Kaiserslautern - Fachbereich Mathematik
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Faculty / Organisational entity
For the last decade, optimization of beam orientations in intensity-modulated radiation therapy (IMRT) has been shown to be successful in improving the treatment plan. Unfortunately, the quality of a set of beam orientations depends heavily on its corresponding beam intensity profiles. Usually, a stochastic selector is used for optimizing beam orientation, and then a single objective inverse treatment planning algorithm is used for the optimization of beam intensity profiles. The overall time needed to solve the inverse planning for every random selection of beam orientations becomes excessive. Recently, considerable improvement has been made in optimizing beam intensity profiles by using multiple objective inverse treatment planning. Such an approach results in a variety of beam intensity profiles for every selection of beam orientations, making the dependence between beam orientations and its intensity profiles less important. This thesis takes advantage of this property to accelerate the optimization process through an approximation of the intensity profiles that are used for multiple selections of beam orientations, saving a considerable amount of calculation time. A dynamic algorithm (DA) and evolutionary algorithm (EA), for beam orientations in IMRT planning will be presented. The DA mimics, automatically, the methods of beam's eye view and observer's view which are recognized in conventional conformal radiation therapy. The EA is based on a dose-volume histogram evaluation function introduced as an attempt to minimize the deviation between the mathematical and clinical optima. To illustrate the efficiency of the algorithms they have been applied to different clinical examples. In comparison to the standard equally spaced beams plans, improvements are reported for both algorithms in all the clinical examples even when, for some cases, fewer beams are used. A smaller number of beams is always desirable without compromising the quality of the treatment plan. It results in a shorter treatment delivery time, which reduces potential errors in terms of patient movements and decreases discomfort.
This thesis deals with the solution of special problems arising in financial engineering or financial mathematics. The main focus lies on commodity indices. Chapter 1 addresses the important issue for the financial engineering practice of developing well-suited models for certain assets (here: commodity indices). Descriptive analysis of the Dow Jones-UBS commodity index compared to the Standard & Poor 500 stock index provides us with first insights of some features of the corresponding distributions. Statistical tests of normality and mean reversion then helps us in setting up a model for commodity indices. Additionally, chapter 1 encompasses a thorough introduction to commodity investment, history of commodities trading and the most important derivatives, namely futures and European options on futures. Chapter 2 proposes a model for commodity indices and derives fair prices for the most important derivatives in the commodity markets. It is a Heston model supplemented with a stochastic convenience yield. The Heston model belongs to the model class of stochastic volatility models and is currently widely used in stock markets. For the application in the commodity markets the stochastic convenience yield is included in the drift of the instantaneous spot return process. Motivated by the results of chapter 1 it seems reasonable to model the convenience yield by a mean reverting Ornstein-Uhlenbeck process. Since trading desks only apply and consider models with closed form solutions for options I derive such formulas for commodity futures by solving the corresponding partial differential equation. Additionally, semi-closed form formulas for European options on futures are determined. The Cauchy problem with respect to these options is more challenging than the first one. A solution can be provided. Unlike equities, which typically entitle the holder to a continuing stake in a corporation, commodity futures contracts normally specify a certain date for the delivery of the underlying physical commodity. In order to avoid the delivery process and maintain a futures position, nearby contracts must be sold and contracts that have not yet reached the delivery period must be purchased (so called rolling). Optimal trading days for selling and buying futures are determined by applying statistical tests for stochastic dominance. Besides the optimization of the rolling procedure for commodity futures we dedicate ourselves in chapter 3 with the optimization of the weightings of the commodity futures that make up the index. To this end, I apply the Markowitz approach or mean-variance optimization. The mean-variance optimization penalizes up-side and down-side risk equally, whereas most investors do not mind up-side risk. To overcome this, I consider in the next step other risk measures, namely Value-at-Risk and Conditional Value-at-Risk. The Conditional Value-at-Risk is generalized to discontinuous cumulative distribution functions of the loss. For continuous loss distributions, the Conditional Value-at-Risk at a given confidence level is defined as the expected loss exceeding the Value-at-Risk. Loss distributions associated with finite sampling or scenario modeling are, however, discontinuous. Various risk measures involving discontinuous loss distributions shall be introduced and compared. I then apply the theoretical results to the field of portfolio optimization with commodity indices. Furthermore, I uncover graphically the behavior of these risk measures. For this purpose, I consider the risk measures as a function of the confidence level. Based on a special discrete loss distribution, the graphs demonstrate the different properties of these risk measures. The goal of the first section of chapter 4 is to apply the mathematical concept of excursions for the creation of optimal highly automated or algorithmic trading strategies. The idea is to consider the gain of the strategy and the excursion time it takes to realize the gain. In this section I calculate formulas for the Ornstein-Uhlenbeck process. I show that the corresponding formulas can be calculated quite fast since the only function appearing in the formulas is the so called imaginary error function. This function is already implemented in many programs, such as in Maple. My main contribution of this topic is the optimization of the trading strategy for Ornstein-Uhlenbeck processes via the Banach fixed-point theorem. The second section of chapter 4 deals with statistical arbitrage strategies, a long horizon trading opportunity that generates a riskless profit. The results of this section provide an investor with a tool to investigate empirically if some strategies (for example momentum strategies) constitute statistical arbitrage opportunities or not.
The application behind the subject of this thesis are multiscale simulations on highly heterogeneous particle-reinforced composites with large jumps in their material coefficients. Such simulations are used, e.g., for the prediction of elastic properties. As the underlying microstructures have very complex geometries, a discretization by means of finite elements typically involves very fine resolved meshes. The latter results in discretized linear systems of more than \(10^8\) unknowns which need to be solved efficiently. However, the variation of the material coefficients even on very small scales reveals the failure of most available methods when solving the arising linear systems. While for scalar elliptic problems of multiscale character, robust domain decomposition methods are developed, their extension and application to 3D elasticity problems needs to be further established.
The focus of the thesis lies in the development and analysis of robust overlapping domain decomposition methods for multiscale problems in linear elasticity. The method combines corrections on local subdomains with a global correction on a coarser grid. As the robustness of the overall method is mainly determined by how well small scale features of the solution can be captured on the coarser grid levels, robust multiscale coarsening strategies need to be developed which properly transfer information between fine and coarse grids.
We carry out a detailed and novel analysis of two-level overlapping domain decomposition methods for the elasticity problems. The study also provides a concept for the construction of multiscale coarsening strategies to robustly solve the discretized linear systems, i.e. with iteration numbers independent of variations in the Young's modulus and the Poisson ratio of the underlying composite. The theory also captures anisotropic elasticity problems and allows applications to multi-phase elastic materials with non-isotropic constituents in two and three spatial dimensions.
Moreover, we develop and construct new multiscale coarsening strategies and show why they should be preferred over standard ones on several model problems. In a parallel implementation (MPI) of the developed methods, we present applications to real composites and robustly solve discretized systems of more than \(200\) million unknowns.
Diese Arbeit beschäftigt sich mit Methoden zur Klassifikation von Ovoiden in quadratischen Räumen. Die Anwendung der dazu entwickelten Algorithmen erfolgt hauptsächlich in achtdimensionalen Räumen speziell über den Körpern GF(7), GF(8) und GF(9). Zu verschiedenen, zumeist kleinen, zyklischen Gruppen werden hier die unter diesen Gruppen invarianten Ovoide bestimmt. Die bei dieser Suche auftretenden Ovoide sind alle bereits bekannt. Es ergeben sich jedoch Restriktionen an die Stabilisatoren gegebenenfalls existierender, unbekannter Ovoide.
The desire to model in ever increasing detail geometrical and physical features has lead to a steady increase in the number of points used in field solvers. While many solvers have been ported to parallel machines, grid generators have left behind. Sequential generation of meshes of large size is extremely problematic both in terms of time and memory requirements. Therefore, the need for developing parallel mesh generation technique is well justified. In this work a novel algorithm is presented for automatic parallel generation of tetrahedral computational meshes based on geometrical domain decomposition. It has a potential to remove this bottleneck. Different domain decomposition approaches and criteria have been investigated. Questions regarding time and memory consumption, efficiency of computations and quality of generated surface and volume meshes have been considered. As a result of the work parTgen (partitioner and parallel tetrahedral mesh generator) software package based on the developed algorithm has been created. Several real-life examples of relatively complex structures involving large meshes (of order 10^7-10^8 elements) are given. It has been shown that high mesh quality is achieved. Memory and time consumption are reduced significantly, and parallel algorithm is efficient.
This thesis introduces so-called cone scalarising functions. They are by construction compatible with a partial order for the outcome space given by a cone. The quality of the parametrisations of the efficient set given by the cone scalarising functions are then investigated. Here, the focus lies on the (weak) efficiency of the generated solutions, the reachability of effiecient points and continuity of the solution set. Based on cone scalarising functions Pareto Navigation a novel, interactive, multiobjective optimisation method is proposed. It changes the ordering cone to realise bounds on partial tradeoffs. Besides, its use of an equality constraint for the changing component of the reference point is a new feature. The efficiency of its solutions, the reachability of efficient solutions and continuity is then analysed. Potential problems are demonstrated using a critical example. Furthermore, the use of Pareto Navigation in a two-phase approach and for nonconvex problems is discussed. Finally, its application for intensity-modulated radiotherapy planning is described. Thereby, its realisation in a graphical user interface is shown.
In der Automobilindustrie muss der Nachweis von Bauteilzuverlässigkeiten auf statistischen Verfahren basieren, da die Bauteilfestigkeit und Kundenbeanspruchung streuen. Die bisherigen Vorgehensweisen der Tests führen häufig Fehlentscheidungen bzgl. der Freigabe, was unnötige Design-Änderungen und somit hohe Kosten bedeuten kann. In vorliegender Arbeit wird der Ansatz der partiellen Durchläuferzählung entwickelt, welche die statische Güte der bisherigen Testverfahren (Success Runs) erhöht.
Pedestrian Flow Models
(2014)
There have been many crowd disasters because of poor planning of the events. Pedestrian models are useful in analysing the behavior of pedestrians in advance to the events so that no pedestrians will be harmed during the event. This thesis deals with pedestrian flow models on microscopic, hydrodynamic and scalar scales. By following the Hughes' approach, who describes the crowd as a thinking fluid, we use the solution of the Eikonal equation to compute the optimal path for pedestrians. We start with the microscopic model for pedestrian flow and then derive the hydrodynamic and scalar models from it. We use particle methods to solve the governing equations. Moreover, we have coupled a mesh free particle method to the fixed grid for solving the Eikonal equation. We consider an example with a large number of pedestrians to investigate our models for different settings of obstacles and for different parameters. We also consider the pedestrian flow in a straight corridor and through T-junction and compare our numerical results with the experiments. A part of this work is devoted for finding a mesh free method to solve the Eikonal equation. Most of the available methods to solve the Eikonal equation are restricted to either cartesian grid or triangulated grid. In this context, we propose a mesh free method to solve the Eikonal equation, which can be applicable to any arbitrary grid and useful for the complex geometries.
In this thesis one considers the periodic homogenization of a linearly coupled magneto-elastic model problem and focuses on the derivation of spectral methods to solve the obtained unit cell problem afterwards. In the beginning, the equations of linear elasticity and magnetism are presented together with the physical quantities used within. After specifying the model assumptions, the system of partial differential equations is rewritten in a weak form for which the existence and uniqueness of solutions is discussed. The model problem then undergoes a homogenization process where the original problem is approximated by a substitute problem with a repeating micro-structural geometry that was generated from a representative volume element (RVE). The following separation of scales, which can be achieved either by an asymptotic expansion or through a two-scale limit process, yields the homogenized problem on the macroscopic scale and the periodic unit cell problem. The latter is further analyzed using Fourier series, leading to periodic Lippmann-Schwinger type equations allowing for the development of matrix-free solvers. It is shown that, while it is possible to craft a scheme for the coupled problem from the purely elastic and magnetic Lippmann-Schwinger equations alone without much additional effort, a more general setting is provided when deriving a Lippmann-Schwinger equation for the coupled system directly. These numerical approaches are then validated with some analytically solvable test problems, before their performance is tested against each other for some more complex examples.
This thesis is devoted to applying symbolic methods to the problems of decoding linear codes and of algebraic cryptanalysis. The paradigm we employ here is as follows. We reformulate the initial problem in terms of systems of polynomial equations over a finite field. The solution(s) of such systems should yield a way to solve the initial problem. Our main tools for handling polynomials and polynomial systems in such a paradigm is the technique of Gröbner bases and normal form reductions. The first part of the thesis is devoted to formulating and solving specific polynomial systems that reduce the problem of decoding linear codes to the problem of polynomial system solving. We analyze the existing methods (mainly for the cyclic codes) and propose an original method for arbitrary linear codes that in some sense generalizes the Newton identities method widely known for cyclic codes. We investigate the structure of the underlying ideals and show how one can solve the decoding problem - both the so-called bounded decoding and more general nearest codeword decoding - by finding reduced Gröbner bases of these ideals. The main feature of the method is that unlike usual methods based on Gröbner bases for "finite field" situations, we do not add the so-called field equations. This tremendously simplifies the underlying ideals, thus making feasible working with quite large parameters of codes. Further we address complexity issues, by giving some insight to the Macaulay matrix of the underlying systems. By making a series of assumptions we are able to provide an upper bound for the complexity coefficient of our method. We address also finding the minimum distance and the weight distribution. We provide solid experimental material and comparisons with some of the existing methods in this area. In the second part we deal with the algebraic cryptanalysis of block iterative ciphers. Namely, we analyze the small-scale variants of the Advanced Encryption Standard (AES), which is a widely used modern block cipher. Here a cryptanalyst composes the polynomial systems which solutions should yield a secret key used by communicating parties in a symmetric cryptosystem. We analyze the systems formulated by researchers for the algebraic cryptanalysis, and identify the problem that conventional systems have many auxiliary variables that are not actually needed for the key recovery. Moreover, having many such auxiliary variables, specific to a given plaintext/ciphertext pair, complicates the use of several pairs which is common in cryptanalysis. We thus provide a new system where the auxiliary variables are eliminated via normal form reductions. The resulting system in key-variables only is then solved. We present experimental evidence that such an approach is quite good for small scaled ciphers. We investigate further our approach and employ the so-called meet-in-the-middle principle to see how far one can go in analyzing just 2-3 rounds of scaled ciphers. Additional "tuning techniques" are discussed together with experimental material. Overall, we believe that the material of this part of the thesis makes a step further in algebraic cryptanalysis of block ciphers.
This thesis covers two important fields in financial mathematics, namely the continuous time portfolio optimisation and credit risk modelling. We analyse optimisation problems of portfolios of Call and Put options on the stock and/or the zero coupon bond issued by a firm with default risk. We use the martingale approach for dynamic optimisation problems. Our findings show that the riskier the option gets, the less proportion of his wealth the investor allocates to the risky asset. Further, we analyse the Credit Default Swap (CDS) market quotes on the Eurobonds issued by Turkish sovereign for building the term structure of the sovereign credit risk. Two methods are introduced and compared for bootstrapping the risk-neutral probabilities of default (PD) in an intensity based (or reduced form) credit risk modelling approach. We compare the market-implied PDs with the actual PDs reported by credit rating agencies based on historical experience. Our results highlight the market price of the sovereign credit risk depending on the assigned rating category in the sampling period. Finally, we find an optimal leverage strategy for delivering the payments promised by a Constant Proportion Debt Obligation (CPDO). The problem is solved via the introduction and explicit solution of a stochastic control problem by transforming the related Hamilton-Jacobi-Bellman Equation into its dual. Contrary to the industry practise, the optimal leverage function we derive is a non-linear function of the CPDO asset value. The simulations show promising behaviour of the optimal leverage function compared with the one popular among practitioners.
This thesis is concerned with stochastic control problems under transaction costs. In particular, we consider a generalized menu cost problem with partially controlled regime switching, general multidimensional running cost problems and the maximization of long-term growth rates in incomplete markets. The first two problems are considered under a general cost structure that includes a fixed cost component, whereas the latter is analyzed under proportional and Morton-Pliska
transaction costs.
For the menu cost problem and the running cost problem we provide an equivalent characterization of the value function by means of a generalized version of the Ito-Dynkin formula instead of the more restrictive, traditional approach via the use of quasi-variational inequalities (QVIs). Based on the finite element method and weak solutions of QVIs in suitable Sobolev spaces, the value function is constructed iteratively. In addition to the analytical results, we study a novel application of the menu cost problem in management science. We consider a company that aims to implement an optimal investment and marketing strategy and must decide when to issue a new version of a product and when and how much
to invest into marketing.
For the long-term growth rate problem we provide a rigorous asymptotic analysis under both proportional and Morton-Pliska transaction costs in a general incomplete market that includes, for instance, the Heston stochastic volatility model and the Kim-Omberg stochastic excess return model as special cases. By means of a dynamic programming approach leading-order optimal strategies are constructed
and the leading-order coefficients in the expansions of the long-term growth rates are determined. Moreover, we analyze the asymptotic performance of Morton-Pliska strategies in settings with proportional transaction costs. Finally, pathwise optimality of the constructed strategies is established.
One crucial assumption of continuous financial mathematics is that the portfolio can be rebalanced continuously and that there are no transaction costs. In reality, this of course does not work. On the one hand, continuous rebalancing is impossible, on the other hand, each transaction causes costs which have to be subtracted from the wealth. Therefore, we focus on trading strategies which are based on discrete rebalancing - in random or equidistant times - and where transaction costs are considered. These strategies are considered for various utility functions and are compared with the optimal ones of continuous trading.
We discuss the portfolio selection problem of an investor/portfolio manager in an arbitrage-free financial market where a money market account, coupon bonds and a stock are traded continuously. We allow for stochastic interest rates and in particular consider one and two-factor Vasicek models for the instantaneous
short rates. In both cases we consider a complete and an incomplete market setting by adding a suitable number of bonds.
The goal of an investor is to find a portfolio which maximizes expected utility
from terminal wealth under budget and present expected short-fall (PESF) risk
constraints. We analyze this portfolio optimization problem in both complete and
incomplete financial markets in three different cases: (a) when the PESF risk is
minimum, (b) when the PESF risk is between minimum and maximum and (c) without risk constraints. (a) corresponds to the portfolio insurer problem, in (b) the risk constraint is binding, i.e., it is satisfied with equality, and (c) corresponds
to the unconstrained Merton investment.
In all cases we find the optimal terminal wealth and portfolio process using the
martingale method and Malliavin calculus respectively. In particular we solve in the incomplete market settings the dual problem explicitly. We compare the
optimal terminal wealth in the cases mentioned using numerical examples. Without
risk constraints, we further compare the investment strategies for complete
and incomplete market numerically.
Die Dissertation "Portfoliooptimierung im Binomialmodell" befasst sich mit der Frage, inwieweit
das Problem der optimalen Portfolioauswahl im Binomialmodell lösbar ist bzw. inwieweit
die Ergebnisse auf das stetige Modell übertragbar sind. Dabei werden neben dem
klassischen Modell ohne Kosten und ohne Veränderung der Marktsituation auch Modellerweiterungen
untersucht.
We introduce and investigate a product pricing model in social networks where the value a possible buyer assigns to a product is influenced by the previous buyers. The selling proceeds in discrete, synchronous rounds for some set price and the individual values are additively altered. Whereas computing the revenue for a given price can be done in polynomial time, we show that the basic problem PPAI, i.e., is there a price generating a requested revenue, is weakly NP-complete. With algorithm Frag we provide a pseudo-polynomial time algorithm checking the range of prices in intervals of common buying behavior we call fragments. In some special cases, e.g., solely positive influences, graphs with bounded in-degree, or graphs with bounded path length, the amount of fragments is polynomial. Since the run-time of Frag is polynomial in the amount of fragments, the algorithm itself is polynomial for these special cases. For graphs with positive influence we show that every buyer does also buy for lower prices, a property that is not inherent for arbitrary graphs. Algorithm FixHighest improves the run-time on these graphs by using the above property.
Furthermore, we introduce variations on this basic model. The version of delaying the propagation of influences and the awareness of the product can be implemented in our basic model by substituting nodes and arcs with simple gadgets. In the chapter on Dynamic Product Pricing we allow price changes, thereby raising the complexity even for graphs with solely positive or negative influences. Concerning Perishable Product Pricing, i.e., the selling of products that are usable for some time and can be rebought afterward, the principal problem is computing the revenue that a given price can generate in some time horizon. In general, the problem is #P-hard and algorithm Break runs in pseudo-polynomial time. For polynomially computable revenue, we investigate once more the complexity to find the best price.
We conclude the thesis with short results in topics of Cooperative Pricing, Initial Value as Parameter, Two Product Pricing, and Bounded Additive Influence.
The main topic of this thesis is to define and analyze a multilevel Monte Carlo algorithm for path-dependent functionals of the solution of a stochastic differential equation (SDE) which is driven by a square integrable, \(d_X\)-dimensional Lévy process \(X\). We work with standard Lipschitz assumptions and denote by \(Y=(Y_t)_{t\in[0,1]}\) the \(d_Y\)-dimensional strong solution of the SDE.
We investigate the computation of expectations \(S(f) = \mathrm{E}[f(Y)]\) using randomized algorithms \(\widehat S\). Thereby, we are interested in the relation of the error and the computational cost of \(\widehat S\), where \(f:D[0,1] \to \mathbb{R}\) ranges in the class \(F\) of measurable functionals on the space of càdlàg functions on \([0,1]\), that are Lipschitz continuous with respect to the supremum norm.
We consider as error \(e(\widehat S)\) the worst case of the root mean square error over the class of functionals \(F\). The computational cost of an algorithm \(\widehat S\), denoted \(\mathrm{cost}(\widehat S)\), should represent the runtime of the algorithm on a computer. We work in the real number model of computation and further suppose that evaluations of \(f\) are possible for piecewise constant functions in time units according to its number of breakpoints.
We state strong error estimates for an approximate Euler scheme on a random time discretization. With this strong error estimates, the multilevel algorithm leads to upper bounds for the convergence order of the error with respect to the computational cost. The main results can be summarized in terms of the Blumenthal-Getoor index of the driving Lévy process, denoted by \(\beta\in[0,2]\). For \(\beta <1\) and no Brownian component present, we almost reach convergence order \(1/2\), which means, that there exists a sequence of multilevel algorithms \((\widehat S_n)_{n\in \mathbb{N}}\) with \(\mathrm{cost}(\widehat S_n) \leq n\) such that \( e(\widehat S_n) \precsim n^{-1/2}\). Here, by \( \precsim\), we denote a weak asymptotic upper bound, i.e. the inequality holds up to an unspecified positive constant. If \(X\) has a Brownian component, the order has an additional logarithmic term, in which case, we reach \( e(\widehat S_n) \precsim n^{-1/2} \, (\log(n))^{3/2}\).
For the special subclass of $Y$ being the Lévy process itself, we also provide a lower bound, which, up to a logarithmic term, recovers the order \(1/2\), i.e., neglecting logarithmic terms, the multilevel algorithm is order optimal for \( \beta <1\).
An empirical error analysis via numerical experiments matches the theoretical results and completes the analysis.
In a recent paper, G. Malle and G. Robinson proposed a modular anologue to Brauer's famous \( k(B) \)-conjecture. If \( B \) is a \( p \)-block of a finite group with defect group \( D \), then they conjecture that \( l(B) \leq p^r \), where \( r \) is the sectional \( p \)-rank of \( D \). Since this conjecture is relatively new, there is obviously still a lot of work to do. This thesis is concerned with proving their conjecture for the finite groups of exceptional Lie type.
This thesis discusses methods for the classification of finite projective planes via exhaustive search. In the main part the author classifies all projective planes of order 16 admitting a large quasiregular group of collineations. This is done by a complete search using the computer algebra system GAP. Computational methods for the construction of relative difference sets are discussed. These methods are implemented in a GAP-package, which is available separately. As another result --found in cooperation with U. Dempwolff-- the projective planes defined by planar monomials are classified. Furthermore the full automorphism group of the non-translation planes defined by planar monomials are classified.
In this thesis we study a variant of the quadrature problem for stochastic differential equations (SDEs), namely the approximation of expectations \(\mathrm{E}(f(X))\), where \(X = (X(t))_{t \in [0,1]}\) is the solution of an SDE and \(f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R}\) is a functional, mapping each realization of \(X\) into the real numbers. The distinctive feature in this work is that we consider randomized (Monte Carlo) algorithms with random bits as their only source of randomness, whereas the algorithms commonly studied in the literature are allowed to sample from the uniform distribution on the unit interval, i.e., they do have access to random numbers from \([0,1]\).
By assumption, all further operations like, e.g., arithmetic operations, evaluations of elementary functions, and oracle calls to evaluate \(f\) are considered within the real number model of computation, i.e., they are carried out exactly.
In the following, we provide a detailed description of the quadrature problem, namely we are interested in the approximation of
\begin{align*}
S(f) = \mathrm{E}(f(X))
\end{align*}
for \(X\) being the \(r\)-dimensional solution of an autonomous SDE of the form
\begin{align*}
\mathrm{d}X(t) = a(X(t)) \, \mathrm{d}t + b(X(t)) \, \mathrm{d}W(t), \quad t \in [0,1],
\end{align*}
with deterministic initial value
\begin{align*}
X(0) = x_0 \in \mathbb{R}^r,
\end{align*}
and driven by a \(d\)-dimensional standard Brownian motion \(W\). Furthermore, the drift coefficient \(a \colon \mathbb{R}^r \to \mathbb{R}^r\) and the diffusion coefficient \(b \colon \mathbb{R}^r \to \mathbb{R}^{r \times d}\) are assumed to be globally Lipschitz continuous.
For the function classes
\begin{align*}
F_{\infty} = \bigl\{f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R} \colon |f(x) - f(y)| \leq \|x-y\|_{\sup}\bigr\}
\end{align*}
and
\begin{align*}
F_p = \bigl\{f \colon C([0,1],\mathbb{R}^r) \to \mathbb{R} \colon |f(x) - f(y)| \leq \|x-y\|_{L_p}\bigr\}, \quad 1 \leq p < \infty.
\end{align*}
we have established the following.
\[\]
\(\textit{Theorem 1.}\)
There exists a random bit multilevel Monte Carlo (MLMC) algorithm \(M\) using
\[
L = L(\varepsilon,F) = \begin{cases}\lceil{\log_2(\varepsilon^{-2}}\rceil, &\text{if} \ F = F_p,\\
\lceil{\log_2(\varepsilon^{-2} + \log_2(\log_2(\varepsilon^{-1}))}\rceil, &\text{if} \ F = F_\infty
\end{cases}
\]
and replication numbers
\[
N_\ell = N_\ell(\varepsilon,F) = \begin{cases}
\lceil{(L+1) \cdot 2^{-\ell} \cdot \varepsilon^{-2}}\rceil, & \text{if} \ F = F_p,\\
\lceil{(L+1) \cdot 2^{-\ell} \cdot \max(\ell,1) \cdot \varepsilon^{-2}}\rceil, & \text{if} \ F=f_\infty
\end{cases}
\]
for \(\ell = 0,\ldots,L\), for which exists a positive constant \(c\) such that
\begin{align*}
\mathrm{error}(M,F) = \sup_{f \in F} \bigl(\mathrm{E}(S(f) - M(f))^2\bigr)^{1/2} \leq c \cdot \varepsilon
\end{align*}
and
\begin{align*}
\mathrm{cost}(M,F) = \sup_{f \in F} \mathrm{E}(\mathrm{cost}(M,f)) \leq c \cdot \varepsilon^{-2} \cdot \begin{cases}
(\ln(\varepsilon^{-1}))^2, &\text{if} \ F=F_p,\\
(\ln(\varepsilon^{-1}))^3, &\text{if} \ F=F_\infty
\end{cases}
\end{align*}
for every \(\varepsilon \in {]0,1/2[}\).
\[\]
Hence, in terms of the \(\varepsilon\)-complexity
\begin{align*}
\mathrm{comp}(\varepsilon,F) = \inf\bigl\{\mathrm{cost}(M,F) \colon M \ \text{is a random bit MC algorithm}, \mathrm{error}(M,F) \leq \varepsilon\bigr\}
\end{align*}
we have established the upper bound
\begin{align*}
\mathrm{comp}(\varepsilon,F) \leq c \cdot \varepsilon^{-2} \cdot \begin{cases}
(\ln(\varepsilon^{-1}))^2, &\text{if} \ F=F_p,\\
(\ln(\varepsilon^{-1}))^3, &\text{if} \ F=F_\infty
\end{cases}
\end{align*}
for some positive constant \(c\). That is, we have shown the same weak asymptotic upper bound as in the case of random numbers from \([0,1]\). Hence, in this sense, random bits are almost as powerful as random numbers for our computational problem.
Moreover, we present numerical results for a non-analyzed adaptive random bit MLMC Euler algorithm, in the particular cases of the Brownian motion, the geometric Brownian motion, the Ornstein-Uhlenbeck SDE and the Cox-Ingersoll-Ross SDE. We also provide a numerical comparison to the corresponding adaptive random number MLMC Euler method.
A key challenge in the analysis of the algorithm in Theorem 1 is the approximation of probability distributions by means of random bits. A problem very closely related to the quantization problem, i.e., the optimal approximation of a given probability measure (on a separable Hilbert space) by means of a probability measure with finite support size.
Though we have shown that the random bit approximation of the standard normal distribution is 'harder' than the corresponding quantization problem (lower weak rate of convergence), we have been able to establish the same weak rate of convergence as for the corresponding quantization problem in the case of the distribution of a Brownian bridge on \(L_2([0,1])\), the distribution of the solution of a scalar SDE on \(L_2([0,1])\), and the distribution of a centered Gaussian random element in a separable Hilbert space.
For computational reasons, the spline interpolation of the Earth's gravitational potential is usually done in a spherical framework. In this work, however, we investigate a spline method with respect to the real Earth. We are concerned with developing the real Earth oriented strategies and methods for the Earth's gravitational potential determination. For this purpose we introduce the reproducing kernel Hilbert space of Newton potentials on and outside given regular surface with reproducing kernel defined as a Newton integral over it's interior. We first give an overview of thus far achieved results considering approximations on regular surfaces using surface potentials (Chapter 3). The main results are contained in the fourth chapter where we give a closer look to the Earth's gravitational potential, the Newton potentials and their characterization in the interior and the exterior space of the Earth. We also present the L2-decomposition for regions in R3 in terms of distributions, as a main strategy to impose the Hilbert space structure on the space of potentials on and outside a given regular surface. The properties of the Newton potential operator are investigated in relation to the closed subspace of harmonic density functions. After these preparations, in the fifth chapter we are able to construct the reproducing kernel Hilbert space of Newton potentials on and outside a regular surface. The spline formulation for the solution to interpolation problems, corresponding to a set of bounded linear functionals is given, and corresponding convergence theorems are proven. The spline formulation reflects the specifics of the Earth's surface, due to the representation of the reproducing kernel (of the solution space) as a Newton integral over the inner space of the Earth. Moreover, the approximating potential functions have the same domain of harmonicity as the actual Earth's gravitational potential, i.e., they are harmonic outside and continuous on the Earth's surface. This is a step forward in comparison to the spherical harmonic spline formulation involving functions harmonic down to the Runge sphere. The sixth chapter deals with the representation of the used kernel in the spherical case. It turns out that in the case of the spherical Earth, this kernel can be considered a kind of generalization to spherically oriented kernels, such as Abel-Poisson or the singularity kernel. We also investigate the existence of the closed expression of the kernel. However, at this point it remains to be unknown to us. So, in Chapter 7, we are led to consider certain discretization methods for integrals over regions in R3, in connection to theory of the multidimensional Euler summation formula for the Laplace operator. We discretize the Newton integral over the real Earth (representing the spline function) and give a priori estimates for approximate integration when using this discretization method. The last chapter summarizes our results and gives some directions for the future research.
The knowledge of structural properties in microscopic materials contributes to a deeper understanding of macroscopic properties. For the study of such materials, several imaging techniques reaching scales in the order of nanometers have been developed. One of the most powerful and sophisticated imaging methods is focused-ion-beam scanning electron
microscopy (FIB-SEM), which combines serial sectioning by an ion beam and imaging by
a scanning electron microscope.
FIB-SEM imaging reaches extraordinary scales below 5 nm with large representative
volumes. However, the complexity of the imaging process results in the addition of artificial distortion and artifacts generating poor-quality images. We introduce a method
for the quality evaluation of images by analyzing general characteristics of the images
as well as artifacts exclusively for FIB-SEM, namely curtaining and charging. For the
evaluation, we propose quality indexes, which are tested on several data sets of porous and non-porous materials with different characteristics and distortions. The quality indexes report objective evaluations in accordance with visual judgment.
Moreover, the acquisition of large volumes at high resolution can be time-consuming. An approach to speed up the imaging is by decreasing the resolution and by considering cuboidal voxel configurations. However, non-isotropic resolutions may lead to errors in the reconstructions. Even if the reconstruction is correct, effects are visible in the analysis.
We study the effects of different voxel settings on the prediction of material and flow properties of reconstructed structures. Results show good agreement between highly resolved cases and ground truths as is expected. Structural anisotropy is reported as
resolution decreases, especially in anisotropic grids. Nevertheless, gray image interpolation
remedies the induced anisotropy. These benefits are visible at flow properties as well.
For highly porous structures, the structural reconstruction is even more difficult as
a consequence of deeper parts of the material visible through the pores. We show as an application example, the reconstruction of two highly porous structures of optical layers, where a typical workflow from image acquisition, preprocessing, reconstruction until a
spatial analysis is performed. The study case shows the advantages of 3D imaging for
optical porous layers. The analysis reveals geometrical structural properties related to the manufacturing processes.
In this thesis, mathematical research questions related to recursive utility and stochastic differential utility (SDU) are explored.
First, a class of backward equations under nonlinear expectations is investigated: Existence and uniqueness of solutions are established, and the issues of stability and discrete-time approximation are addressed. It is then shown that backward equations of this class naturally appear as a continuous-time limit in the context of recursive utility with nonlinear expectations.
Then, the Epstein-Zin parametrization of SDU is studied. The focus is on specifications with both relative risk aversion and elasitcity of intertemporal substitution greater that one. A concave utility functional is constructed and a utility gradient inequality is established.
Finally, consumption-portfolio problems with recursive preferences and unspanned risk are investigated. The investor's optimal strategies are characterized by a specific semilinear partial differential equation. The solution of this equation is constructed by a fixed point argument, and a corresponding efficient and accurate method to calculate optimal strategies numerically is given.
This thesis consists of two parts, i.e. the theoretical background of (R)ABSDE including basic theorems, theoretical proofs and properties (Chapter 2-4), as well as numerical algorithms and simulations for (R)ABSDES (Chapter 5). For the theoretical part, we study ABSDEs (Chapter 2), RABSDEs with one obstacle (Chapter 3)and RABSDEs with two obstacles (Chapter 4) in the defaultable setting respectively, including the existence and uniqueness theorems, applications, the comparison theorem for ABSDEs, their relations with PDEs and stochastic differential delay equations (SDDE). The numerical algorithm part (Chapter 5) introduces two main algorithms, a discrete penalization scheme and a discrete reflected scheme based on a random walk approximation of the Brownian motion as well as a discrete approximation of the default martingale; we give the convergence results of the algorithms, provide a numerical example and an application in American game options in order to illustrate the performance of the algorithms.
In this thesis, we investigate several upcoming issues occurring in the context of conceiving and building a decision support system. We elaborate new algorithms for computing representative systems with special quality guarantees, provide concepts for supporting the decision makers after a representative system was computed, and consider a methodology of combining two optimization problems.
We review the original Box-Algorithm for two objectives by Hamacher et al. (2007) and discuss several extensions regarding coverage, uniformity, the enumeration of the whole nondominated set, and necessary modifications if the underlying scalarization problem cannot be solved to optimality. In a next step, the original Box-Algorithm is extended to the case of three objective functions to compute a representative system with desired coverage error. Besides the investigation of several theoretical properties, we prove the correctness of the algorithm, derive a bound on the number of iterations needed by the algorithm to meet the desired coverage error, and propose some ideas for possible extensions.
Furthermore, we investigate the problem of selecting a subset with desired cardinality from the computed representative system, the Hypervolume Subset Selection Problem (HSSP). We provide two new formulations for the bicriteria HSSP, a linear programming formulation and a \(k\)-link shortest path formulation. For the latter formulation, we propose an algorithm for which we obtain the currently best known complexity bound for solving the bicriteria HSSP. For the tricriteria HSSP, we propose an integer programming formulation with a corresponding branch-and-bound scheme.
Moreover, we address the issue of how to present the whole set of computed representative points to the decision makers. Based on common illustration methods, we elaborate an algorithm guiding the decision makers in choosing their preferred solution.
Finally, we step back and look from a meta-level on the issue of how to combine two given optimization problems and how the resulting combinations can be related to each other. We come up with several different combined formulations and give some ideas for the practical approach.
In this thesis the combinatorial framework of toric geometry is extended to equivariant sheaves over toric varieties. The central questions are how to extract combinatorial information from the so developed description and whether equivariant sheaves can, like toric varieties, be considered as purely combinatorial objects. The thesis consists of three main parts. In the first part, by systematically extending the framework of toric geometry, a formalism is developed for describing equivariant sheaves by certain configurations of vector spaces. In the second part, homological properties of a certain class of equivariant sheaves are investigated, namely that of reflexive equivariant sheaves. Several kinds of resolutions for these sheaves are constructed which depend only on the configuration of their associated vector spaces. Thus a partially positive answer to the question of combinatorial representability is given. As a particular result, a new way for computing minimal resolutions for Z^n - graded modules over polynomial rings is obtained. In the third part a complete classification of the simplest nontrivial sheaves, equivariant vector bundles of rank two over smooth toric surfaces, is given. A combinatorial characterization is given and parameter spaces (moduli spaces) are constructed which depend only on this characterization. In appendices a outlook on equivariant sheaves and the relation of Chern classes to their combinatorial classification is given, particularly focussing on the case of the projective plane. A classification of equivariant vector bundles of rank three over the projective plane is given.
An autoregressive-ARCH model with possible exogeneous variables is treated. We estimate the conditional volatility of the model by applying feedforward networks to the residuals and prove consistency and asymptotic normality for the estimates under the rate of feedforward networks complexity. Recurrent neural networks estimates of GARCH and value-at-risk is studied. We prove consistency and asymptotic normality for the recurrent neural networks ARMA estimator under the rate of recurrent networks complexity. We also overcome the estimation problem in stochastic variance models in discrete time by feedforward networks and the introduction of a new distributions on the innovations. We use the method to calculate market risk such as expected shortfall and Value-at risk. We tested this distribution together with other new distributions on the GARCH family models against other common distributions on the financial market such as Normal Inverse Gaussian, normal and the Student's t- distributions. As an application of the models, some German stocks are studied and the different approaches are compared together with the most common method of GARCH(1,1) fit.
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.
In this work we focus on the regression models with asymmetrical error distribution,
more precisely, with extreme value error distributions. This thesis arises in the framework
of the project "Robust Risk Estimation". Starting from July 2011, this project won
three years funding by the Volkswagen foundation in the call "Extreme Events: Modelling,
Analysis, and Prediction" within the initiative "New Conceptual Approaches to
Modelling and Simulation of Complex Systems". The project involves applications in
Financial Mathematics (Operational and Liquidity Risk), Medicine (length of stay and
cost), and Hydrology (river discharge data). These applications are bridged by the
common use of robustness and extreme value statistics.
Within the project, in each of these applications arise issues, which can be dealt with by
means of Extreme Value Theory adding extra information in the form of the regression
models. The particular challenge in this context concerns asymmetric error distributions,
which significantly complicate the computations and make desired robustification
extremely difficult. To this end, this thesis makes a contribution.
This work consists of three main parts. The first part is focused on the basic notions
and it gives an overview of the existing results in the Robust Statistics and Extreme
Value Theory. We also provide some diagnostics, which is an important achievement of
our project work. The second part of the thesis presents deeper analysis of the basic
models and tools, used to achieve the main results of the research.
The second part is the most important part of the thesis, which contains our personal
contributions. First, in Chapter 5, we develop robust procedures for the risk management
of complex systems in the presence of extreme events. Mentioned applications use time
structure (e.g. hydrology), therefore we provide extreme value theory methods with time
dynamics. To this end, in the framework of the project we considered two strategies. In
the first one, we capture dynamic with the state-space model and apply extreme value
theory to the residuals, and in the second one, we integrate the dynamics by means of
autoregressive models, where the regressors are described by generalized linear models.
More precisely, since the classical procedures are not appropriate to the case of outlier
presence, for the first strategy we rework classical Kalman smoother and extended
Kalman procedures in a robust way for different types of outliers and illustrate the performance
of the new procedures in a GPS application and a stylized outlier situation.
To apply approach to shrinking neighborhoods we need some smoothness, therefore for
the second strategy, we derive smoothness of the generalized linear model in terms of
L2 differentiability and create sufficient conditions for it in the cases of stochastic and
deterministic regressors. Moreover, we set the time dependence in these models by
linking the distribution parameters to the own past observations. The advantage of
our approach is its applicability to the error distributions with the higher dimensional
parameter and case of regressors of possibly different length for each parameter. Further,
we apply our results to the models with generalized Pareto and generalized extreme value
error distributions.
Finally, we create the exemplary implementation of the fixed point iteration algorithm
for the computation of the optimally robust in
uence curve in R. Here we do not aim to
provide the most
exible implementation, but rather sketch how it should be done and
retain points of particular importance. In the third part of the thesis we discuss three applications,
operational risk, hospitalization times and hydrological river discharge data,
and apply our code to the real data set taken from Jena university hospital ICU and
provide reader with the various illustrations and detailed conclusions.
A significant step to engineering design is to take into account uncertainties and to
develop optimal designs that are robust with respect to perturbations. Furthermore, it
is often of interest to optimize for different conflicting objective functions describing the
quality of a design, leading to a multi-objective optimization problem. In this context,
generating methods for solving multi-objective optimization problems seek to find a
representative set of solutions fulfilling the concept of Pareto optimality. When multiple
uncertain objective functions are involved, it is essential to define suitable measures for
robustness that account for a combined effect of uncertainties in objective space. Many
tasks in engineering design include the solution of an underlying partial differential
equation that can be computationally expensive. Thus, it is of interest to use efficient
strategies for finding optimal designs. This research aims to present suitable measures
for robustness in a multi-objective context, as well as optimization strategies for multi-
objective robust design.
This work introduces new ideas for robustness measures in the context of multi-
objective robust design. Losses and expected losses based on distances in objective space
are used to describe robustness. A direct formulation and a two-phase formulation based
on expected losses are proposed for finding a set of robust optimal solutions.
Furthermore, suitable optimization strategies for solving the resulting multi-objective
robust design problem are formulated and analyzed. The multi-objective optimization
problem is solved with a constraint-based approach that is based on solving several
constrained single-objective optimization problems with a hybrid optimization strategy.
The hybrid method combines a global search method on a surrogate model with adjoint-
based optimization methods. In the context of optimization with an underlying partial
differential equation, a one-shot approach is extended to handle additional constraints.
The developed concepts for multi-objective robust design and the proposed optimiza-
tion strategies are applied to an aerodynamic shape optimization problem. The drag
coefficient and the lift coefficient are optimized under the consideration of uncertain-
ties in the operational conditions and geometrical uncertainties. The uncertainties are
propagated with the help of a non-intrusive polynomial chaos approach. For increasing
the efficiency when considering a higher-dimensional random space, it is made use of a
Karhunen-Loève expansion and a dimension-adaptive sparse grid quadrature.
Semiparametric estimation of conditional quantiles for time series, with applications in finance
(2003)
The estimation of conditional quantiles has become an increasingly important issue in insurance and financial risk management. The stylized facts of financial time series data has rendered direct applications of extreme value theory methodologies, in the estimation of extreme conditional quantiles, inappropriate. On the other hand, quantile regression based procedures work well in nonextreme parts of a given data but breaks down in extreme probability levels. In order to solve this problem, we combine nonparametric regressions for time series and extreme value theory approaches in the estimation of extreme conditional quantiles for financial time series. To do so, a class of time series models that is similar to nonparametric AR-(G)ARCH models but which does not depend on distributional and moments assumptions, is introduced. We discuss estimation procedures for the nonextreme levels using the models and consider the estimates obtained by inverting conditional distribution estimators and by direct estimation using Koenker-Basset (1978) version for kernels. Under some regularity conditions, the asymptotic normality and uniform convergence, with rates, of the conditional quantile estimator for strong mixing time series, are established. We study the estimation of scale function in the introduced models using similar procedures and show that under some regularity conditions, the scale estimate is weakly consistent and asymptotically normal. The application of introduced models in the estimation of extreme conditional quantiles is achieved by augmenting them with methods in extreme value theory. It is shown that the overal extreme conditional quantiles estimator is consistent. A Monte Carlo study is carried out to illustrate the good performance of the estimates and real data are used to demonstrate the estimation of Value-at-Risk and conditional expected shortfall in financial risk management and their multiperiod predictions discussed.
By using Gröbner bases of ideals of polynomial algebras over a field, many implemented algorithms manage to give exciting examples and counter examples in Commutative Algebra and Algebraic Geometry. Part A of this thesis will focus on extending the concept of Gröbner bases and Standard bases for polynomial algebras over the ring of integers and its factors \(\mathbb{Z}_m[x]\). Moreover we implemented two algorithms for this case in Singular which use different approaches in detecting useless computations, the classical Buchberger algorithm and a F5 signature based algorithm. Part B includes two algorithms that compute the graded Hilbert depth of a graded module over a polynomial algebra \(R\) over a field, as well as the depth and the multigraded Stanley depth of a factor of monomial ideals of \(R\). The two algorithms provide faster computations and examples that lead B. Ichim and A. Zarojanu to a counter example of a question of J. Herzog. A. Duval, B. Goeckner, C. Klivans and J. Martin have recently discovered a counter example for the Stanley Conjecture. We prove in this thesis that the Stanley Conjecture holds in some special cases. Part D explores the General Neron Desingularization in the frame of Noetherian local domains of dimension 1. We have constructed and implemented in Singular and algorithm that computes a strong Artin Approximation for Cohen-Macaulay local rings of dimension 1.
Standard bases are one of the main tools in computational commutative algebra. In 1965
Buchberger presented a criterion for such bases and thus was able to introduce a first approach for their computation. Since the basic version of this algorithm is rather inefficient
due to the fact that it processes lots of useless data during its execution, active research for
improvements of those kind of algorithms is quite important.
In this thesis we introduce the reader to the area of computational commutative algebra with a focus on so-called signature-based standard basis algorithms. We do not only
present the basic version of Buchberger’s algorithm, but give an extensive discussion of different attempts optimizing standard basis computations, from several sorting algorithms
for internal data up to different reduction processes. Afterwards the reader gets a complete
introduction to the origin of signature-based algorithms in general, explaining the under-
lying ideas in detail. Furthermore, we give an extensive discussion in terms of correctness,
termination, and efficiency, presenting various different variants of signature-based standard basis algorithms.
Whereas Buchberger and others found criteria to discard useless computations which
are completely based on the polynomial structure of the elements considered, Faugère presented a first signature-based algorithm in 2002, the F5 Algorithm. This algorithm is famous for generating much less computational overhead during its execution. Within this
thesis we not only present Faugère’s ideas, we also generalize them and end up with several
different, optimized variants of his criteria for detecting redundant data.
Being not completely focussed on theory, we also present information about practical
aspects, comparing the performance of various implementations of those algorithms in the
computer algebra system Singular over a wide range of example sets.
In the end we give a rather extensive overview of recent research in this area of computational commutative algebra.
Lithium-ion batteries are increasingly becoming an ubiquitous part of our everyday life - they are present in mobile phones, laptops, tools, cars, etc. However, there are still many concerns about their longevity and their safety. In this work we focus on the simulation of several degradation mechanisms on the microscopic scale, where one can resolve the active materials inside the electrodes of the lithium-ion batteries as porous structures. We mainly study two aspects - heat generation and mechanical stress. For the former we consider an electrochemical non-isothermal model on the spatially resolved porous scale to observe the temperature increase inside a battery cell, as well as to observe the individual heat sources to assess their contributions to the total heat generation. As a result from our experiments, we determined that the temperature has very small spatial variance for our test cases and thus allows for an ODE formulation of the heat equation.
The second aspect that we consider is the generation of mechanical stress as a result of the insertion of lithium ions in the electrode materials. We study two approaches - using small strain models and finite strain models. For the small strain models, the initial geometry and the current geometry coincide. The model considers a diffusion equation for the lithium ions and equilibrium equation for the mechanical stress. First, we test a single perforated cylindrical particle using different boundary conditions for the displacement and with Neumann boundary conditions for the diffusion equation. We also test for cylindrical particles, but with boundary conditions for the diffusion equation in the electrodes coming from an isothermal electrochemical model for the whole battery cell. For the finite strain models we take in consideration the deformation of the initial geometry as a result of the intercalation and the mechanical stress. We compare two elastic models to study the sensitivity of the predicted elastic behavior on the specific model used. We also consider a softening of the active material dependent on the concentration of the lithium ions and using data for silicon electrodes. We recover the general behavior of the stress from known physical experiments.
Some models, like the mechanical models we use, depend on the local values of the concentration to predict the mechanical stress. In that sense we perform a short comparative study between the Finite Element Method with tetrahedral elements and the Finite Volume Method with voxel volumes for an isothermal electrochemical model.
The spatial discretizations of the PDEs are done using the Finite Element Method. For some models we have discontinuous quantities where we adapt the FEM accordingly. The time derivatives are discretized using the implicit Backward Euler method. The nonlinear systems are linearized using the Newton method. All of the discretized models are implemented in a C++ framework developed during the thesis.
In this thesis, we consider a problem from modular representation theory of finite groups. Lluís Puig asked the question whether the order of the defect groups of a block \( B \) of the group algebra of a given finite group \( G \) can always be bounded in terms of the order of the vertices of an arbitrary simple module lying in \( B \).
In characteristic \( 2 \), there are examples showing that this is not possible in general, whereas in odd characteristic, no such examples are known. For instance, it is known that the answer to Puig's question is positive in case that \( G \) is a symmetric group, by work of Danz, Külshammer, and Puig.
Motivated by this, we study the cases where \( G \) is a finite classical group in non-defining characteristic or one of the finite groups \( G_2(q) \) or \( ³D_4(q) \) of Lie type, again in non-defining characteristic. Here, we generalize Puig's original question by replacing the vertices occurring in his question by arbitrary self-centralizing subgroups of the defect groups. We derive positive and negative answers to this generalized question.
\[\]
In addition to that, we determine the vertices of the unipotent simple \( GL_2(q) \)-module labeled by the partition \( (1,1) \) in characteristic \( 2 \). This is done using a method known as Brauer construction.
The nowadays increasing number of fields where large quantities of data are collected generates an emergent demand for methods for extracting relevant information from huge databases. Amongst the various existing data mining models, decision trees are widely used since they represent a good trade-off between accuracy and interpretability. However, one of their main problems is that they are very instable, which complicates the process of the knowledge discovery because the users are disturbed by the different decision trees generated from almost the same input learning samples. In the current work, binary tree classifiers are analyzed and partially improved. The analysis of tree classifiers goes from their topology from the graph theory point of view to the creation of a new tree classification model by means of combining decision trees and soft comparison operators (Mlynski, 2003) with the purpose to not only overcome the well known instability problem of decision trees, but also in order to confer the ability of dealing with uncertainty. In order to study and compare the structural stability of tree classifiers, we propose an instability coefficient which is based on the notion of Lipschitz continuity and offer a metric to measure the proximity between decision trees. This thesis converges towards its main part with the presentation of our model ``Soft Operators Decision Tree\'\' (SODT). Mainly, we describe its construction, application and the consistency of the mathematical formulation behind this. Finally we show the results of the implementation of SODT and compare numerically the stability and accuracy of a SODT and a crisp DT. The numerical simulations support the stability hypothesis and a smaller tendency to overfitting the training data with SODT than with crisp DT is observed. A further aspect of this inclusion of soft operators is that we choose them in a way so that the resulting goodness function (used by this method) is differentiable and thus allows to calculate the best split points by means of gradient descent methods. The main drawback of SODT is the incorporation of the unpreciseness factor, which increases the complexity of the algorithm.
Solving probabilistic-robust optimization problems using methods from semi-infinite optimization
(2023)
Optimization under uncertainty is one field of mathematics which is strongly inspired by real world problems. To handle uncertainties several models have arisen. One of these is the probust model where a combination of probabilistic and worst-case uncertainty is considered. So far, just problem instances with a special structure can be dealt with. In this thesis, we introduce solving techniques applicable for any probust optimization problem. On the one hand, we create upper bounds for the solution value by solving a sequence of chance constrained optimization problems. These bounds are based on discretization schemes which are inspired by semi-infinite optimization. On the other hand, we create lower bounds by solving a sequence of set-approximation problems. Here, we substitute the original event set by an appropriate family of sets. We examine the performance of the corresponding algorithms on simple packing problems where we can provide the probust solution analytically. Afterwards, we solve a water reservoir and a distillation problem and compare the probust solutions with solutions arising from other uncertainty models.
In this text we survey some large deviation results for diffusion processes. The first chapters present results from the literature such as the Freidlin-Wentzell theorem for diffusions with small noise. We use these results to prove a new large deviation theorem about diffusion processes with strong drift. This is the main result of the thesis. In the later chapters we give another application of large deviation results, namely to determine the exponential decay rate for the Bayes risk when separating two different processes. The final chapter presents techniques which help to experiment with rare events for diffusion processes by means of computer simulations.
The main two problems of continuous-time financial mathematics are option pricing and portfolio optimization. In this thesis, various new aspects of these major topics of financial mathematics will be discussed. In all our considerations we will assume the standard diffusion type setting for securitiy prices which is today well-know under the term "Black-Scholes model". This setting and the basic results of option pricing and portfolio optimization are surveyed in the first chapter. The next three chapters deal with generalizations of the standard portfolio problem, also know as "Merton's problem". Here, we will always use the stochastic control approach as introduced in the seminal papers by Merton (1969, 1971, 1990). One such problem is the very realistic setting of an investor who is faced with fixed monetary streams. More precisely, in addition to maximizing the utility from final wealth via choosing an investment strategy, the investor also has to fulfill certain consumption needs. Also the opposite situation, an additional income stream can now be taken into account in our portfolio optimization problem. We consider various examples and solve them on one hand via classical stochastic control methods and on the other hand by our new separation theorem. This together with some numerical examples forms Chapter 2. Chapter 3 is mainly concerned with the portfolio problem if the investor has different lending and borrowing rates. We give explicit solutions (where possible) and numerical methods to calculate the optimal strategy in the cases of log utility and HARA utility for three different modelling approaches of the dependence of the borrowing rate on the fraction of wealth financed by a credit. The further generalization of the standard Merton problem in Chapter 4 consists in considering simultaneously the possibilities for continuous and discrete consumption. In our general approach there is a possibility for assigning the different consumption times different weights which is a generalization of the usual way of making them comparable via discounting. Chapter 5 deals with the special case of pricing basket options. Here, the main problem is not path-dependence but the multi-dimensionality which makes it impossible to give usuefull analytical representations of the option price. We review the literature and compare six different numerical methods in a systematic way. Thereby we also look at the influence of various parameters such as strike, correlation, forwards or volatilities on the erformance of the different numerical methods. The problem of pricing Asian options on average spot with average strike is the topic of Chapter 6. We here apply the bivariate normal distribution to obtain an approximate option price. This method proves to be very reliable and e±cient for the valuation of different variants of Asian options on average spot with average strike.
We discuss some first steps towards experimental design for neural network regression which, at present, is too complex to treat fully in general. We encounter two difficulties: the nonlinearity of the models together with the high parameter dimension on one hand, and the common misspecification of the models on the other hand.
Regarding the first problem, we restrict our consideration to neural networks with only one and two neurons in the hidden layer and a univariate input variable. We prove some results regarding locally D-optimal designs, and present a numerical study using the concept of maximin optimal designs.
In respect of the second problem, we have a look at the effects of misspecification on optimal experimental designs.
In modern textile manufacturing industries, the function of human eyes to detect disturbances in the production processes which yield defective products is switched to cameras. The camera images are analyzed with various methods to detect these disturbances automatically. There are, however, still problems with in particular semi-regular textures which are typical for weaving patterns. We study three parts of that problem of automatic texture analysis: image smoothing, texture synthesis and defect detection. In image smoothing, we develop a two dimensional kernel smoothing method with locally and directionally adaptive bandwidths allowing correlation in the errors. Two approaches are used in synthesising texture. The first is based on constructing a generalized Ising energy function in the Markov Random Field setup, and for the second, we use two-dimensional periodic bootstrap methods for semi-regular texture synthesis. We treat defect detection as multihypothesis testing problem with the null hypothesis representing the absence of defects and the other hypotheses representing various types of defects. We develop a test based on a nonparametric regression setup, and we use the bootstrap for approximating the distribution of our test statistic.
This thesis investigates the constrained form of the spherical Minimax location problem and the spherical Weber location problem. Specifically, we consider the problem of locating a new facility on the surface of the unit sphere in the presence of convex spherical polygonal restricted regions and forbidden regions such that the maximum weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized and the sum of the weighted distance from the new facility on the surface of the unit sphere to m existing facilities is minimized. It is assumed that a forbidden region is an area on the surface of the unit sphere where travel and facility location are not permitted and that distance is measured using the great circle arc distance. We represent a polynomial time algorithm for the spherical Minimax location problem for the special case where all the existing facilities are located on the surface of a hemisphere. Further, we have developed algorithms for spherical Weber location problem using barrier distance on a hemisphere as well as on the unit sphere.
The new international capital standard for credit institutions (“Basel II”) allows banks to use internal rating systems in order to determine the risk weights that are relevant for the calculation of capital charge. Therefore, it is necessary to develop a system that enfolds the main practices and methods existing in the context of credit rating. The aim of this thesis is to give a suggestion of setting up a credit rating system, where the main techniques used in practice are analyzed, presenting some alternatives and considering the problems that can arise from a statistical point of view. Finally, we will set up some guidelines on how to accomplish the challenge of credit scoring. The judgement of the quality of a credit with respect to the probability of default is called credit rating. A method based on a multi-dimensional criterion seems to be natural, due to the numerous effects that can influence this rating. However, owing to governmental rules, the tendency is that typically one-dimensional criteria will be required in the future as a measure for the credit worthiness or for the quality of a credit. The problem as described above can be resolved via transformation of a multi-dimensional data set into a one-dimensional one while keeping some monotonicity properties and also keeping the loss of information (due to the loss of dimensionality) at a minimum level.
This dissertation presents a generalization of the generalized grey Brownian motion with componentwise independence, called a vector-valued generalized grey Brownian motion (vggBm), and builds a framework of mathematical analysis around this process with the aim of solving stochastic differential equations with respect to this process. Similar to that of the one-dimensional case, the construction of vggBm starts with selecting the appropriate nuclear triple, and construct the corresponding probability measure on the co-nuclear space. Since independence of components are essential in constructing vggBm, a natural way to achieve this is to use the nuclear triple of product spaces: \[ \mathcal{S}_d(\mathbb{R}) \subset L^2_d(\mathbb{R}) \subset \mathcal{S}_d'(\mathbb{R}), \]
where \( L^2_d(\mathbb{R}) \) is the real separable Hilbert space of \( \mathbb{R}^d \)-valued square integrable functions on \( \mathbb{R} \) with respect to the Lebesgue measure, \( \mathcal{S}_d(\mathbb{R}) \) is the external direct sum of \(d\) copies of the nuclear space \(\mathcal{S}(\mathbb{R})\) of Schwartz test functions, and \(\mathcal{S}_d'(\mathbb{R})\) is the dual space of \(\mathcal{S}_d(\mathbb{R})\).
The probability measure used is the the \(d\)-fold product measure of the Mittag-Leffler measure, denoted by \(\mu_{\beta}^{\otimes d}\), whose characteristic function is given by \[ \int_{\mathcal{S}_d'(\mathbb{R})} e^{i\langle\omega,\varphi\rangle}\,\text{d}\mu_{\beta}^{\otimes d}(\omega) = \prod_{k=1}^{d}E_\beta\left(-\frac{1}{2}\langle\varphi_k,\varphi_k\rangle\right),\qquad \varphi\in \mathcal{S}_d(\mathbb{R}), \]
where \( \beta\in(0,1] \), and \( E_\beta \) is the Mittag-Leffler function. Vector-valued generalized grey Brownian motion, denoted by \( B^{\beta,\alpha}_{d}:=(B^{\beta,\alpha}_{d,t})_{t\geq 0}\), is then defined as a process taking values in \( L^2(\mu_{\beta}^{\otimes d};\mathbb{R}^d) \) given by
\[ B^{\beta,\alpha}_{d,t}(\omega) := (\langle\omega_1,M^{\alpha/2}_{-}1\!\!1_{[0,t)}\rangle,\dots,\langle\omega_d,M^{\alpha/2}_{-}1\!\!1_{[0,t)}\rangle),\quad \omega\in\mathcal{S}_d'(\mathbb{R}), \]
where \( M^{\alpha/2} \) is an appropriate fractional operator indexed by \( \alpha\in(0,2) \) and \( 1\!\!1_{[0,t)} \) is the indicator function on the interval \( [0,t) \). This process is, in general, not the aforementioned \(d\)-dimensional analogues of ggBm for \(d\geq 2\), since componentwise independence of the latter process holds only in the Gaussian case.
The study of analysis around vggBm starts with accessibility to Appell systems, so that characterizations and tools for the analysis of the corresponding distribution spaces are established. Then, explicit examples of the use of these characterizations and tools are given: the construction of Donsker's delta function, the existence of local times and self-intersection local times of vggBm, the existence of the derivative of vggBm in the sense of distributions, and the existence of solutions to linear stochastic differential equations with respect to vggBm.
Nonwoven materials are used as filter media which are the key component of automotive filters such as air filters, oil filters, and fuel filters. Today, the advanced engine technologies require innovative filter media with higher performances. A virtual microstructure of the nonwoven filter medium, which has similar filter properties as the existing material, can be used to design new filter media from existing media. Nonwoven materials considered in this thesis prominently feature non-overlapping fibers, curved fibers, fibers with circular cross section, fibers of apparently infinite length, and fiber bundles. To this end, as part of this thesis, we extend the Altendorf-Jeulin individual fiber model to incorporate all the above mentioned features. The resulting novel stochastic 3D fiber model can generate geometries with good visual resemblance of real filter media. Furthermore, pressure drop, which is one of the important physical properties of the filter, simulated numerically on the computed tomography (CT) data of the real nonwoven material agrees well (with a relative error of 8%) with the pressure drop simulated in the generated microstructure realizations from our model.
Generally, filter properties for the CT data and generated microstructure realizations are computed using numerical simulations. Since numerical simulations require extensive system memory and computation time, it is important to find the representative domain size of the generated microstructure for a required filter property. As part of this thesis, simulation and a statistical approach are used to estimate the representative domain size of our microstructure model. Precisely, the representative domain size with respect to the packing density, the pore size distribution, and the pressure drop are considered. It turns out that the statistical approach can be used to estimate the representative domain size for the given property more precisely and using less generated microstructures than the purely simulation based approach.
Among the various properties of fibrous filter media, fiber thickness and orientation are important characteristics which should be considered in design and quality assurance of filter media. Automatic analysis of images from scanning electron microscopy (SEM) is a suitable tool in that context. Yet, the accuracy of such image analysis tools cannot be judged based on images of real filter media since their true fiber thickness and orientation can never be known accurately. A solution is to employ synthetically generated models for evaluation. By combining our 3D fiber system model with simulation of the SEM imaging process, quantitative evaluation of the fiber thickness and orientation measurements becomes feasible. We evaluate the state-of-the-art automatic thickness and orientation estimation method that way.
Continuous stochastic control theory has found many applications in optimal investment. However, it lacks some reality, as it is based on the assumption that interventions are costless, which yields optimal strategies where the controller has to intervene at every time instant. This thesis consists of the examination of two types of more realistic control methods with possible applications. In the first chapter, we study the stochastic impulse control of a diffusion process. We suppose that the controller minimizes expected discounted costs accumulating as running and controlling cost, respectively. Each control action causes costs which are bounded from below by some positive constant. This makes a continuous control impossible as it would lead to an immediate ruin of the controller. We give a rigorous development of the relevant theory, where our guideline is to establish verification and convergence results under minimal assumptions, without focusing on the existence of solutions to the corresponding (quasi-)variational inequalities. If the impulse control problem can be characterized or approximated by (quasi-)variational inequalities, it remains to solve these equations. In Section 1.2, we solve the stochastic impulse control problem for a one-dimensional diffusion process with constant coefficients and convex running costs. Further, in Section 1.3, we solve a particular multi-dimensional example, where the uncontrolled process is given by an at least two-dimensional Brownian motion and the cost functions are rotationally symmetric. By symmetry, this problem can be reduced to a one-dimensional problem. In the last section of the first chapter, we suggest a new impulse control problem, where the controller is in addition allowed to invest his initial capital into a market consisting of a money market account and a risky asset. The costs which arise upon controlling the diffusion process and upon trading in this market have to be paid out of the controller's bond holdings. The aim of the controller is to minimize the running costs, caused by the abstract diffusion process, without getting ruined. The second chapter is based on a paper which is joint work with Holger Kraft and Frank Seifried. We analyze the portfolio decision of an investor trading in a market where the economy switches randomly between two possible states, a normal state where trading takes place continuously, and an illiquidity state where trading is not allowed at all. We allow for jumps in the market prices at the beginning and at the end of a trading interruption. Section 2.1 provides an explicit representation of the investor's portfolio dynamics in the illiquidity state in an abstract market consisting of two assets. In Section 2.2 we specify this market model and assume that the investor maximizes expected utility from terminal wealth. We establish convergence results, if the maximal number of liquidity breakdowns goes to infinity. In the Markovian framework of Section 2.3, we provide the corresponding Hamilton-Jacobi-Bellman equations and prove a verification result. We apply these results to study the portfolio problem for a logarithmic investor and an investor with a power utility function, respectively. Further, we extend this model to an economy with three regimes. For instance, the third state could model an additional financial crisis where trading is still possible, but the excess return is lower and the volatility is higher than in the normal state.
In some processes for spinning synthetic fibers the filaments are exposed to highly turbulent air flows to achieve a high degree of stretching (elongation). The quality of the resulting filaments, namely thickness and uniformity, is thus determined essentially by the aerodynamic force coming from the turbulent flow. Up to now, there is a gap between the elongation measured in experiments and the elongation obtained by numerical simulations available in the literature.
The main focus of this thesis is the development of an efficient and sufficiently accurate simulation algorithm for the velocity of a turbulent air flow and the application in turbulent spinning processes.
In stochastic turbulence models the velocity is described by an \(\mathbb{R}^3\)-valued random field. Based on an appropriate description of the random field by Marheineke, we have developed an algorithm that fulfills our requirements of efficiency and accuracy. Applying a resulting stochastic aerodynamic drag force on the fibers then allows the simulation of the fiber dynamics modeled by a random partial differential algebraic equation system as well as a quantization of the elongation in a simplified random ordinary differential equation model for turbulent spinning. The numerical results are very promising: whereas the numerical results available in the literature can only predict elongations up to order \(10^4\) we get an order of \(10^5\), which is closer to the elongations of order \(10^6\) measured in experiments.
Multiphase materials combine properties of several materials, which makes them interesting for high-performing components. This thesis considers a certain set of multiphase materials, namely silicon-carbide (SiC) particle-reinforced aluminium (Al) metal matrix composites and their modelling based on stochastic geometry models.
Stochastic modelling can be used for the generation of virtual material samples: Once we have fitted a model to the material statistics, we can obtain independent three-dimensional “samples” of the material under investigation without the need of any actual imaging. Additionally, by changing the model parameters, we can easily simulate a new material composition.
The materials under investigation have a rather complicated microstructure, as the system of SiC particles has many degrees of freedom: Size, shape, orientation and spatial distribution. Based on FIB-SEM images, that yield three-dimensional image data, we extract the SiC particle structure using methods of image analysis. Then we model the SiC particles by anisotropically rescaled cells of a random Laguerre tessellation that was fitted to the shapes of isotropically rescaled particles. We fit a log-normal distribution for the volume distribution of the SiC particles. Additionally, we propose models for the Al grain structure and the Aluminium-Copper (\({Al}_2{Cu}\)) precipitations occurring on the grain boundaries and on SiC-Al phase boundaries.
Finally, we show how we can estimate the parameters of the volume-distribution based on two-dimensional SEM images. This estimation is applied to two samples with different mean SiC particle diameters and to a random section through the model. The stereological estimations are within acceptable agreement with the parameters estimated from three-dimensional image data
as well as with the parameters of the model.
This thesis is devoted to deal with the stochastic optimization problems in various situations with the aid of the Martingale method. Chapter 2 discusses the Martingale method and its applications to the basic optimization problems, which are well addressed in the literature (for example, [15], [23] and [24]). In Chapter 3, we study the problem of maximizing expected utility of real terminal wealth in the presence of an index bond. Chapter 4, which is a modification of the original research paper joint with Korn and Ewald [39], investigates an optimization problem faced by a DC pension fund manager under inflationary risk. Although the problem is addressed in the context of a pension fund, it presents a way of how to deal with the optimization problem, in the case there is a (positive) endowment. In Chapter 5, we turn to a situation where the additional income, other than the income from returns on investment, is gained by supplying labor. Chapter 6 concerns a situation where the market considered is incomplete. A trick of completing an incomplete market is presented there. The general theory which supports the discussion followed is summarized in the first chapter.
Structure and Construction of Instanton Bundles on P3