The high complexity of civil engineering structures makes it difficult to satisfactorily evaluate their reliability. However, a good risk assessment of such structures is incredibly important to avert dangers and possible disasters for public life. For this purpose, we need algorithms that reliably deliver estimates for their failure probabilities with high efficiency and whose results enable a better understanding of their reliability. This is a major challenge, especially when dynamics, for example due to uncertainties or time-dependent states, must be included in the model. The contributions are centered around Subset Simulation, a very popular adaptive Monte Carlo method for reliability analysis in the engineering sciences. It particularly well estimates small failure probabilities in high dimensions and is therefore tailored to the demands of many complex problems. We modify Subset Simulation and couple it with interpolation methods in order to keep its remarkable properties and receive all conditional failure probabilities with respect to one variable of the structural reliability model. This covers many sorts of model dynamics with several model constellations, such as time-dependent modeling, sensitivity and uncertainty, in an efficient way, requiring similar computational demands as a static reliability analysis for one model constellation by Subset Simulation. The algorithm offers many new opportunities for reliability evaluation and can even be used to verify results of Subset Simulation by artificially manipulating the geometry of the underlying limit state in numerous ways, allowing to provide correct results where Subset Simulation systematically fails. To improve understanding and further account for model uncertainties, we present a new visualization technique that matches the extensive information on reliability we get as a result from the novel algorithm. In addition to these extensions, we are also dedicated to the fundamental analysis of Subset Simulation, partially bridging the gap between theory and results by simulation where inconsistencies exist. Based on these findings, we also extend practical recommendations on selection of the intermediate probability with respect to the implementation of the algorithm and derive a formula for correction of the bias. For a better understanding, we also provide another stochastic interpretation of the algorithm and offer alternative implementations which stick to the theoretical assumptions, typically made in analysis.

Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around one or several sites of vaso-occlusion are typical for GBM and hint on poor prognosis of patient survival. This thesis focuses on studying the establishment and maintenance of these histological patterns specific to GBM with the aim of modeling the microlocal tumor environment under the influence of acidity, tissue anisotropy and hypoxia-induced angiogenesis. This aim is reached with two classes of models: multiscale and multiphase. Each of them features a reaction-diffusion equation (RDE) for the acidity acting as a chemorepellent and inhibitor of growth, coupled in a nonlinear way to a reaction-diffusion-taxis equation (RDTE) for glioma dynamics. The numerical simulations of the resulting systems are able to reproduce pseudopalisade-like patterns. The effect of tumor vascularization on these patterns is studied through a flux-limited model belonging to the multiscale class. Thereby, PDEs of reaction-diffusion-taxis type are deduced for glioma and endothelial cell (EC) densities with flux-limited pH-taxis for the tumor and chemotaxis towards vascular endothelial growth factor (VEGF) for ECs. These, in turn, are coupled to RDEs for acidity and VEGF produced by tumor. The numerical simulations of the obtained system show pattern disruption and transient behavior due to hypoxia-induced angiogenesis. Moreover, comparing two upscaling techniques through numerical simulations, we observe that the macroscopic PDEs obtained via parabolic scaling (directed tissue) are able to reproduce glioma patterns, while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related.

Linear algebra, together with polynomial arithmetic, is the foundation of computer algebra. The algorithms have improved over the last 20 years, and the current state of the art algorithms for matrix inverse, solution of a linear system and determinants have a theoretical sub-cubic complexity. This thesis presents fast and practical algorithms for some classical problems in linear algebra over number fields and polynomial rings. Here, a number field is a finite extension of the field of rational numbers, and the polynomial rings we considered in this thesis are over finite fields. One of the key problems of symbolic computation is intermediate coefficient swell: the bit length of intermediate results can grow during the computation compared to those in the input and output. The standard strategy to overcome this is not to compute the number directly but to compute it modulo some other numbers, using either the Chinese remainder theorem (CRT) or a variation of Newton-Hensel lifting. Often, the final step of these algorithms is combined with reconstruction methods such as rational reconstruction to convert the integral result into the rational solution. Here, we present reconstruction methods over number fields with a fast and simple vector-reconstruction algorithm. The state of the art method for computing the determinant over integers is due to Storjohann. When generalizing his method over number field, we encountered the problem that modules generated by the rows of a matrix over number fields are in general not free, thus Strojohann's method cannot be used directly. Therefore, we have used the theory of pseudo-matrices to overcome this problem. As a sub-problem of this application, we generalized a unimodular certification method for pseudo-matrices: similar to the integer case, we check whether the determinant of the given pseudo matrix is a unit by testing the integrality of the corresponding dual module using higher-order lifting. One of the main algorithms in linear algebra is the Dixon solver for linear system solving due to Dixon. Traditionally this algorithm is used only for square systems having a unique solution. Here we generalized Dixon algorithm for non-square linear system solving. As the solution is not unique, we have used a basis of the kernel to normalize the solution. The implementation is accompanied by a fast kernel computation algorithm that also extends to compute the reduced-row-echelon form of a matrix over integers and number fields. The fast implementations for computing the characteristic polynomial and minimal polynomial over number fields use the CRT-based modular approach. Finally, we extended Storjohann's determinant computation algorithm over polynomial ring over finite fields, with its sub-algorithms for reconstructions and unimodular certification. In this case, we face the problem of intermediate degree swell. To avoid this phenomenon, we used higher-order lifting techniques in the unimodular certification algorithm. We have successfully used the half-gcd approach to optimize the rational polynomial reconstruction.

Die Felix-Klein-Modellierungswoche ist eine Projektwoche für Schülerinnen und Schüler der gymnasialen Oberstufe. Wir beschreiben hier die Projekte, die während der Modellierungswoche im Februar 2021 durchgeführt wurden.