Kaiserslautern - Fachbereich Mathematik
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Faculty / Organisational entity
In 2006 Jeffrey Achter proved that the distribution of divisor class groups of degree 0 of function fields with a fixed genus and the distribution of eigenspaces in symplectic similitude groups are closely related to each other. Gunter Malle proposed that there should be a similar correspondence between the distribution of class groups of number fields and the distribution of eigenspaces in ceratin matrix groups. Motivated by these results and suggestions we study the distribution of eigenspaces corresponding to the eigenvalue one in some special subgroups of the general linear group over factor rings of rings of integers of number fields and derive some conjectural statements about the distribution of \(p\)-parts of class groups of number fields over a base field \(K_{0}\). Where our main interest lies in the case that \(K_{0}\) contains the \(p\)th roots of unity, because in this situation the \(p\)-parts of class groups seem to behave in an other way like predicted by the popular conjectures of Henri Cohen and Jacques Martinet. In 2010 based on computational data Malle has succeeded in formulating a conjecture in the spirit of Cohen and Martinet for this case. Here using our investigations about the distribution in matrixgroups we generalize the conjecture of Malle to a more abstract level and establish a theoretical backup for these statements.
The construction of number fields with given Galois group fits into the framework of the inverse Galois problem. This problem remains still unsolved, although many partial results have been obtained over the last century.
Shafarevich proved in 1954 that every solvable group is realizable as the Galois group of a number field. Unfortunately, the proof does not provide a method to explicitly find such a field.
This work aims at producing a constructive version of the theorem by solving the following task: given a solvable group $G$ and a $B\in \mathbf N$, construct all normal number fields with Galois group $G$ and absolute discriminant bounded by $B$.
Since a field with solvable Galois group can be realized as a tower of abelian extensions, the main role in our algorithm is played by class field theory, which is the subject of the first part of this work.
The second half is devoted to the study of the relation between the group structure and the field through Galois correspondence.
In particular, we study the existence of obstructions to embedding problems and some criteria to predict the Galois group of an extension.
On the complexity and approximability of optimization problems with Minimum Quantity Constraints
(2020)
During the last couple of years, there has been a variety of publications on the topic of
minimum quantity constraints. In general, a minimum quantity constraint is a lower bound
constraint on an entity of an optimization problem that only has to be fulfilled if the entity is
“used” in the respective solution. For example, if a minimum quantity \(q_e\) is defined on an
edge \(e\) of a flow network, the edge flow on \(e\) may either be \(0\) or at least \(q_e\) units of flow.
Minimum quantity constraints have already been applied to problem classes such as flow, bin
packing, assignment, scheduling and matching problems. A result that is common to all these
problem classes is that in the majority of cases problems with minimum quantity constraints
are NP-hard, even if the problem without minimum quantity constraints but with fixed lower
bounds can be solved in polynomial time. For instance, the maximum flow problem is known
to be solvable in polynomial time, but becomes NP-hard once minimum quantity constraints
are added.
In this thesis we consider flow, bin packing, scheduling and matching problems with minimum
quantity constraints. For each of these problem classes we provide a summary of the
definitions and results that exist to date. In addition, we define new problems by applying
minimum quantity constraints to the maximum-weight b-matching problem and to open
shop scheduling problems. We contribute results to each of the four problem classes: We
show NP-hardness for a variety of problems with minimum quantity constraints that have
not been considered so far. If possible, we restrict NP-hard problems to special cases that
can be solved in polynomial time. In addition, we consider approximability of the problems:
For most problems it turns out that, unless P=NP, there cannot be any polynomial-time
approximation algorithm. Hence, we consider bicriteria approximation algorithms that allow
the constraints of the problem to be violated up to a certain degree. This approach proves to
be very helpful and we provide a polynomial-time bicriteria approximation algorithm for at
least one problem of each of the four problem classes we consider. For problems defined on
graphs, the class of series parallel graphs supports this approach very well.
We end the thesis with a summary of the results and several suggestions for future research
on minimum quantity constraints.
In this thesis, we deal with the finite group of Lie type \(F_4(2^n)\). The aim is to find information on the \(l\)-decomposition numbers of \(F_4(2^n)\) on unipotent blocks for \(l\neq2\) and \(n\in \mathbb{N}\) arbitrary and on the irreducible characters of the Sylow \(2\)-subgroup of \(F_4(2^n)\).
S. M. Goodwin, T. Le, K. Magaard and A. Paolini have found a parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), a Sylow \(2\)-subgroup of \(F_4(q)\), of \(F_4(p^n)\), \(p\) a prime, for the case \(p\neq2\).
We managed to adapt their methods for the parametrization of the irreducible characters of the Sylow \(2\)-subgroup for the case \(p=2\) for the group \(F_4(q)\), \(q=p^n\). This gives a nearly complete parametrization of the irreducible characters of the unipotent subgroup \(U\) of \(F_4(q)\), namely of all irreducible characters of \(U\) arising from so-called abelian cores.
The general strategy we have applied to obtain information about the \(l\)-decomposition numbers on unipotent blocks is to induce characters of the unipotent subgroup \(U\) of \(F_4(q)\) and Harish-Chandra induce projective characters of proper Levi subgroups of \(F_4(q)\) to obtain projective characters of \(F_4(q)\). Via Brauer reciprocity, the multiplicities of the ordinary irreducible unipotent characters in these projective characters give us information on the \(l\)-decomposition numbers of the unipotent characters of \(F_4(q)\).
Sadly, the projective characters of \(F_4(q)\) we obtained were not sufficient to give the shape of the entire decomposition matrix.
It is considered an analytical model of defaultable bond portfolio in terms of its face value process. The face value process dynamically evolves with time and incorporates changes caused by recovery payment on default followed by purchasing of new bonds. The further studies involve properties, distribution and control of the face value process.
Many real life problems have multiple spatial scales. In addition to the multiscale nature one has to take uncertainty into account. In this work we consider multiscale problems with stochastic coefficients.
We combine multiscale methods, e.g., mixed multiscale finite elements or homogenization, which are used for deterministic problems with stochastic methods, such as multi-level Monte Carlo or polynomial chaos methods.
The work is divided into three parts.
In the first two parts we study homogenization with different stochastic methods. Therefore we consider elliptic stationary diffusion equations with stochastic coefficients.
The last part is devoted to the study of mixed multiscale finite elements in combination with multi-level Monte Carlo methods. In the third part we consider multi-phase flow and transport equations.
Symplectic linear quotient singularities belong to the class of symplectic singularities introduced by Beauville in 2000.
They are linear quotients by a group preserving a symplectic form on the vector space and are necessarily singular by a classical theorem of Chevalley-Serre-Shephard-Todd.
We study \(\mathbb Q\)-factorial terminalizations of such quotient singularities, that is, crepant partial resolutions that are allowed to have mild singularities.
The only symplectic linear quotients that can possibly admit a smooth \(\mathbb Q\)-factorial terminalization are by a theorem of Verbitsky those by symplectic reflection groups.
A smooth \(\mathbb Q\)-factorial terminalization is in this context referred to as a symplectic resolution and over the past two decades, there is an ongoing effort to classify exactly which symplectic reflection groups give rise to quotients that admit symplectic resolutions.
We reduce this classification to finitely many, precisely 45, open cases by proving that for almost all quotients by symplectically primitive symplectic reflection groups no such resolution exists.
Concentrating on the groups themselves, we prove that a parabolic subgroup of a symplectic reflection group is generated by symplectic reflections as well.
This is a direct analogue of a theorem of Steinberg for complex reflection groups.
We further study divisor class groups of \(\mathbb Q\)-factorial terminalizations of linear quotients by finite subgroups \(G\) of the special linear group and prove that such a class group is completely controlled by the symplectic reflections - or more generally junior elements - contained in \(G\).
We finally discuss our implementation of an algorithm by Yamagishi for the computation of the Cox ring of a \(\mathbb Q\)-factorial terminalization of a linear quotient in the computer algebra system OSCAR.
We use this algorithm to construct a generating system of the Cox ring corresponding to the quotient by a dihedral group of order \(2d\) with \(d\) odd acting by symplectic reflections.
Although our argument follows the algorithm, the proof does not logically depend on computer calculations.
We are able to derive the \(\mathbb Q\)-factorial terminalization itself from the Cox ring in this case.
This thesis deals with modeling aspects of generalized Newtonian and of non-Newtonian fluids, as well as with development and validation of algorithms used in simulation of such fluids. The main contribution in the modeling part are the introduction and analysis of a new model for the generalized Newtonian fluids, where constitutive equation is of an algebraic form. Distinction between shear and extensional viscosities leads to anisotropic viscosity model. It can be considered as a natural extension of the well known (isotropic viscosity) Carreau model, which deals only with shear viscosity properties of the fluid. The proposed model takes additionally into account extensional viscosity properties. Numerical results show that the anisotropic viscosity model gives much better agreement with experimental observations than the isotropic one. Another contribution of the thesis consists of the development and analysis of robust and reliable algorithms for simulation of generalized Newtonian fluids. For such fluids the momentum equations are strongly coupled through mixed derivatives appearing in the viscous term (unlike the case of Newtonian fluids). It is shown in this thesis, that a careful treatment of those derivatives is essential in deriving robust algorithms. A modification of a standard SIMPLE-like algorithm is given, where all the viscous terms from the momentum equations are discretized in an implicit manner. Moreover, it is shown that a block diagonal preconditioner to the viscous operator is good enough to be used in simulations. Furthermore, different solution techniques, namely projection type methods (consists of solving momentum equations and pressure correction equation) and fully coupled methods (momentum and continuity equations are solved together), are compared. It is shown, that explicit discretization of the mixed derivatives lead to stability problems. Further, analytical estimates of eigenvalue distribution for three different preconditioners, applied to the transformed system arising after discretization and linearization of the momentum and continuity equations, are provided. We propose to apply a block Gauss-Seidel preconditioner to the transformed system. The analysis shows, that this preconditioner is able to cluster eigenvalues around unity independent of the transformation step. It is not the case for other preconditioners applied to the transformed system as discussed in the thesis. The block Gauss-Seidel preconditioner has also shown the best behavior (among all preconditioners discussed in the thesis) in numerical experiments. Further contribution consists of comparison and validation of numerical algorithms applied in simulations of non-Newtonian fluids modeled by time integral constitutive equations. Numerical results from simulations of dilute polymer solutions, described by the integral Oldroyd B model, have shown very good quantitative agreement with the results obtained by differential Oldroyd B counterpart in 4:1 planar contraction domain at low Weissenberg numbers. In this case, the Weissenberg number is changed by changing the relaxation time. However, contrary to the differential Oldroyd B model, the integral one allows to perform stable simulations also in the range of high Weissenberg numbers. Moreover, very good agreement with experimental observations has been achieved. Simulations of concentrated polymer solutions (polystyrene and polybutadiene solutions), modeled by the integral Doi Edwards model, supplemented by chain length fluctuations, have shown very good qualitative agreement with the results obtained by its differential approximation in 4:1:4 constriction domain. Again, much higher Weissenberg numbers can be achieved when the integral model is used. Moreover, very good quantitative results with experimental data of polystyrene solution for the first normal stress difference and shear viscosity defined here as the quotient of a shear stress and a shear rate. Finally, comparison of the two methods used for approximating the time integral constitutive equation, namely Deformation Field Method (DFM) and Backward Lagrangian Particle Method (BLPM), is performed. In BLPM the particle paths are recalculated at every time step of the simulations, what has never been tried before. The results have shown, that in the considered geometries both methods give similar results.
The goal of this work is the development and investigation of an interdisciplinary and in itself closed hydrodynamic approach to the simulation of dilute and dense granular flow. The definition of “granular flow” is a nontrivial task in itself. We say that it is either the flow of grains in a vacuum or in a fluid. A grain is an observable piece of a certain material, for example stone when we mean the flow of sand. Choosing a hydrodynamic view on granular flow, we treat the granular material as a fluid. A hydrodynamic model is developed, that describes the process of flowing granular material. This is done through a system of partial differential equations and algebraic relations. This system is derived by the kinetic theory of granular gases which is characterized by inelastic collisions extended with approaches from soil mechanics. Solutions to the system have to be obtained to understand the process. The equations are so difficult to solve that an analytical solution is out of reach. So approximate solutions must be obtained. Hence the next step is the choice or development of a numerical algorithm to obtain approximate solutions of the model. Common to every problem in numerical simulation, these two steps do not lead to a result without implementation of the algorithm. Hence the author attempts to present this work in the following frame, to participate in and contribute to the three areas Physics, Mathematics and Software implementation and approach the simulation of granular flow in a combined and interdisciplinary way. This work is structured as follows. A continuum model for granular flow which covers the regime of fast dilute flow as well as slow dense flow up to vanishing velocity is presented in the first chapter. This model is strongly nonlinear in the dependence of viscosity and other coefficients on the hydrodynamic variables and it is singular because some coefficients diverge towards the maximum packing fraction of grains. Hence the second difficulty, the challenging task of numerically obtaining approximate solutions for this model is faced in the second chapter. In the third chapter we aim at the validation of both the model and the numerical algorithm through numerical experiments and investigations and show their application to industrial problems. There we focus intensively on the shear flow experiment from the experimental and analytical work of Bocquet et al. which serves well to demonstrate the algorithm, all boundary conditions involved and provides a setting for analytical studies to compare our results. The fourth chapter rounds up the work with the implementation of both the model and the numerical algorithm in a software framework for the solution of complex rheology problems developed as part of this thesis.
The fast development of the financial markets in the last decade has lead to the creation of a variety of innovative interest rate related products that require advanced numerical pricing methods. Examples in this respect are products with a complicated strong path-dependence such as a Target Redemption Note, a Ratchet Cap, a Ladder Swap and others. On the other side, the usage of the standard in the literature one-factor Hull and White (1990) type of short rate models allows only for a perfect correlation between all continuously compounded spot rates or Libor rates and thus are not suited for pricing innovative products depending on several Libor rates such as for example a "steepener" option. One possible solution to this problem deliver the two-factor short rate models and in this thesis we consider a two-factor Hull and White (1990) type of a short rate process derived from the Heath, Jarrow, Morton (1992) framework by limiting the volatility structure of the forward rate process to a deterministic one. In this thesis, we often choose to use a variety of modified (binomial, trinomial and quadrinomial) tree constructions as a main numerical pricing tool due to their flexibility and fast convergence and (when there is no closed-form solution) compare their results with fine grid Monte Carlo simulations. For the purpose of pricing the already mentioned innovative short-rate related products, in this thesis we offer and examine two different lattice construction methods for the two-factor Hull-White type of a short rate process which are able to deal easily both with modeling of the mean-reversion of the underlying process and with the strong path-dependence of the priced options. Additionally, we prove that the so-called rotated lattice construction method overcomes the typical for the existing two-factor tree constructions problem with obtaining negative "risk-neutral probabilities". With a variety of numerical examples, we show that this leads to a stability in the results especially in cases of high volatility parameters and negative correlation between the base factors (which is typically the case in reality). Further, noticing that Chan et al (1992) and Ritchken and Sankarasubramanian (1995) showed that option prices are sensitive to the level of the short rate volatility, we examine the pricing of European and American options where the short rate process has a volatility structure of a Cheyette (1994) type. In this relation, we examine the application of the two offered lattice construction methods and compare their results with the Monte Carlo simulation ones for a variety of examples. Additionally, for the pricing of American options with the Monte Carlo method we expand and implement the simulation algorithm of Longstaff and Schwartz (2000). With a variety of numerical examples we compare again the stability and the convergence of the different lattice construction methods. Dealing with the problems of pricing strongly path-dependent options, we come across the cumulative Parisian barrier option pricing problem. We notice that in their classical form, the cumulative Parisian barrier options have been priced both analytically (in a quasi closed form) and with a tree approximation (based on the Forward Shooting Grid algorithm, see e.g. Hull and White (1993), Kwok and Lau (2001) and others). However, we offer an additional tree construction method which can be seen as a direct binomial tree integration that uses the analytically calculated conditional survival probabilities. The advantage of the offered method is on one side that the conditional survival probabilities are easier to calculate than the closed-form solution itself and on the other side that this tree construction is very flexible in the sense that it allows easy incorporation of additional features such as e.g a forward starting one. The obtained results are better than the Forward Shooting Grid tree ones and are very close to the analytical quasi closed form solution. Finally, we pay our attention to pricing another type of innovative interest rate alike products - namely the Longevity bond - whose coupon payments depend on the survival function of a given cohort. Due to the lack of a market for mortality, for the pricing of the Longevity bonds we develop (following Korn, Natcheva and Zipperer (2006)) a framework that contains principles from both Insurance and Financial mathematic. Further on, we calibrate the existing models for the stochastic mortality dynamics to historical German data and additionally offer new stochastic extensions of the classical (deterministic) models of mortality such as the Gompertz and the Makeham one. Finally, we compare and analyze the results of the application of all considered models to the pricing of a Longevity bond on the longevity of the German males.
In the context of inverse optimization, inverse versions of maximum flow and minimum cost flow problems have thoroughly been investigated. In these network flow problems there are two important problem parameters: flow capacities of the arcs and costs incurred by sending a unit flow on these arcs. Capacity changes for maximum flow problems and cost changes for minimum cost flow problems have been studied under several distance measures such as rectilinear, Chebyshev, and Hamming distances. This thesis also deals with inverse network flow problems and their counterparts tension problems under the aforementioned distance measures. The major goals are to enrich the inverse optimization theory by introducing new inverse network problems that have not yet been treated in the literature, and to extend the well-known combinatorial results of inverse network flows for more general classes of problems with inherent combinatorial properties such as matroid flows on regular matroids and monotropic programming. To accomplish the first objective, the inverse maximum flow problem under Chebyshev norm is analyzed and the capacity inverse minimum cost flow problem, in which only arc capacities are perturbed, is introduced. In this way, it is attempted to close the gap between the capacity perturbing inverse network problems and the cost perturbing ones. The foremost purpose of studying inverse tension problems on networks is to achieve a well-established generalization of the inverse network problems. Since tensions are duals of network flows, carrying the theoretical results of network flows over to tensions follows quite intuitively. Using this intuitive link between network flows and tensions, a generalization to matroid flows and monotropic programs is built gradually up.
On Gyroscopic Stabilization
(2012)
This thesis deals with systems of the form
\(
M\ddot x+D\dot x+Kx=0\;, \; x \in \mathbb R^n\;,
\)
with a positive definite mass matrix \(M\), a symmetric damping matrix \(D\) and a positive definite stiffness
matrix \(K\).
If the equilibrium in the system is unstable, a small disturbance is enough to set the system in motion again. The motion of the system sustains itself, an effect which is called self-excitation or self-induced vibration. The reason behind this effect is the presence of negative damping, which results for example from dry friction.
Negative damping implies that the damping matrix \(D\) is indefinite or negative definite. Throughout our work, we assume \(D\) to be indefinite, and that the system possesses both stable and unstable modes and thus is unstable.
It is now the idea of gyroscopic stabilization to mix the modes of a system with indefinite damping such
that the system is stabilized without introducing further
dissipation. This is done by adding gyroscopic forces \(G\dot x\) with a suitable
skew-symmetric matrix \(G\) to the left-hand side. We call \(G=-G^T\in\mathbb R^{n\times n}\) a gyroscopic stabilizer for
the unstable system, if
\(
M\ddot x+(D+ G)\dot x+Kx=0
\)
is asymptotically stable. We show the existence of \(G\) in space dimensions three and four.
This thesis deals with the numerical study of multiscale problems arising in the modelling of processes of the flow of fluid in plain and porous media. Many of these processes, governed by partial differential equations, are relevant in engineering, industry, and environmental studies. The overall task of modelling and simulating the filtration-related multiscale processes becomes interdisciplinary as it employs physics, mathematics and computer programming to reach its aim. Keeping the challenges in mind, the main focus is to overcome the limitations of accuracy, speed and memory and to develop novel efficient numerical algorithms which could, in part or whole, be utilized by those working in the field of porous media. This work has essentially four parts. A single grid basic algorithm and a corresponding parallel algorithm to solve the macroscopic Navier-Stokes-Brinkmann model is discussed. An upscaling subgrid algorithm is derived and numerically tested for the same model. Moving a step further in the line of multiscale methods, an iterative Mutliscale Finite Volume (iMSFV) method is developed for the Stokes-Darcy system. Additionally, the last part of the thesis deals with ways to incorporate changes occurring at different (meso) scale level. The flow equations are coupled with the Convection-Diffusion-Reaction (CDR) equation, which models the transport and capturing of particle concentrations. By employing the numerical method for the coupled flow and transport problem, we understand the interplay between the flow velocity and filtration.
Many open problems in graph theory aim to verify that a specific class of graphs has a certain property.
One example, which we study extensively in this thesis, is the 3-decomposition conjecture.
It states that every cubic graph can be decomposed into a spanning tree, cycles, and a matching.
Our most noteworthy contributions to this conjecture are a proof that graphs which are star-like satisfy the conjecture and that several small graphs, which we call forbidden subgraphs, cannot be part of minimal counterexamples.
These star-like graphs are a natural generalisation of Hamiltonian graphs in this context and encompass an infinite family of graphs for which the conjecture was not known previously.
Moreover, we use the forbidden subgraphs we determined to deduce that 3-connected cubic graphs of path-width at most 4 satisfy the 3-decomposition conjecture:
we do this by showing that the path-width restriction causes one of these forbidden subgraphs to appear.
In the second part of this thesis, we delve deeper into two steps of the proof that 3-connected cubic graphs of path-width 4 satisfy the conjecture.
These steps involve a significant amount of case distinctions and, as such, are impractical to extend to larger path-width values.
We show how to formalise the techniques used in such a way that they can be implemented and solved algorithmically.
As a result, only the work that is "interesting" to do remains and the many "straightforward" parts can now be done by a computer.
While one step is specific to the 3-decomposition conjecture, we derive a general algorithm for the other.
This algorithm takes a class of graphs \(\mathcal G\) as an input, together with a set of graphs \(\mathcal U\), and a path-width bound \(k\).
It then attempts to answer the following question:
does any graph in \(\mathcal G\) that has path-width at most \(k\) contain a subgraph in \(\mathcal U\)?
We show that this problem is undecidable in general, so our algorithm does not always terminate, but we also provide a general criterion that guarantees termination.
In the final part of this thesis we investigate two connectivity problems on directed graphs.
We prove that verifying the existence of an \(st\)-path in a local certification setting, cannot be achieved with a constant number of bits.
More precisely, we show that a proof labelling scheme needs \(\Theta(\log \Delta)\) many bits, where \(\Delta\) denotes the maximum degree.
Furthermore, we investigate the complexity of the separating by forbidden pairs problem, which asks for the smallest number of arc pairs that are needed such that any \(st\)-path completely contains at least one such pair.
We show that the corresponding decision problem in \(\mathsf{\Sigma_2P}\)-complete.
The thesis at hand deals with the numerical solution of multiscale problems arising in the modeling of processes in fluid and thermo dynamics. Many of these processes, governed by partial differential equations, are relevant in engineering, geoscience, and environmental studies. More precisely, this thesis discusses the efficient numerical computation of effective macroscopic thermal conductivity tensors of high-contrast composite materials. The term "high-contrast" refers to large variations in the conductivities of the constituents of the composite. Additionally, this thesis deals with the numerical solution of Brinkman's equations. This system of equations adequately models viscous flows in (highly) permeable media. It was introduced by Brinkman in 1947 to reduce the deviations between the measurements for flows in such media and the predictions according to Darcy's model.
The present thesis deals with coupled steady state laminar flows of isothermal incompressible viscous Newtonian fluids in plain and in porous media. The flow in the pure fluid region is usually described by the (Navier-)Stokes system of equations. The most popular models for the flow in the porous media are those suggested by Darcy and by Brinkman. Interface conditions, proposed in the mathematical literature for coupling Darcy and Navier-Stokes equations, are shortly reviewed in the thesis. The coupling of Navier-Stokes and Brinkman equations in the literature is based on the so called continuous stress tensor interface conditions. One of the main tasks of this thesis is to investigate another type of interface conditions, namely, the recently suggested stress tensor jump interface conditions. The mathematical models based on these interface conditions were not carefully investigated from the mathematical point of view, and also their validity was a subject of discussions. The considerations within this thesis are a step toward better understanding of these interface conditions. Several aspects of the numerical simulations of such coupled flows are considered: -the choice of proper interface conditions between the plain and porous media -analysis of the well-posedness of the arising systems of partial differential equations; -developing numerical algorithm for the stress tensor jump interface conditions, coupling Navier-Stokes equations in the pure liquid media with the Navier-Stokes-Brinkman equations in the porous media; -validation of the macroscale mathematical models on the base of a comparison with the results from a direct numerical simulation of model representative problems, allowing for grid resolution of the pore level geometry; -developing software and performing numerical simulation of 3-D industrial flows, namely of oil flows through car filters.
We study a multi-scale model for growth of malignant gliomas in the human brain.
Interactions of individual glioma cells with their environment determine the gross tumor shape.
We connect models on different time and length scales to derive a practical description of tumor growth that takes these microscopic interactions into account.
From a simple subcellular model for haptotactic interactions of glioma cells with the white matter we derive a microscopic particle system, which leads to a meso-scale model for the distribution of particles, and finally to a macroscopic description of the cell density.
The main body of this work is dedicated to the development and study of numerical methods adequate for the meso-scale transport model and its transition to the macroscopic limit.
Composite materials are used in many modern tools and engineering applications and
consist of two or more materials that are intermixed. Features like inclusions in a matrix
material are often very small compared to the overall structure. Volume elements that
are characteristic for the microstructure can be simulated and their elastic properties are
then used as a homogeneous material on the macroscopic scale.
Moulinec and Suquet [2] solve the so-called Lippmann-Schwinger equation, a reformulation of the equations of elasticity in periodic homogenization, using truncated
trigonometric polynomials on a tensor product grid as ansatz functions.
In this thesis, we generalize their approach to anisotropic lattices and extend it to
anisotropic translation invariant spaces. We discretize the partial differential equation
on these spaces and prove the convergence rate. The speed of convergence depends on
the smoothness of the coefficients and the regularity of the ansatz space. The spaces of
translates unify the ansatz of Moulinec and Suquet with de la Vallée Poussin means and
periodic Box splines, including the constant finite element discretization of Brisard and
Dormieux [1].
For finely resolved images, sampling on a coarser lattice reduces the computational
effort. We introduce mixing rules as the means to transfer fine-grid information to the
smaller lattice.
Finally, we show the effect of the anisotropic pattern, the space of translates, and the
convergence of the method, and mixing rules on two- and three-dimensional examples.
References
[1] S. Brisard and L. Dormieux. “FFT-based methods for the mechanics of composites:
A general variational framework”. In: Computational Materials Science 49.3 (2010),
pp. 663–671. doi: 10.1016/j.commatsci.2010.06.009.
[2] H. Moulinec and P. Suquet. “A numerical method for computing the overall response
of nonlinear composites with complex microstructure”. In: Computer Methods in
Applied Mechanics and Engineering 157.1-2 (1998), pp. 69–94. doi: 10.1016/s00457825(97)00218-1.
A modular level set algorithm is developed to study the interface and its movement for free moving boundary problems. The algorithm is divided into three basic modules : initialization, propagation and contouring. Initialization is the process of finding the signed distance function from closed objects. We discuss here, a methodology to find an accurate signed distance function from a closed, simply connected surface discretized by triangulation. We compute the signed distance function using the direct method and it is stored efficiently in the neighborhood of the interface by a narrow band level set method. A novel approach is employed to determine the correct sign of the distance function at convex-concave junctions of the surface. The accuracy and convergence of the method with respect to the surface resolution is studied. It is shown that the efficient organization of surface and narrow band data structures enables the solution of large industrial problems. We also compare the accuracy of the signed distance function by direct approach with Fast Marching Method (FMM). It is found that the direct approach is more accurate than FMM. Contouring is performed through a variant of the marching cube algorithm used for the isosurface construction from volumetric data sets. The algorithm is designed to keep foreground and background information consistent, contrary to the neutrality principle followed for surface rendering in computer graphics. The algorithm ensures that the isosurface triangulation is closed, non-degenerate and non-ambiguous. The constructed triangulation has desirable properties required for the generation of good volume meshes. These volume meshes are used in the boundary element method for the study of linear electrostatics. For estimating surface properties like interface position, normal and curvature accurately from a discrete level set function, a method based on higher order weighted least squares is developed. It is found that least squares approach is more accurate than finite difference approximation. Furthermore, the method of least squares requires a more compact stencil than those of finite difference schemes. The accuracy and convergence of the method depends on the surface resolution and the discrete mesh width. This approach is used in propagation for the study of mean curvature flow and bubble dynamics. The advantage of this approach is that the curvature is not discretized explicitly on the grid and is estimated on the interface. The method of constant velocity extension is employed for the propagation of the interface. With least squares approach, the mean curvature flow has considerable reduction in mass loss compared to finite difference techniques. In the bubble dynamics, the modules are used for the study of a bubble under the influence of surface tension forces to validate Young-Laplace law. It is found that the order of curvature estimation plays a crucial role for calculating accurate pressure difference between inside and outside of the bubble. Further, we study the coalescence of two bubbles under surface tension force. The application of these modules to various industrial problems is discussed.
The immiscible lattice BGK method for solving the two-phase incompressible Navier-Stokes equations is analysed in great detail. Equivalent moment analysis and local differential geometry are applied to examine how interface motion is determined and how surface tension effects can be included such that consistency to the two-phase incompressible Navier-Stokes equations can be expected. The results obtained from theoretical analysis are verified by numerical experiments. Since the intrinsic interface tracking scheme of immiscible lattice BGK is found to produce unsatisfactory results in two-dimensional simulations several approaches to improving it are discussed but all of them turn out to yield no substantial improvement. Furthermore, the intrinsic interface tracking scheme of immiscible lattice BGK is found to be closely connected to the well-known conservative volume tracking method. This result suggests to couple the conservative volume tracking method for determining interface motion with the Navier-Stokes solver of immiscible lattice BGK. Applied to simple flow fields, this coupled method yields much better results than plain immiscible lattice BGK.
Numerical Algorithms in Algebraic Geometry with Implementation in Computer Algebra System SINGULAR
(2011)
Polynomial systems arise in many applications: robotics, kinematics, chemical kinetics,
computer vision, truss design, geometric modeling, and many others. Many polynomial
systems have solutions sets, called algebraic varieties, having several irreducible
components. A fundamental problem of the numerical algebraic geometry is to decompose
such an algebraic variety into its irreducible components. The witness point sets are
the natural numerical data structure to encode irreducible algebraic varieties.
Sommese, Verschelde and Wampler represented the irreducible algebraic decomposition of
an affine algebraic variety \(X\) as a union of finite disjoint sets \(\cup_{i=0}^{d}W_i=\cup_{i=0}^{d}\left(\cup_{j=1}^{d_i}W_{ij}\right)\) called numerical irreducible decomposition. The \(W_i\) correspond to the pure i-dimensional components, and the \(W_{ij}\) represent the i-dimensional irreducible components. The numerical irreducible decomposition is implemented in BERTINI.
We modify this concept using partially Gröbner bases, triangular sets, local dimension, and
the so-called zero sum relation. We present in the second chapter the corresponding
algorithms and their implementations in SINGULAR. We give some examples and timings,
which show that the modified algorithms are more efficient if the number of variables is not
too large. For a large number of variables BERTINI is more efficient.
Leykin presented an algorithm to compute the embedded components of an algebraic variety
based on the concept of the deflation of an algebraic variety.
Depending on the modified algorithm mentioned above, we will present in the third chapter an
algorithm and its implementation in SINGULAR to compute the embedded components.
The irreducible decomposition of algebraic varieties allows us to formulate in the fourth
chapter some numerical algebraic algorithms.
In the last chapter we present two SINGULAR libraries. The first library is used to compute
the numerical irreducible decomposition and the embedded components of an algebraic variety.
The second library contains the procedures of the algorithms in the last Chapter to test
inclusion, equality of two algebraic varieties, to compute the degree of a pure i-dimensional
component, and the local dimension.
The thesis studies change points in absolute time for censored survival data with some contributions to the more common analysis of change points with respect to survival time. We first introduce the notions and estimates of survival analysis, in particular the hazard function and censoring mechanisms. Then, we discuss change point models for survival data. In the literature, usually change points with respect to survival time are studied. Typical examples are piecewise constant and piecewise linear hazard functions. For that kind of models, we propose a new algorithm for numerical calculation of maximum likelihood estimates based on a cross entropy approach which in our simulations outperforms the common Nelder-Mead algorithm.
Our original motivation was the study of censored survival data (e.g., after diagnosis of breast cancer) over several decades. We wanted to investigate if the hazard functions differ between various time periods due, e.g., to progress in cancer treatment. This is a change point problem in the spirit of classical change point analysis. Horváth (1998) proposed a suitable change point test based on estimates of the cumulative hazard function. As an alternative, we propose similar tests based on nonparametric estimates of the hazard function. For one class of tests related to kernel probability density estimates, we develop fully the asymptotic theory for the change point tests. For the other class of estimates, which are versions of the Watson-Leadbetter estimate with censoring taken into account and which are related to the Nelson-Aalen estimate, we discuss some steps towards developing the full asymptotic theory. We close by applying the change point tests to simulated and real data, in particular to the breast cancer survival data from the SEER study.
In many medical, financial, industrial, e.t.c. applications of statistics, the model parameters may undergo changes at unknown moment of time. In this thesis, we consider change point analysis in a regression setting for dichotomous responses, i.e. they can be modeled as Bernoulli or 0-1 variables. Applications are widespread including credit scoring in financial statistics and dose-response relations in biometry. The model parameters are estimated using neural network method. We show that the parameter estimates are identifiable up to a given family of transformations and derive the consistency and asymptotic normality of the network parameter estimates using the results in Franke and Neumann Franke Neumann (2000). We use a neural network based likelihood ratio test statistic to detect a change point in a given set of data and derive the limit distribution of the estimator using the results in Gombay and Horvath (1994,1996) under the assumption that the model is properly specified. For the misspecified case, we develop a scaled test statistic for the case of one-dimensional parameter. Through simulation, we show that the sample size, change point location and the size of change influence change point detection. In this work, the maximum likelihood estimation method is used to estimate a change point when it has been detected. Through simulation, we show that change point estimation is influenced by the sample size, change point location and the size of change. We present two methods for determining the change point confidence intervals: Profile log-likelihood ratio and Percentile bootstrap methods. Through simulation, the Percentile bootstrap method is shown to be superior to profile log-likelihood ratio method.
The aim of the thesis is the numerical investigation of saturated, stationary, incompressible Newtonian flow in porous media when inertia is not negligible. We focus our attention to the Navier-Stokes system with two pressures derived by two-scale homogenization. The thesis is subdivided into five Chapters. After the introductory remarks on porous media, filtration laws and upscaling methods, the first chapter is closed by stating the basic terminology and mathematical fundamentals. In Chapter 2, we start by formulating the Navier-Stokes equations on a periodic porous medium. By two-scale expansions of the velocity and pressure, we formally derive the Navier-Stokes system with two pressures. For the sake of completeness, known existence and uniqueness results are repeated and a convergence proof is given. Finally, we consider Stokes and Navier-Stokes systems with two pressures with respect to their relation to Darcy's law. Chapter 3 and Chapter 4 are devoted to the numerical solution of the nonlinear two pressure system. Therefore, we follow two approaches. The first approach which is developed in Chapter 3 is based on a splitting of the Navier-Stokes system with two pressures into micro and macro problems. The splitting is achieved by Taylor expanding the permeability function or by discretely computing the permeability function. The problems to be solved are a series of Stokes and Navier-Stokes problems on the periodicity cell. The Stokes problems are solved by an Uzawa conjugate gradient method. The Navier-Stokes equations are linearized by a least-squares conjugate gradient method, which leads to the solution of a sequence of Stokes problems. The macro problem consists of solving a nonlinear uniformly elliptic equation of second order. The least-squares linearization is applied to the macro problem leading to a sequence of Poisson problems. All equations will be discretized by finite elements. Numerical results are presented at the end of Chapter 3. The second approach presented in Chapter 4 relies on the variational formulation in a certain Hilbert space setting of the Navier-Stokes system with two pressures. The nonlinear problem is again linearized by the least-squares conjugate gradient method. We obtain a sequence of Stokes systems with two pressures. For the latter systems, we propose a fast solution method which relies on pre-computing Stokes systems on the periodicity cell for finite element basis functions acting as right hand sides. Finally, numerical results are discussed. In Chapter 5 we are concerned with modeling and simulation of the pressing section of a paper machine. We state a two-dimensional model of a press nip which takes into account elasticity and flow phenomena. Nonlinear filtration laws are incorporated into the flow model. We present a numerical solution algorithm and the chapter is closed by a numerical investigation of the model with special focus on inertia effects.
Nonlinear diffusion filtering of images using the topological gradient approach to edges detection
(2007)
In this thesis, the problem of nonlinear diffusion filtering of gray-scale images is theoretically and numerically investigated. In the first part of the thesis, we derive the topological asymptotic expansion of the Mumford-Shah like functional. We show that the dominant term of this expansion can be regarded as a criterion to edges detection in an image. In the numerical part, we propose the finite volume discretization for the Catté et al. and the Weickert diffusion filter models. The proposed discretization is based on the integro-interpolation method introduced by Samarskii. The numerical schemes are derived for the case of uniform and nonuniform cell-centered grids of the computational domain \(\Omega \subset \mathbb{R}^2\). In order to generate a nonuniform grid, the adaptive coarsening technique is proposed.
Non-commutative polynomial algebras appear in a wide range of applications, from quantum groups and theoretical physics to linear differential and difference equations. In the thesis, we have developed a framework, unifying many important algebras in the classes of \(G\)- and \(GR\)-algebras and studied their ring-theoretic properties. Let \(A\) be a \(G\)-algebra in \(n\) variables. We establish necessary and sufficient conditions for \(A\) to have a Poincar'e-Birkhoff-Witt (PBW) basis. Further on, we show that besides the existence of a PBW basis, \(A\) shares some other properties with the commutative polynomial ring \(\mathbb{K}[x_1,\ldots,x_n]\). In particular, \(A\) is a Noetherian integral domain of Gel'fand-Kirillov dimension \(n\). Both Krull and global homological dimension of \(A\) are bounded by \(n\); we provide examples of \(G\)-algebras where these inequalities are strict. Finally, we prove that \(A\) is Auslander-regular and a Cohen-Macaulay algebra. In order to perform symbolic computations with modules over \(GR\)-algebras, we generalize Gröbner bases theory, develop and respectively enhance new and existing algorithms. We unite the most fundamental algorithms in a suite of applications, called "Gröbner basics" in the literature. Furthermore, we discuss algorithms appearing in the non-commutative case only, among others two-sided Gröbner bases for bimodules, annihilators of left modules and operations with opposite algebras. An important role in Representation Theory is played by various subalgebras, like the center and the Gel'fand-Zetlin subalgebra. We discuss their properties and their relations to Gröbner bases, and briefly comment some aspects of their computation. We proceed with these subalgebras in the chapter devoted to the algorithmic study of morphisms between \(GR\)-algebras. We provide new results and algorithms for computing the preimage of a left ideal under a morphism of \(GR\)-algebras and show both merits and limitations of several methods that we propose. We use this technique for the computation of the kernel of a morphism, decomposition of a module into central characters and algebraic dependence of pairwise commuting elements. We give an algorithm for computing the set of one-dimensional representations of a \(G\)-algebra \(A\), and prove, moreover, that if the set of finite dimensional representations of \(A\) over a ground field \(K\) is not empty, then the homological dimension of \(A\) equals \(n\). All the algorithms are implemented in a kernel extension Plural of the computer algebra system Singular. We discuss the efficiency of computations and provide a comparison with other computer algebra systems. We propose a collection of benchmarks for testing the performance of algorithms; the comparison of timings shows that our implementation outperforms all of the modern systems with the combination of both broad functionality and fast implementation. In the thesis, there are many new non-trivial examples, and also the solutions to various problems, arising in different fields of mathematics. All of them were obtained with the developed theory and the implementation in Plural, most of them are treated computationally in this thesis for the first time.
In group theory, a big and important family of infinite groups is given by the algebraic groups. These groups and their structures are already well-understood. In representation theory, the study of the unipotent variety in algebraic groups - and by extension the study of the nilpotent variety in the associated Lie algebra - is of particular interest.
Let \( G \) be a connected reductive algebraic group over an algebraically closed field \(\mathbf{k}\), and let \(\operatorname{Lie}(G)\) be its associated Lie algebra. By now, the orbits in the nilpotent and unipotent variety under the action of \(G\) are completely known and can be found for example in a book of Liebeck and Seitz. There exists, however, no uniform description of these orbits that holds in both good and bad characteristic. With this in mind, Lusztig defined a partition of the unipotent variety of \(G\) in 2011. Equivalently, one can consider certain subsets of the nilpotent variety of \(\operatorname{Lie}(G)\) called the nilpotent pieces. This approach appears in the same paper by Lusztig in which he explicitly determines the nilpotent pieces for simple algebraic groups of classical type.
The nilpotent pieces for the exceptional groups of type \(G_2, F_4, E_6, E_7,\) and \(E_8\) in bad characteristic have not yet been determined.
This thesis gives an introduction to the definition of the nilpotent pieces and presents a solution to this problem for groups of type \(G_2, F_4, E_6\), and partly for \(E_7\). The solution relies heavily on computational work which we elaborate on in later chapters.
This thesis focuses on dealing with some new aspects of continuous time portfolio optimization by using the stochastic control method.
First, we extend the Busch-Korn-Seifried model for a large investor by using the Vasicek model for the short rate, and that problem is solved explicitly for two types of intensity functions.
Next, we justify the existence of the constant proportion portfolio insurance (CPPI) strategy in a framework containing a stochastic short rate and a Markov switching parameter. The effect of Vasicek short rate on the CPPI strategy has been studied by Horsky (2012). This part of the thesis extends his research by including a Markov switching parameter, and the generalization is based on the B\"{a}uerle-Rieder investment problem. The explicit solutions are obtained for the portfolio problem without the Money Market Account as well as the portfolio problem with the Money Market Account.
Finally, we apply the method used in Busch-Korn-Seifried investment problem to explicitly solve the portfolio optimization with a stochastic benchmark.
Inflation modeling is a very important tool for conducting an efficient monetary policy. This doctoral thesis reviewed inflation models, in particular the Phillips curve models of inflation dynamics. We focused on a well known and widely used model, the so-called three equation new Keynesian model which is a system of equations consisting of a new Keynesian Phillips curve (NKPC), an investment and saving (IS) curve and an interest rate rule.
We gave a detailed derivation of these equations. The interest rate rule used in this model is normally determined by using a Lagrangian method to solve an optimal control problem constrained by a standard discrete time NKPC which describes the inflation dynamics and an IS curve that represents the output gaps dynamics. In contrast to the real world, this method assumes that the policy makers intervene continuously. This means that the costs resulting from the change in the interest rates are ignored. We showed also that there are approximation errors made, when one log-linearizes non linear equations, by doing the derivation of the standard discrete time NKPC.
We agreed with other researchers as mentioned in this thesis, that errors which result from ignoring such log-linear approximation errors and the costs of altering interest rates by determining interest rate rule, can lead to a suboptimal interest rate rule and hence to non-optimal paths of output gaps and inflation rate.
To overcome such a problem, we proposed a stochastic optimal impulse control method. We formulated the problem as a stochastic optimal impulse control problem by considering the costs of change in interest rates and the approximation error terms. In order to formulate this problem, we first transform the standard discrete time NKPC and the IS curve into their high-frequency versions and hence into their continuous time versions where error terms are described by a zero mean Gaussian white noise with a finite and constant variance. After formulating this problem, we use the quasi-variational inequality approach to solve analytically a special case of the central bank problem, where an inflation rate is supposed to be on target and a central bank has to optimally control output gap dynamics. This method gives an optimal control band in which output gap process has to be maintained and an optimal control strategy, which includes the optimal size of intervention and optimal intervention time, that can be used to keep the process into the optimal control band.
Finally, using a numerical example, we examined the impact of some model parameters on optimal control strategy. The results show that an increase in the output gap volatility as well as in the fixed and proportional costs of the change in interest rate lead to an increase in the width of the optimal control band. In this case, the optimal intervention requires the central bank to wait longer before undertaking another control action.
In this work we present and estimate an explanatory model with a predefined system of explanatory equations, a so called lag dependent model. We present a locally optimal, on blocked neural network based lag estimator and theorems about consistensy. We define the change points in context of lag dependent model, and present a powerfull algorithm for change point detection in high dimensional high dynamical systems. We present a special kind of bootstrap for approximating the distribution of statistics of interest in dependent processes.
Das zentrale Thema dieser Arbeit sind vollständig gekoppelte reflektierte Vorwärts-Rückwärts-Stochastische-Differentialgleichungen (FBSDE). Zunächst wird ein Spezialfall, die teilweise gekoppelten FBSDE, betrachtet und deren Verbindung zur Bewertung Amerikanischer Optionen aufgezeigt. Für die Lösung dieser Gleichung wird Monte-Carlo-Simulation benötigt, daher werden verschiedene Varianzreduktionsmaßnahmen erarbeitet und miteinander verglichen. Im Folgenden wird der allgemeinere Fall der vollständig gekoppelten reflektierten FBSDE behandelt. Es wird gezeigt, wie das Problem der Lösung dieser Gleichungen in ein Optimierungsproblem übertragen werden kann und infolgedessen mit numerischen Methoden aus diesem Bereich der Mathematik bearbeitet werden kann. Abschließend folgen Vergleiche der verschiedenen numerischen Ansätze mit bereits existierenden Verfahren.
Zwei zentrale Probleme der modernen Finanzmathematik sind die Portfolio-Optimierung und die Optionsbewertung. Während es bei der Portfolio-Optimierung darum geht, das Vermögen optimal auf verschiedene Anlagemöglichkeiten zu verteilen, versucht die Optionsbewertung faire Preise von derivativen Finanzinstrumenten zu bestimmen. In dieser Arbeit werden Fragestellungen aus beiden dieser Themenbereiche bearbeitet. Die Arbeit beginnt mit einem Kapitel über Grundlagen, in dem zum Beispiel das Portfolio-Problem von Merton dargestellt und die Black/Scholes-Formel zur Optionsbewertung hergeleitet wird. In Kapitel 2 wird das Portfolio-Problem von Morton und Pliska betrachtet, die in das Merton-Modell fixe Transaktionskosten eingeführt haben. Dabei muß der Investor bei jeder Transaktion einen fixen Anteil vom derzeitigen Vermögen als Kosten abführen. Es wird die asymptotische Approximation dieses Modells von Atkinson und Wilmott vorgestellt und die optimale Portfoliostrategie aus den Marktparametern hergeleitet. Danach werden die tatsächlichen Transaktionskosten abgeschätzt und ein User Guide zur praktischen Anwendung dieses Transaktionskostenmodells angegeben. Zum Schluß wird das Modell numerisch analysiert, indem unter anderem die erwartete Handelszeit und die Güte der Abschätzung der tatsächlichen Transaktionskosten berechnet werden. Ein Portfolio-Problem mit internationalen Märkten wird in Kapitel 3 vorgestellt. Dem Investor steht zusätzlich zu seinem Heimatland noch ein weiteres Land für seine Vermögensanlagen zur Verfügung. Dabei werden die Preisprozesse für die ausländischen Wertpapiere mit einem stochastischen Wechselkurs in die Heimatwährung umgerechnet. In einer statischen Analyse wird unter anderem berechnet, wieviel weniger Vermögen der Investor benötigt, um das gleiche erwartete Endvermögen zu erhalten wie in dem Fall, wenn ihm keine Auslandsanlagen zur Verfügung stehen. Kapitel 4 behandelt drei verschiedene Portfolio-Probleme mit Sprung-Diffusions-Prozessen. Nach der Herleitung eines Verifikationssatzes wird das Problem bei Anlagemöglichkeit in eine Aktie und in ein Geldmarktkonto jeweils für eine konstante und eine stochastische Zinsrate untersucht. Im ersten Fall wird eine implizite Darstellung für den optimalen Portfolioprozeß und eine Bedingung angegeben, unter der diese Darstellung eindeutig lösbar ist. Außerdem wird der optimale Portfolioprozeß für verschiedene Verteilungen für die Sprunghöhe untersucht. Im Falle einer stochastischen Zinsrate kann nur ein Kandidat für den optimalen Lösungsprozeß angeben werden. Dieser hat wieder eine implizite Darstellung. Das letzte Portfolio-Problem ist eine Abwandlung des Modells aus Kapitel 3. Wird dort der Wechselkurs durch eine geometrisch Brownsche Bewegung modelliert, ist er hier ein reiner Sprungprozeß. Es wird wieder der optimale Portfolioprozeß hergeleitet, wobei ein Anteil davon unter Umständen nur numerisch lösbar ist. Eine hinreichende Bedingung für die Lösbarkeit wird angegeben. In Kapitel 5 werden verschiedene Bewertungsansätze für Optionen auf Bondindizes präsentiert. Es wird eine Methode vorgestellt, mit der die Optionen anhand von Marktpreisen bewertet werden können. Für den Fall, daß es nicht genug Marktpreise gibt, wird ein Verfahren angegeben, um den Bondindex realitätsnah zu simulieren und künstliche Marktpreise zu erzeugen. Diese Preise können dann für eine Kalibrierung verwendet werden.
Diversification is one of the main pillars of investment strategies. The prominent 1/N portfolio, which puts equal weight on each asset is, apart from its simplicity, a method which is hard to outperform in realistic settings, as many studies have shown. However, depending on the number of considered assets, this method can lead to very large portfolios. On the other hand, optimization methods like the mean-variance portfolio suffer from estimation errors, which often destroy the theoretical benefits. We investigate the performance of the equal weight portfolio when using fewer assets. For this we explore different naive portfolios, from selecting the best Sharpe ratio assets to exploiting knowledge about correlation structures using clustering methods. The clustering techniques separate the possible assets into non-overlapping clusters and the assets within a cluster are ordered by their Sharpe ratio. Then the best asset of each portfolio is chosen to be a member of the new portfolio with equal weights, the cluster portfolio. We show that this portfolio inherits the advantages of the 1/N portfolio and can even outperform it empirically. For this we use real data and several simulation models. We prove these findings from a statistical point of view using the framework by DeMiguel, Garlappi and Uppal (2009). Moreover, we show the superiority regarding the Sharpe ratio in a setting, where in each cluster the assets are comonotonic. In addition, we recommend the consideration of a diversification-risk ratio to evaluate the performance of different portfolios.
This thesis deals with generalized inverses, multivariate polynomial interpolation and approximation of scattered data. Moreover, it covers the lifting scheme, which basically links the aforementioned topics. For instance, determining filters for the lifting scheme is connected to multivariate polynomial interpolation. More precisely, sets of interpolation sites are required that can be interpolated by a unique polynomial of a certain degree. In this thesis a new class of such sets is introduced and elements from this class are used to construct new and computationally more efficient filters for the lifting scheme.
Furthermore, a method to approximate multidimensional scattered data is introduced which is based on the lifting scheme. A major task in this method is to solve an ordinary linear least squares problem which possesses a special structure. Exploiting this structure yields better approximations and therefore this particular least squares problem is analyzed in detail. This leads to a characterization of special generalized inverses with partially prescribed image spaces.
Gliomas are one of the most common types of primary brain tumors. Among
those, high grade astrocytomas - so-called glioblastoma multiforme - are the
most aggressive type of cancer originating in the brain, leaving patients a median survival time of 15 to 20 months after diagnosis. The invasive behavior
of the tumor leads to considerable difficulties regarding the localization of all
tumor cells, and thus impedes successful therapy. Here, mathematical models
can help to enhance the assessment of the tumor’s extent.
In this thesis, we set up a multiscale model for the evolution of a glioblastoma.
Starting on the microscopic level, we model subcellular binding processes and
velocity dynamics of single cancer cells. From the resulting mesoscopic equation, we derive a macroscopic equation via scaling methods. Combining this
equation with macroscopic descriptions of the tumor environment, a nonlinear
PDE-ODE-system is obtained. We consider several variations of the derived
model, amongst others introducing a new model for therapy by gliadel wafers,
a treatment approach indicated i.a. for recurrent glioblastoma.
We prove global existence of a weak solution to a version of the developed
PDE-ODE-system, containing degenerate diffusion and flux limitation in the
taxis terms of the tumor equation. The nonnegativity and boundedness of all
components of the solution by their biological carrying capacities is shown.
Finally, 2D-simulations are performed, illustrating the influence of different
parts of the model on tumor evolution. The effects of treatment by gliadel
wafers are compared to the therapy outcomes of classical chemotherapy in different settings.
Cell migration is essential for embryogenesis, wound healing, immune surveillance, and
progression of diseases, such as cancer metastasis. For the migration to occur, cellular
structures such as actomyosin cables and cell-substrate adhesion clusters must interact.
As cell trajectories exhibit a random character, so must such interactions. Furthermore,
migration often occurs in a crowded environment, where the collision outcome is deter-
mined by altered regulation of the aforementioned structures. In this work, guided by a
few fundamental attributes of cell motility, we construct a minimal stochastic cell migration
model from ground-up. The resulting model couples a deterministic actomyosin contrac-
tility mechanism with stochastic cell-substrate adhesion kinetics, and yields a well-defined
piecewise deterministic process. The signaling pathways regulating the contractility and
adhesion are considered as well. The model is extended to include cell collectives. Numer-
ical simulations of single cell migration reproduce several experimentally observed results,
including anomalous diffusion, tactic migration, and contact guidance. The simulations
of colliding cells explain the observed outcomes in terms of contact induced modification
of contractility and adhesion dynamics. These explained outcomes include modulation
of collision response and group behavior in the presence of an external signal, as well as
invasive and dispersive migration. Moreover, from the single cell model we deduce a pop-
ulation scale formulation for the migration of non-interacting cells. In this formulation,
the relationships concerning actomyosin contractility and adhesion clusters are maintained.
Thus, we construct a multiscale description of cell migration, whereby single, collective,
and population scale formulations are deduced from the relationships on the subcellular
level in a mathematically consistent way.
Different aspects of geomagnetic field modelling from satellite data are examined in the framework of modern multiscale approximation. The thesis is mostly concerned with wavelet techniques, i.e. multiscale methods based on certain classes of kernel functions which are able to realize a multiscale analysis of the funtion (data) space under consideration. It is thus possible to break up complicated functions like the geomagnetic field, electric current densities or geopotentials into different pieces and study these pieces separately. Based on a general approach to scalar and vectorial multiscale methods, topics include multiscale denoising, crustal field approximation and downward continuation, wavelet-parametrizations of the magnetic field in Mie-representation as well as multiscale-methods for the analysis of time-dependent spherical vector fields. For each subject the necessary theoretical framework is established and numerical applications examine and illustrate the practical aspects.
Multilevel Constructions
(2014)
The thesis consists of the two chapters.
The first chapter is addressed to make a deep investigation of the MLMC method. In particular we take an optimisation view at the estimate. Rather than fixing the number of discretisation points \(n_i\) to be a geometric sequence, we are trying to find an optimal set up for \(n_i\) such that for a fixed error the estimate can be computed within a minimal time.
In the second chapter we propose to enhance the MLMC estimate with the weak extrapolation technique. This technique helps to improve order of a weak convergence of a scheme and as a result reduce CC of an estimate. In particular we study high order weak extrapolation approach, which is know not be inefficient in the standard settings. However, a combination of the MLMC and the weak extrapolation yields an improvement of the MLMC.
This thesis is divided into two parts. Both cope with multi-class image segmentation and utilize
non-smooth optimization algorithms.
The topic of the first part, namely unsupervised segmentation, is the application of clustering
to image pixels. Therefore, we start with an introduction of the biconvex center-based clustering
algorithms c-means and fuzzy c-means, where c denotes the number of classes. We show that
fuzzy c-means can be seen as an approximation of c-means in terms of power means.
Since noise is omnipresent in our image data, these simple clustering models are not suitable
for its segmentation. To this end, we introduce a general and finite dimensional segmentation
model that consists of a data term stemming from the aforementioned clustering models plus a
continuous regularization term. We tackle this optimization model via an alternating minimiza-
tion approach called regularized c-centers (RcC). Thereby, we fix the centers and optimize the
segment membership of the pixels and vice versa. In this general setting, we prove convergence
in the sense of set-valued algorithms using Zangwill’s Theory [172].
Further, we present a segmentation model with a total variation regularizer. While updating
the cluster centers is straightforward for fixed segment memberships of the pixels, updating the
segment membership can be solved iteratively via non-smooth, convex optimization. Thereby,
we do not iterate a convex optimization algorithm until convergence. Instead, we stop as soon as
we have a certain amount of decrease in the objective functional to increase the efficiency. This
algorithm is a particular implementation of RcC providing also the corresponding convergence
theory. Moreover, we show the good performance of our method in various examples such as
simulated 2d images of brain tissue and 3d volumes of two materials, namely a multi-filament
composite superconductor and a carbon fiber reinforced silicon carbide ceramics. Thereby, we
exploit the property of the latter material that two components have no common boundary in
our adapted model.
The second part of the thesis is concerned with supervised segmentation. We leave the area
of center based models and investigate convex approaches related to graph p-Laplacians and
reproducing kernel Hilbert spaces (RKHSs). We study the effect of different weights used to
construct the graph. In practical experiments we show on the one hand image types that
are better segmented by the p-Laplacian model and on the other hand images that are better
segmented by the RKHS-based approach. This is due to the fact that the p-Laplacian approach
provides smoother results, while the RKHS approach provides often more accurate and detailed
segmentations. Finally, we propose a novel combination of both approaches to benefit from the
advantages of both models and study the performance on challenging medical image data.
This thesis is concerned with tropical moduli spaces, which are an important tool in tropical enumerative geometry. The main result is a construction of tropical moduli spaces of rational tropical covers of smooth tropical curves and of tropical lines in smooth tropical surfaces. The construction of a moduli space of tropical curves in a smooth tropical variety is reduced to the case of smooth fans. Furthermore, we point out relations to intersection theory on suitable moduli spaces on algebraic curves.
This thesis deals with the following question. Given a moduli space of coherent sheaves on a projective variety with a fixed Hilbert polynomial, to find a natural construction that replaces the subvariety of the sheaves that are not locally free on their support (we call such sheaves singular) by some variety consisting of sheaves that are locally free on their support. We consider this problem on the example of the coherent sheaves on \(\mathbb P_2\) with Hilbert polynomial 3m+1.
Given a singular coherent sheaf \(\mathcal F\) with singular curve C as its support we replace \(\mathcal F\) by locally free sheaves \(\mathcal E\) supported on a reducible curve \(C_0\cup C_1\), where \(C_0\) is a partial normalization of C and \(C_1\) is an extra curve bearing the degree of \(\mathcal E\). These bundles resemble the bundles considered by Nagaraj and Seshadri. Many properties of the singular 3m+1 sheaves are inherited by the new sheaves we introduce in this thesis (we call them R-bundles). We consider R-bundles as natural replacements of the singular sheaves. R-bundles refine the information about 3m+1 sheaves on \(\mathbb P_2\). Namely, for every isomorphism class of singular 3m+1 sheaves there are \(\mathbb P_1\) many equivalence classes of R-bundles. There is a variety \(\tilde M\) of dimension 10 that may be considered as the space of all the isomorphism classes of the non-singular 3m+1 sheaves on \(\mathbb P_2\) together with all the equivalence classes of all R-bundles. This variety is obtained by blowing up the moduli space of 3m+1 sheaves on \(\mathbb P_2\) along the subvariety of singular sheaves. We modify the definition of a 3m+1 family and obtain a notion of a new family over an arbitrary variety S. In particular 3m+1 families of the non-singular sheaves on \(\mathbb P_2\) are families in this sense. New families over one point are either non-singular 3m+1 sheaves or R-bundles. For every variety S we introduce an equivalence relation on the set of all new families over S. The notion of equivalence for families over one point coincides with isomorphism for non-singular 3m+1 sheaves and with equivalence for R-bundles. We obtain a moduli functor \(\tilde{\mathcal M}:(Sch) \rightarrow (Sets)\) that assigns to every variety S the set of the equivalence classes of the new families over S. There is a natural transformation of functors \(\tilde{\mathcal M}\rightarrow \mathcal M\) that establishes a relation between \(\tilde{\mathcal M}\) and the moduli functor \(\mathcal M\) of the 3m+1 moduli problem on \(\mathbb P_2\). There is also a natural transformation \(\tilde{\mathcal M} \rightarrow Hom(\__ ,\tilde M)\), inducing a bijection \(\tilde{\mathcal M}(pt)\cong \tilde M\), which means that \(\tilde M\) is a coarse moduli space of the moduli problem \(\tilde{\mathcal M}\).
The question of how to model dependence structures between financial assets was revolutionized since the last decade when the copula concept was introduced in financial research. Even though the concept of splitting marginal behavior and dependence structure (described by a copula) of multidimensional distributions already goes back to Sklar (1955) and Hoeffding (1940), there were very little empirical efforts done to check out the potentials of this approach. The aim of this thesis is to figure out the possibilities of copulas for modelling, estimating and validating purposes. Therefore we extend the class of Archimedean Copulas via a transformation rule to new classes and come up with an explicit suggestion covering the Frank and Gumbel family. We introduce a copula based mapping rule leading to joint independence and as results of this mapping we present an easy method of multidimensional chi²-testing and a new estimate for high dimensional parametric distributions functions. Different ways of estimating the tail dependence coefficient, describing the asymptotic probability of joint extremes, are compared and improved. The limitations of elliptical distributions are carried out and a generalized form of them, preserving their applicability, is developed. We state a method to split a (generalized) elliptical distribution into its radial and angular part. This leads to a positive definite robust estimate of the dispersion matrix (here only given as a theoretical outlook). The impact of our findings is stated by modelling and testing the return distributions of stock- and currency portfolios furthermore of oil related commodities- and LME metal baskets. In addition we show the crash stability of real estate based firms and the existence of nonlinear dependence in between the yield curve.
This dissertation deals with the optimization of the web formation in a spunbond process for the production of artificial fabrics. A mathematical model of the process is presented. Based on the model, two kind of attributes to be optimized are considered, those related with the quality of the fabric and those describing the stability of the production process. The problem falls in the multicriteria and decision making framework. The functions involved on the model of the process are non linear, non convex and non differentiable. A strategy in two steps; exploration and continuation, is proposed to approximate numerically the Pareto frontier and alternative methods are proposed to navigate the set and support the decision making process. The proposed strategy is applied to a particular production process and numerical results are presented.
A vehicles fatigue damage is a highly relevant figure in the complete vehicle design process.
Long term observations and statistical experiments help to determine the influence of differnt parts of the vehicle, the driver and the surrounding environment.
This work is focussing on modeling one of the most important influence factors of the environment: road roughness. The quality of the road is highly dependant on several surrounding factors which can be used to create mathematical models.
Such models can be used for the extrapolation of information and an estimation of the environment for statistical studies.
The target quantity we focus on in this work ist the discrete International Roughness Index or discrete IRI. The class of models we use and evaluate is a discriminative classification model called Conditional Random Field.
We develop a suitable model specification and show new variants of stochastic optimizations to train the model efficiently.
The model is also applied to simulated and real world data to show the strengths of our approach.
Since its invention by Sir Allistair Pilkington in 1952, the float glass process has been used to manufacture long thin flat sheets of glass. Today, float glass is very popular due to its high quality and relatively low production costs. When producing thinner glass the main concern is to retain its optical quality, which can be deteriorated during the manufacturing process. The most important stage of this process is the floating part, hence is considered to be responsible for the loss in the optical quality. A series of investigations performed on the finite products showed the existence of many short wave patterns, which strongly affect the optical quality of the glass. Our work is concerned with finding the mechanism for wave development, taking into account all possible factors. In this thesis, we model the floating part of the process by an theoretical study of the stability of two superposed fluids confined between two infinite plates and subjected to a large horizontal temperature gradient. Our approach is to take into account the mixed convection effects (viscous shear and buoyancy), neglecting on the other hand the thermo-capillarity effects due to the length of our domain and the presence of a small stabilizing vertical temperature gradient. Both fluids are treated as Newtonian with constant viscosity. They are immiscible, incompressible, have very different properties and have a free surface between them. The lower fluid is a liquid metal with a very small kinematic viscosity, whereas the upper fluid is less dense. The two fluids move with different velocities: the speed of the upper fluid is imposed, whereas the lower fluid moves as a result of buoyancy effects. We examine the problem by means of small perturbation analysis, and obtain a system of two Orr-Sommerfeld equations coupled with two energy equations, and general interface and boundary conditions. We solve the system analytically in the long- and short- wave limit, by using asymptotic expansions with respect to the wave number. Moreover, we write the system in the form of a general eigenvalue problem and we solve the system numerically by using Chebyshev spectral methods for fluid dynamics. The results (both analytical and numerical) show the existence of the small-amplitude travelling waves, which move with constant velocity for wave numbers in the intermediate range. We show that the stability of the system is ensured in the long wave limit, a fact which is in agreement with the real float glass process. We analyze the stability for a wide range of wave numbers, Reynolds, Weber and Grashof number, and explain the physical implications on the dynamics of the problem. The consequences of the linear stability results are discussed. In reality in the float glass process, the temperature strongly influences the viscosity of both molten metal and hot glass, which will have direct consequences on the stability of the system. We investigate the linear stability of two superposed fluids with temperature dependent viscosities by considering a different model for the viscosity dependence of each fluid. Although, the temperature-viscosity relationships for glass and metal are more complex than those used in our computations, our intention is to emphasize the effects of this dependence on the stability of the system. It is known from the literature that in the case of one fluid, the heat, which causes viscosity to decrease along the domain, usually destabilizes the flow. For the two superposed fluids problem we investigate this behaviour and discuss the consequences of the linear stability in this new case.
We present a numerical scheme to simulate a moving rigid body with arbitrary shape suspended in a rarefied gas micro flows, in view of applications to complex computations of moving structures in micro or vacuum systems. The rarefied gas is simulated by solving the Boltzmann equation using a DSMC particle method. The motion of the rigid body is governed by the Newton-Euler equations, where the force and the torque on the rigid body is computed from the momentum transfer of the gas molecules colliding with the body. The resulting motion of the rigid body affects in turn again the gas flow in the surroundings. This means that a two-way coupling has been modeled. We validate the scheme by performing various numerical experiments in 1-, 2- and 3-dimensional computational domains. We have presented 1-dimensional actuator problem, 2-dimensional cavity driven flow problem, Brownian diffusion of a spherical particle both with translational and rotational motions, and finally thermophoresis on a spherical particles. We compare the numerical results obtained from the numerical simulations with the existing theories in each test examples.
The work consists of two parts.
In the first part an optimization problem of structures of linear elastic material with contact modeled by Robin-type boundary conditions is considered. The structures model textile-like materials and possess certain quasiperiodicity properties. The homogenization method is used to represent the structures by homogeneous elastic bodies and is essential for formulations of the effective stress and Poisson's ratio optimization problems. At the micro-level, the classical one-dimensional Euler-Bernoulli beam model extended with jump conditions at contact interfaces is used. The stress optimization problem is of a PDE-constrained optimization type, and the adjoint approach is exploited. Several numerical results are provided.
In the second part a non-linear model for simulation of textiles is proposed. The yarns are modeled by hyperelastic law and have no bending stiffness. The friction is modeled by the Capstan equation. The model is formulated as a problem with the rate-independent dissipation, and the basic continuity and convexity properties are investigated. The part ends with numerical experiments and a comparison of the results to a real measurement.
This thesis deals with modeling and simulation of district heating networks (DHN) and the mathematical analysis of the proposed DHN model. We provide a detailed derivation of the complete system of governing equations, starting from a brief exposition of the physical quantities of interest, continued with the components to set up a graph based network model accounting for fluxes and coupling conditions, the transport equations for water and thermal energy in pipelines, and the terms representing consumers and producers. On this basis, we perform an analysis of the solvability of the model equations, starting from the scalar advection problem in a single–consumer single–producer network, to a generalized problem suitable to model simple networks without loops. We also derive an abstract formulation of the problem, which serves as a rigorous mathematical model that can be utilized for optimization problems. The theoretical results can be utilized to perform tran- sient simulations of real world DHN and optimize their performance by optimal control, as indicated in a case study.
Model uncertainty is a challenge that is inherent in many applications of mathematical models in various areas, for instance in mathematical finance and stochastic control. Optimization procedures in general take place under a particular model. This model, however, might be misspecified due to statistical estimation errors and incomplete information. In that sense, any specified model must be understood as an approximation of the unknown "true" model. Difficulties arise since a strategy which is optimal under the approximating model might perform rather bad in the true model. A natural way to deal with model uncertainty is to consider worst-case optimization.
The optimization problems that we are interested in are utility maximization problems in continuous-time financial markets. It is well known that drift parameters in such markets are notoriously difficult to estimate. To obtain strategies that are robust with respect to a possible misspecification of the drift we consider a worst-case utility maximization problem with ellipsoidal uncertainty sets for the drift parameter and with a constraint on the strategies that prevents a pure bond investment.
By a dual approach we derive an explicit representation of the optimal strategy and prove a minimax theorem. This enables us to show that the optimal strategy converges to a generalized uniform diversification strategy as uncertainty increases.
To come up with a reasonable uncertainty set, investors can use filtering techniques to estimate the drift of asset returns based on return observations as well as external sources of information, so-called expert opinions. In a Black-Scholes type financial market with a Gaussian drift process we investigate the asymptotic behavior of the filter as the frequency of expert opinions tends to infinity. We derive limit theorems stating that the information obtained from observing the discrete-time expert opinions is asymptotically the same as that from observing a certain diffusion process which can be interpreted as a continuous-time expert. Our convergence results carry over to convergence of the value function in a portfolio optimization problem with logarithmic utility.
Lastly, we use our observations about how expert opinions improve drift estimates for our robust utility maximization problem. We show that our duality approach carries over to a financial market with non-constant drift and time-dependence in the uncertainty set. A time-dependent uncertainty set can then be defined based on a generic filter. We apply this to various investor filtrations and investigate which effect expert opinions have on the robust strategies.
This thesis shows an approach to combine the advantages of MBS tyre models and FEM models for the use in full vehicle simulations. The procedure proposed in this thesis aims to describe a nonlinear structure with a Finite Element approach combined with nonlinear model reduction methods. Unlike most model reduction methods - as the frequently used Craig-Bampton approach - the method of Proper Orthogonal Decomposition (POD) offers a projection basis suitable for nonlinear models. For the linear wave equation, the POD method is studied comparing two different choices of snapshot sets. Set 1 consists of deformation snapshots, and set 2 additionally contains velocities and accelerations. An error analysis proves no convergence guarantee for deformations only. For inclusion of derivatives it yields an error bound diminishing for small time steps. The numerical results show a better behaviour for the derivative snapshot method, as long as the sum of the left-over eigenvalues is significant. For the reduction of nonlinear systems - especially when using commercial software - it is necessary to decouple the reduced surrogate system from the full model. To achieve this, a lookup table approach is presented. It makes use of the preceding computation step with the full model necessary to set up the POD basis (training step). The nonlinear term of inner forces and the stiffness matrix are output and stored in a lookup table for the reduced system. Numerical examples include a nonlinear string in Matlab and an airspring computed in Abaqus. Both examples show that effort reductions of two orders of magnitude are possible within a reasonable error tolerance. The lookup approaches perform faster than the Trajectory Piecewise Linear (TPWL) method and produce comparable errors. Furthermore, the Abaqus example shows the influence of training excitation on the quality of the reduced model.
Mixed Isogeometric Methods for Hodge–Laplace Problems induced by Second-Order Hilbert Complexes
(2024)
Partial differential equations (PDEs) play a crucial role in mathematics and physics to describe numerous physical processes. In numerical computations within the scope of PDE problems, the transition from classical to weak solutions is often meaningful. The latter may not precisely satisfy the original PDE, but they fulfill a weak variational formulation, which, in turn, is suitable for the discretization concept of Finite Elements (FE). A central concept in this context is the
well-posed problem. A class of PDE problems for which not only well-posedness statements but also suitable weak formulations are known are the so-called abstract Hodge–Laplace problems. These can be derived from Hilbert complexes and constitute a central aspect of the Finite Element Exterior Calculus (FEEC).
This thesis addresses the discretization of mixed formulations of Hodge-Laplace problems, focusing on two key aspects. Firstly, we utilize Isogeometric Analysis (IGA) as a specific paradigm for discretization, combining geometric representations with Non-Uniform Rational B-Splines (NURBS) and Finite Element discretizations.
Secondly, we primarily concentrate on mixed formulations exhibiting a saddle-point structure and generated from Hilbert complexes with second-order derivative operators. We go beyond the well-known case of the classical de Rham
complex, considering complexes such as the Hessian or elasticity complex. The BGG (Bernstein–Gelfand–Gelfand) method is employed to define and examine these second-order complexes. The main results include proofs of discrete well-posedness and a priori error estimates for two different discretization approaches. One approach demonstrates, through the introduction of a Lagrange multiplier, how the so-called isogeometric discrete differential forms can be reused.
A second method addresses the question of how standard NURBS basis functions, through a modification of the mixed formulation, can also lead to convergent procedures. Numerical tests and examples, conducted using MATLAB and the open-source software GeoPDEs, illustrate the theoretical findings. Our primary application extends to linear elasticity theory, extensively
discussing mixed methods with and without strong symmetry of the stress tensor.
The work demonstrates the potential of IGA in numerical computations, particularly in the challenging scenario of second-order Hilbert complexes. It also provides insights into how IGA and FEEC can be meaningfully combined, even for non-de Rham complexes.
Methods for scale and orientation invariant analysis of lower dimensional structures in 3d images
(2023)
This thesis is motivated by two groups of scientific disciplines: engineering sciences and mathematics. On the one hand, engineering sciences such as civil engineering want to design sustainable and cost-effective materials with desirable mechanical properties. The material behaviour depends on physical properties and production parameters. Therefore, physical properties are measured experimentally from real samples. In our case, computed tomography (CT) is used to non-destructively gain insight into the materials’ microstructure. This results in large 3d images which yield information on geometric microstructure characteristics. On the other hand, mathematical sciences are interested in designing methods with suitable and guaranteed properties. For example, a natural assumption of human vision is to analyse images regardless of object position, orientation, or scale. This assumption is formalized through the concepts of equivariance and invariance.
In Part I, we deal with oriented structures in materials such as concrete or fiber-reinforced composites. In image processing, knowledge of the local structure orientation can be used for various tasks, e.g. structure enhancement. The idea of using banks of directed filters parameterized in the orientation space is effective in 2d. However, this class of methods is prohibitive in 3d due to the high computational burden of filtering when using a fine discretization of the unit sphere. Hence, we introduce a method for 3d pixel-wise orientation estimation and directional filtering inspired by the idea of adaptive refinement in discretized settings. Furthermore, an operator for distinction between isotropic and anisotropic structures is defined based on our method. Finally, usefulness of the method is shown on 3d CT images in three different tasks on a fiber-reinforced polymer, concrete with cracks, and partially closed foams. Additionally, our method is extended to construct line granulometry and characterize fiber length and orientation distributions in fiber-reinforced polymers produced by either 3d printing or by injection moulding.
In Part II, we investigate how to introduce scale invariance for neural networks by using the Riesz transform. In classical convolutional neural networks, scale invariance is typically achieved by data augmentation. However, when presented with a scale far outside the range covered by the training set, the network may fail to generalize. Here, we introduce the Riesz network, a novel scale invariant neural network. Instead of standard 2d or 3d convolutions for combining spatial information, the Riesz network is based on the Riesz transform, a scale equivariant operator. As a consequence, this network naturally generalizes to unseen or even arbitrary scales in a single forward pass. As an application example, we consider segmenting cracks in CT images of concrete. In this context, 'scale' refers to the crack thickness which may vary strongly even within the same sample. To prove its scale invariance, the Riesz network is trained on one fixed crack width. We then validate its performance in segmenting simulated and real CT images featuring a wide range of crack widths. As an alternative to deep learning models, the Riesz transform is utilized to construct a scale equivariant scattering network, which does not require a lengthy training procedure and works with very few training examples. Mathematical foundations behind this representation are laid out and analyzed. We show that this representation with 4 times less features than the original scattering networks from Mallat performs comparably well on texture classification and gives superior performance when dealing with scales outside the training set distribution.
Over the last decades, mathematical modeling has reached nearly all fields of natural science. The abstraction and reduction to a mathematical model has proven to be a powerful tool to gain a deeper insight into physical and technical processes. The increasing computing power has made numerical simulations available for many industrial applications. In recent years, mathematicians and engineers have turned there attention to model solid materials. New challenges have been found in the simulation of solids and fluid-structure interactions. In this context, it is indispensable to study the dynamics of elastic solids. Elasticity is a main feature of solid bodies while demanding a great deal of the numerical treatment. There exists a multitude of commercial tools to simulate the behavior of elastic solids. Anyhow, the majority of these software packages consider quasi-stationary problems. In the present work, we are interested in highly dynamical problems, e.g. the rotation of a solid. The applicability to free-boundary problems is a further emphasis of our considerations. In the last years, meshless or particle methods have attracted more and more attention. In many fields of numerical simulation these methods are on a par with classical methods or superior to them. In this work, we present the Finite Pointset Method (FPM) which uses a moving least squares particle approximation operator. The application of this method to various industrial problems at the Fraunhofer ITWM has shown that FPM is particularly suitable for highly dynamical problems with free surfaces and strongly changing geometries. Thereby, FPM offers exactly the features that we require for the analysis of the dynamics of solid bodies. In the present work, we provide a numerical scheme capable to simulate the behavior of elastic solids. We present the system of partial differential equations describing the dynamics of elastic solids and show its hyperbolic character. In particular, we focus our attention to the constitutive law for the stress tensor and provide evolution equations for the deviatoric part of the stress tensor in order to circumvent limitations of the classical Hooke's law. Furthermore, we present the basic principle of the Finite Pointset Method. In particular, we provide the concept of upwinding in a given direction as a key ingredient for stabilizing hyperbolic systems. The main part of this work describes the design of a numerical scheme based on FPM and an operator splitting to take the different processes within a solid body into account. Each resulting subsystem is treated separately in an adequate way. Hereby, we introduce the notion of system-inherent directions and dimensional upwinding. Finally, a coupling strategy for the subsystems and results are presented. We close this work with some final conclusions and an outlook on future work.
Understanding human crowd behaviour has been an intriguing topic of interdisciplinary research in recent decades. Modelling of crowd dynamics using differential equations is an indispensable approach to unraveling the various complex dynamics involved in such interacting particle systems. Numerical simulation of pedestrian crowd via these mathematical models allows us to study different realistic scenarios beyond the limitations of studies via controlled experiments.
In this thesis, the main objective is to understand and analyse the dynamics in a domain shared by both pedestrians and moving obstacles. We model pedestrian motion by combining the social force concept with the idea of optimal path computation. This leads to a system of ordinary differential equations governing the dynamics of individual pedestrians via the interaction forces (social forces) between them. Additionally, a non-local force term involving the optimal path and desired velocity governs the pedestrian trajectory. The optimal path computation involves solving a time-independent Eikonal equation, which is coupled to the system of ODEs. A hydrodynamic model is developed from this microscopic model via the mean-field limit.
To consider the interaction with moving obstacles in the domain, we model a set of kinematic equations for the obstacle motion. Two kinds of obstacles are considered - "passive", which move in their predefined trajectories and have only a one-way interaction with pedestrians, and "dynamic", which have a feedback interaction with pedestrians and have their trajectories changing dynamically. The coupled model of pedestrians and obstacles is used to discern pedestrian collision avoidance behaviour in different computational scenarios in a long rectangular domain. We observe that pedestrians avoid collisions through route choice strategies that involve changes in speed and path. We extend this model to consider the interaction between pedestrians and vehicular traffic. We appropriately model the interactions of vehicles, following lane traffic, based on the car-following approach. We observe how the deceleration and braking mechanism of vehicles is executed at pedestrian crossings depending on the right of way on the roads.
As a second objective, we study the disease contagion in moving crowds. We consider the influence of the crowd motion in a complex dynamical environment on the course of infection of pedestrians. A hydrodynamic model for multi-group pedestrian flow is derived from the kinetic equations based on a social force model. It is coupled along with an Eikonal equation to a non-local SEIS contagion model for disease spread. Here, apart from the description of local contacts, the influence of contact times has also been modelled. We observe that the nature of the flow and the geometry of the domain lead to changes in density which affect the contact time and, consequently, the rate of spread of infection.
Finally, the social force model is compared to a variable speed based rational behaviour pedestrian model. We derive a hierarchy of the heuristics-based model from microscopic to macroscopic scales and numerically investigate these models in different density scenarios. Various numerical test cases are considered, including uni- and bi-directional flows and scenarios with and without obstacles. We observe that in low-density scenarios, collision avoidance forces arising from the behavioural heuristics give valid results. Whereas in high-density scenarios, repulsive force terms are essential.
The numerical simulations of all the models are carried out using a mesh-free particle method based on least square approximations. The meshfree numerical framework provides an efficient and elegant way to handle complex geometric situations involving boundaries and stationary or moving obstacles.
This thesis introduces a novel deformation method for computational meshes. It is based on the numerical path following for the equations of nonlinear elasticity. By employing a logarithmic variation of the neo-Hookean hyperelastic material law, the method guarantees that the mesh elements do not become inverted and remain well-shaped. In order to demonstrate the performance of the method, this thesis addresses two areas of active research in isogeometric analysis: volumetric domain parametrization and fluid-structure interaction. The former concerns itself with the construction of a parametrization for a given computational domain provided only a parametrization of the domain’s boundary. The proposed mesh deformation method gives rise to a novel solution approach to this problem. Within it, the domain parametrization is constructed as a deformed configuration of a simplified domain. In order to obtain the simplified domain, the boundary of the target domain is projected in the \(L^2\)-sense onto a coarse NURBS basis. Then, the Coons patch is applied to parametrize the simplified domain. As a range of 2D and 3D examples demonstrates, the mesh deformation approach is able to produce high-quality parametrizations for complex domains where many state-of-the-art methods either fail or become unstable and inefficient. In the context of fluid-structure interaction, the proposed mesh deformation method is applied to robustly update the computational mesh in situations when the fluid domain undergoes large deformations. In comparison to the state-of-the-art mesh update methods, it is able to handle larger deformations and does not result in an eventual reduction of mesh quality. The performance of the method is demonstrated on a classic 2D fluid-structure interaction benchmark reproduced by using an isogeometric partitioned solver with strong coupling.
In this thesis we consider the problem of maximizing the growth rate with proportional and fixed costs in a framework with one bond and one stock, which is modeled as a jump diffusion with compound Poisson jumps. Following the approach from [1], we prove that in this framework it is optimal for an investor to follow a CB-strategy. The boundaries depend only on the parameters of the underlying stock and bond. Now it is natural to ask for the investor who follows a CB-strategy which is given by the stopping times \((\tau_i)_{i\in\mathbb N}\) and impulses \((\eta_i)_{i\in\mathbb N}\) how often he has to rebalance. In other words we want to obtain the limit of the inter trading times
\[
\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n(\tau_{i+1}-\tau_{i}).
\]
We are able to obtain this limit which is given by the expected first exit time of the risky fraction process from some interval under the invariant measure of the Markov chain \((\eta_i)_{i\in\mathbb N}\) using the Ergodic Theorem from von Neumann and Birkhoff. In general, it is difficult to obtain the expectation of the first exit time for the process with jumps. Because of the jump part, when the process crosses the boundaries of the interval an overshoot may occur which makes it difficult to obtain the distribution. Nevertheless we can obtain the first exit time if the process has only negative jumps using scale functions. The main difficulty of this approach is that the scale functions are known only up to their Laplace transforms. In [2] and [3] the closed-form expression for the scale function of the Levy process with phase-type distributed jumps is obtained. Phase-type distributions build a rich class of positive-valued distributions: the exponential, hyperexponential, Erlang, hyper-Erlang and Coxian distributions. Since the scale function is given as a function in a closed form we can differentiate to obtain the expected first exit time using the fluctuation identities explicitly.
[1] Irle, A. and Sass,J.: Optimal portfolio policies under fixed and proportional transaction costs, Advances in Applied Probability 38, 916-942.
[2] Egami, M., Yamazaki, K.: On scale functions of spectrally negative Levy processes with phase-type jumps, working paper, July 3.
[3]Egami, M., Yamazaki, K.: Precautionary measures for credit risk management in jump models, working paper, June 17.
In the first part of this work, called Simple node singularity, are computed matrix factorizations of all isomorphism classes, up to shiftings, of rank one and two, graded, indecomposable maximal Cohen--Macaulay (shortly MCM) modules over the affine cone of the simple node singularity. The subsection 2.2 contains a description of all rank two graded MCM R-modules with stable sheafification on the projective cone of R, by their matrix factorizations. It is given also a general description of such modules, of any rank, over a projective curve of arithmetic genus 1, using their matrix factorizations. The non-locally free rank two MCM modules are computed using an alghorithm presented in the Introduction of this work, that gives a matrix factorization of any extension of two MCM modules over a hypersurface. In the second part, called Fermat surface, are classified all graded, rank two, MCM modules over the affine cone of the Fermat surface. For the classification of the orientable rank two graded MCM R-modules, is used a description of the orientable modules (over normal rings) with the help of codimension two Gorenstein ideals, realized by Herzog and Kühl. It is proven (in section 4), that they have skew symmetric matrix factorizations (over any normal hypersurface ring). For the classification of the non-orientable rank two MCM R-modules, we use a similar idea as in the case of the orientable ones, only that the ideal is not any more Gorenstein.
In this thesis we have discussed the problem of decomposing an integer matrix \(A\) into a weighted sum \(A=\sum_{k \in {\mathcal K}} \alpha_k Y^k\) of 0-1 matrices with the strict consecutive ones property. We have developed algorithms to find decompositions which minimize the decomposition time \(\sum_{k \in {\mathcal K}} \alpha_k\) and the decomposition cardinality \(|\{ k \in {\mathcal K}: \alpha_k > 0\}|\). In the absence of additional constraints on the 0-1 matrices \(Y^k\) we have given an algorithm that finds the minimal decomposition time in \({\mathcal O}(NM)\) time. For the case that the matrices \(Y^k\) are restricted to shape matrices -- a restriction which is important in the application of our results in radiotherapy -- we have given an \({\mathcal O}(NM^2)\) algorithm. This is achieved by solving an integer programming formulation of the problem by a very efficient combinatorial algorithm. In addition, we have shown that the problem of minimizing decomposition cardinality is strongly NP-hard, even for matrices with one row (and thus for the unconstrained as well as the shape matrix decomposition). Our greedy heuristics are based on the results for the decomposition time problem and produce better results than previously published algorithms.
Matrix Compression Methods for the Numerical Solution of Radiative Transfer in Scattering Media
(2002)
Radiative transfer in scattering media is usually described by the radiative transfer equation, an integro-differential equation which describes the propagation of the radiative intensity along a ray. The high dimensionality of the equation leads to a very large number of unknowns when discretizing the equation. This is the major difficulty in its numerical solution. In case of isotropic scattering and diffuse boundaries, the radiative transfer equation can be reformulated into a system of integral equations of the second kind, where the position is the only independent variable. By employing the so-called momentum equation, we derive an integral equation, which is also valid in case of linear anisotropic scattering. This equation is very similar to the equation for the isotropic case: no additional unknowns are introduced and the integral operators involved have very similar mapping properties. The discretization of an integral operator leads to a full matrix. Therefore, due to the large dimension of the matrix in practical applcation, it is not feasible to assemble and store the entire matrix. The so-called matrix compression methods circumvent the assembly of the matrix. Instead, the matrix-vector multiplications needed by iterative solvers are performed only approximately, thus, reducing, the computational complexity tremendously. The kernels of the integral equation describing the radiative transfer are very similar to the kernels of the integral equations occuring in the boundary element method. Therefore, with only slight modifications, the matrix compression methods, developed for the latter are readily applicable to the former. As apposed to the boundary element method, the integral kernels for radiative transfer in absorbing and scattering media involve an exponential decay term. We examine how this decay influences the efficiency of the matrix compression methods. Further, a comparison with the discrete ordinate method shows that discretizing the integral equation may lead to reductions in CPU time and to an improved accuracy especially in case of small absorption and scattering coefficients or if local sources are present.
We encounter directional data in numerous application areas such as astronomy, biology or engineering. Examples include the direction of arrival of cosmic rays, the direction of flight of migratory birds or the orientation of steel fibres in fibre-reinforced concrete.
In part I, we define and apply morphological operators, quantiles and depths for directional data. The morphological operators are defined for \(\mathcal{S}^{d−1}\)-valued images with \(\mathcal{S}^{d−1} = \{x \in \mathbb{R}^d :\sqrt{x^T x} = 1\}\) , \(d \geq 2\). Since an ordered structure is necessary for a definition of these operators, which is not naturally given between vectors, an order is determined with the help of the theory of statistical depth functionals.
This allows for defining the basic operators erosion and dilation as well as morphological (multi-scale) operators for \(\mathcal{S}^{d−1}\)-valued images based on them. The operators introduced are related to their grey value counterparts. Furthermore, quantiles and the "angular Mahalanobis" depth for directional data introduced by Ley
et al. (2014) are extended. The concept of Ley et al. (2014) provides useful geometric properties of the depth contours (such as convexity and rotational equivariance) and a Bahadur-type representation of the quantiles. Their concept is canonical for rotationally symmetric depth contours. However, it also produces rotationally symmetric depth contours when the underlying distribution is not rotationally
symmetric. We solve this lack of flexibility for distributions with elliptical depth contours. The basic idea is to deform the elliptic contours by a diffeomorphic mapping to rotationally symmetric contours, thus reverting to the canonical case in Ley et al. (2014). Our results are confirmed by a Monte Carlo simulation study and applied to the analysis of fibre directions in fibre-reinforced concrete. In Part II, we elaborate interdisciplinary results of statistical analysis and stochastic modelling in civil
engineering. Our statistical analysis of the correlation between production parameters (fibre length, fibre diameter, fibre volume fraction as well as casting method, superplasticiser content and specimen size) of ultra-high performance fibre reinforced concrete and the fibre system (spatial arrangement and orientation of the fibres) provides users with a better understanding of this relatively new composite material. The fibre system is modelled by a Boolean model and the fibre orientation by a one-parameter distribution. In addition, the behaviour under tensile loading is modelled.
In this thesis we outline the Kerner's 3-phase traffic flow theory, which states that the flow of vehicular traffic occur in three phases i.e. free flow, synchronized flow and wide moving jam phases.
A macroscopic 3-phase traffic model of the Aw-Rascle type is derived from the microscopic Speed Adaptation 3-phase traffic model
developed by Kerner and Klenov [J. Phys. A: Math. Gen., 39(2006), pp. 1775-1809 ].
We derive the same macroscopic model from the kinetic traffic flow model of Klar and Wegener [SIAM J. Appl. Math., 60(2000), pp. 1749-1766 ] as well as that of Illner, Klar and Materne [Comm. Math. Sci., 1(2003), pp. 1-12 ].
In the above stated derivations, the 3-phase traffic theory is constituted in the macroscopic model through a relaxation term.
This serves as an incentive to modify the relaxation term of the `switching curve' model of Greenberg,
Klar and Rascle [SIAM J. Appl. Math.,63(2003), pp.818-833 ] to obtain another macroscopic 3-phase traffic model, which is still of the Aw-Rascle type.
By specifying the relaxation term differently we obtain three kinds of models, namely the macroscopic Speed Adaptation,
the Switching Curve and the modified Switching Curve models.
To demonstrate the capability of the derived macroscopic traffic models to reproduce the features of 3-phase traffic theory, we simulate a
multi-lane road that has a bottleneck. We consider a stationary and a moving bottleneck.
The results of the simulations for the three models are compared.
Non–woven materials consist of many thousands of fibres laid down on a conveyor belt
under the influence of a turbulent air stream. To improve industrial processes for the
production of non–woven materials, we develop and explore novel mathematical fibre and
material models.
In Part I of this thesis we improve existing mathematical models describing the fibres on the
belt in the meltspinning process. In contrast to existing models, we include the fibre–fibre
interaction caused by the fibres’ thickness which prevents the intersection of the fibres and,
hence, results in a more accurate mathematical description. We start from a microscopic
characterisation, where each fibre is described by a stochastic functional differential
equation and include the interaction along the whole fibre path, which is described by a
delay term. As many fibres are required for the production of a non–woven material, we
consider the corresponding mean–field equation, which describes the evolution of the fibre
distribution with respect to fibre position and orientation. To analyse the particular case of
large turbulences in the air stream, we develop the diffusion approximation which yields a
distribution describing the fibre position. Considering the convergence to equilibrium on
an analytical level, as well as performing numerical experiments, gives an insight into the
influence of the novel interaction term in the equations.
In Part II of this thesis we model the industrial airlay process, which is a production method
whereby many short fibres build a three–dimensional non–woven material. We focus on
the development of a material model based on original fibre properties, machine data and
micro computer tomography. A possible linking of these models to other simulation tools,
for example virtual tensile tests, is discussed.
The models and methods presented in this thesis promise to further the field in mathematical
modelling and computational simulation of non–woven materials.
In this dissertation we apply financial mathematical modelling to electricity markets. Electricity is different from any other underlying of financial contracts: it is not storable. This means that electrical energy in one time point cannot be transferred to another. As a consequence, power contracts with disjoint delivery time spans basically have a different underlying. The main idea throughout this thesis is exactly this two-dimensionality of time: every electricity contract is not only characterized by its trading time but also by its delivery time.
The basis of this dissertation are four scientific papers corresponding to the Chapters 3 to 6, two of which have already been published in peer-reviewed journals. Throughout this thesis two model classes play a significant role: factor models and structural models. All ideas are applied to or supported by these two model classes. All empirical studies in this dissertation are conducted on electricity price data from the German market and Chapter 4 in particular studies an intraday derivative unique to the German market. Therefore, electricity market design is introduced by the example of Germany in Chapter 1. Subsequently, Chapter 2 introduces the general mathematical theory necessary for modelling electricity prices, such as Lévy processes and the Esscher transform. This chapter is the mathematical basis of the Chapters 3 to 6.
Chapter 3 studies factor models applied to the German day-ahead spot prices. We introduce a qualitative measure for seasonality functions based on three requirements. Furthermore, we introduce a relation of factor models to ARMA processes, which induces a new method to estimate the mean reversion speed.
Chapter 4 conducts a theoretical and empirical study of a pricing method for a new electricity derivative: the German intraday cap and floor futures. We introduce the general theory of derivative pricing and propose a method based on the Hull-White model of interest rate modelling, which is a one-factor model. We include week futures prices to generate a price forward curve (PFC), which is then used instead of a fixed deterministic seasonality function. The idea that we can combine all market prices, and in particular futures prices, to improve the model quality also plays the major role in Chapter 5 and Chapter 6.
In Chapter 5 we develop a Heath-Jarrow-Morton (HJM) framework that models intraday, day-ahead, and futures prices. This approach is based on two stochastic processes motivated by economic interpretations and separates the stochastic dynamics in trading and delivery time. Furthermore, this framework allows for the use of classical day-ahead spot price models such as the ones of Schwartz and Smith (2000), Lucia and Schwartz (2002) and includes many model classes such as structural models and factor models.
Chapter 6 unifies the classical theory of storage and the concept of a risk premium through the introduction of an unobservable intrinsic electricity price. Since all tradable electricity contracts are derivatives of this actual intrinsic price, their prices should all be derived as conditional expectation under the risk-neutral measure. Through the intrinsic electricity price we develop a framework, which also includes many existing modelling approaches, such as the HJM framework of Chapter 5.
Mechanistic disease spread models for different vector borne diseases have been studied from the 19th century. The relevance of mathematical modeling and numerical simulation of disease spread is increasing nowadays. This thesis focuses on the compartmental models of the vector-borne diseases that are also transmitted directly among humans. An example of such an arboviral disease that falls under this category is the Zika Virus disease. The study begins with a compartmental SIRUV model and its mathematical analysis. The non-trivial relationship between the basic reproduction number obtained through two methods have been discussed. The analytical results that are mathematically proven for this model are numerically verified. Another SIRUV model is presented by considering a different formulation of the model parameters and the newly obtained model is shown to be clearly incorporating the dependence on the ratio of mosquito population size to human population size in the disease spread. In order to incorporate the spatial as well as temporal dynamics of the disease spread, a meta-population model based on the SIRUV model was developed. The space domain under consideration are divided into patches which may denote mutually exclusive spatial entities like administrative areas, districts, provinces, cities, states or even countries. The research focused only on the short term movements or commuting behavior of humans across the patches. This is incorportated in the multi-patch meta-population model using a matrix of residence time fractions of humans in each patches. Mathematically simplified analytical results are deduced by which it is shown that, for an exemplary scenario that is numerically studied, the multi-patch model also admits the threshold properties that the single patch SIRUV model holds. The relevance of commuting behavior of humans in the disease spread has been presented using the numerical results from this model. The local and non-local commuting are incorporated into the meta-population model in a numerical example. Later, a PDE model is developed from the multi-patch model.
Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around one or several sites of vaso-occlusion are typical for GBM and hint on poor prognosis of patient survival.
This thesis focuses on studying the establishment and maintenance of these histological patterns specific to GBM with the aim of modeling the microlocal tumor environment under the influence of acidity, tissue anisotropy and hypoxia-induced angiogenesis. This aim is reached with two classes of models: multiscale and multiphase. Each of them features a reaction-diffusion equation (RDE) for the acidity acting as a chemorepellent and inhibitor of growth, coupled in a nonlinear way to a reaction-diffusion-taxis equation (RDTE) for glioma dynamics. The numerical simulations of the resulting systems are able to reproduce pseudopalisade-like patterns. The effect of tumor vascularization on these patterns is studied through a flux-limited model belonging to the multiscale class. Thereby, PDEs of reaction-diffusion-taxis type are deduced for glioma and endothelial cell (EC) densities with flux-limited pH-taxis for the tumor and chemotaxis towards vascular endothelial growth factor (VEGF) for ECs. These, in turn, are coupled to RDEs for acidity and VEGF produced by tumor. The numerical simulations of the obtained system show pattern disruption and transient behavior due to hypoxia-induced angiogenesis. Moreover, comparing two upscaling techniques through numerical simulations, we observe that the macroscopic PDEs obtained via parabolic scaling (directed tissue) are able to reproduce glioma patterns, while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related.
Paper production is a problem with significant importance for the society and it is a challenging topic for scientific investigations. This study is concerned with the simulations of the pressing section of a paper machine. We aim at the development of an advanced mathematical model of the pressing section, which is able to recover the behavior of the fluid flow within the paper felt sandwich obtained in laboratory experiments.
From the modeling point of view the pressing of the paper-felt sandwich is a complex process since one has to deal with the two-phase flow in moving and deformable porous media. To account for the solid deformations, we use developments from the PhD thesis by S. Rief where the elasticity model is stated and discussed in detail. The flow model which accounts for the movement of water within the paper-felt sandwich is described with the help of two flow regimes: single-phase water flow and two-phase air-water flow. The model for the saturated flow is presented by the Darcy's law and the mass conservation. The second regime is described by the Richards' approach together with dynamic capillary effects. The model for the dynamic capillary pressure - saturation relation proposed by Hassanizadeh and Gray is adapted for the needs of the paper manufacturing process.
We have started the development of the flow model with the mathematical modeling in one-dimensional case. The one-dimensional flow model is derived from a two-dimensional one by an averaging procedure in vertical direction. The model is numerically studied and verified in comparison with measurements. Some theoretical investigations are performed to prove the convergence of the discrete solution to the continuous one. For completeness of the studies, the models with the static and dynamic capillary pressure–saturation relations are considered. Existence, compactness and convergence results are obtained for both models.
Then, a two-dimensional model is developed, which accounts for a multilayer computational domain and formation of the fully saturated zones. For discretization we use a non-orthogonal grid resolving the layer interfaces and the multipoint flux approximation O-method. The numerical experiments are carried out for parameters which are typical for the production process. The static and dynamic capillary pressure-saturation relations are tested to evaluate the influence of the dynamic capillary effect.
The last part of the thesis is an investigation of the validity range of the Richards’ assumption for the two-dimensional flow model with the static capillary pressure-saturation relation. Numerical experiments show that the Richards’ assumption is not the best choice in simulating processes in the pressing section.
Magnetoelastic coupling describes the mutual dependence of the elastic and magnetic fields and can be observed in certain types of materials, among which are the so-called "magnetostrictive materials". They belong to the large class of "smart materials", which change their shape, dimensions or material properties under the influence of an external field. The mechanical strain or deformation a material experiences due to an externally applied magnetic field is referred to as magnetostriction; the reciprocal effect, i.e. the change of the magnetization of a body subjected to mechanical stress is called inverse magnetostriction. The coupling of mechanical and electromagnetic fields is particularly observed in "giant magnetostrictive materials", alloys of ferromagnetic materials that can exhibit several thousand times greater magnitudes of magnetostriction (measured as the ratio of the change in length of the material to its original length) than the common magnetostrictive materials. These materials have wide applications areas: They are used as variable-stiffness devices, as sensors and actuators in mechanical systems or as artificial muscles. Possible application fields also include robotics, vibration control, hydraulics and sonar systems.
Although the computational treatment of coupled problems has seen great advances over the last decade, the underlying problem structure is often not fully understood nor taken into account when using black box simulation codes. A thorough analysis of the properties of coupled systems is thus an important task.
The thesis focuses on the mathematical modeling and analysis of the coupling effects in magnetostrictive materials. Under the assumption of linear and reversible material behavior with no magnetic hysteresis effects, a coupled magnetoelastic problem is set up using two different approaches: the magnetic scalar potential and vector potential formulations. On the basis of a minimum energy principle, a system of partial differential equations is derived and analyzed for both approaches. While the scalar potential model involves only stationary elastic and magnetic fields, the model using the magnetic vector potential accounts for different settings such as the eddy current approximation or the full Maxwell system in the frequency domain.
The distinctive feature of this work is the analysis of the obtained coupled magnetoelastic problems with regard to their structure, strong and weak formulations, the corresponding function spaces and the existence and uniqueness of the solutions. We show that the model based on the magnetic scalar potential constitutes a coupled saddle point problem with a penalty term. The main focus in proving the unique solvability of this problem lies on the verification of an inf-sup condition in the continuous and discrete cases. Furthermore, we discuss the impact of the reformulation of the coupled constitutive equations on the structure of the coupled problem and show that in contrast to the scalar potential approach, the vector potential formulation yields a symmetric system of PDEs. The dependence of the problem structure on the chosen formulation of the constitutive equations arises from the distinction of the energy and coenergy terms in the Lagrangian of the system. While certain combinations of the elastic and magnetic variables lead to a coupled magnetoelastic energy function yielding a symmetric problem, the use of their dual variables results in a coupled coenergy function for which a mixed problem is obtained.
The presented models are supplemented with numerical simulations carried out with MATLAB for different examples including a 1D Euler-Bernoulli beam under magnetic influence and a 2D magnetostrictive plate in the state of plane stress. The simulations are based on material data of Terfenol-D, a giant magnetostrictive materials used in many industrial applications.
The central theme in this thesis concerns the development of enhanced methods and algorithms for appraising market and credit risks and their application within the context of standard and more advanced market models. Generally, methods and algorithms for analysing market risk of complex portfolios involve detailed knowledge of option sensitivities, the so-called "Greeks". Based on an analysis of symmetries in financial market models, relations between option sensitivities are obtained, which can be used for the efficient valuation of the Greeks. Mainly, the relations are derived within the Black Scholes model, however, some relations are also valid for more general models, for instance the Heston model. Portfolios are usually influenced by lots of underlyings, so it is necessary to characterise the dependencies of these basic instruments. It is usual to describe such dependencies by correlation matrices. However, estimations of correlation matrices in practice are disturbed by statistical noise and usually have the problem of rank deficiency due to missing data. A fast algorithm is presented which performs a generalized Cholesky decomposition of a perturbed correlation matrix. In contrast to the standard Cholesky algorithm, an advantage of the generalized method is that it works for semi-positive, rank deficient matrices as well. Moreover, it gives an approximative decomposition when the input matrix is indefinite. A comparison with known algorithms with similar features is performed and it turns out, that the new algorithm can be recommended in situations where computation time is the critical issue. The determination of a profit and loss distribution by Fourier inversion of its characteristic function is a powerful tool, but it can break down when the characteristic function is not integrable. In this thesis, methods for Fourier inversion of non-integrable characteristic functions are studied. In this respect, two theorems are obtained which are based on a suitable approximation of the unknown distribution with known density and characteristic function. Further it will be shown, that straightforward Fast Fourier inversion works, when the according density lives on a bounded interval. The above techniques are of crucial importance to determine the profit and loss distribution (P&L) of large portfolios efficiently. The so-called Delta Gamma normal approach has become industrial standard for the estimation of market risk. It is shown, that the performance of the Delta Gamma normal approach can be improved substantially by application of the developed methods. The same optimization procedure also applies to the Delta Gamma Student model. A standard tool for computing the P&L distribution of a loan portfolio is the CreditRisk+ model. Basically, the CreditRisk+ distribution is a discrete distribution which can be computed from its probability generating function. For this a numerically stable method is presented and as an alternative, a new algorithm based on Fourier inversion is proposed. Finally, an extension of the CreditRisk+ model to market risk is developed, which distribution can be obtained efficiently by the presented Fourier inversion methods as well.
In this dissertation we present analysis of macroscopic models for slow dense granular flow. Models are derived from plasticity theory with yield condition and flow rule. Corner stone equations are conservation of mass and conservation of momentum with special constitutive law. Such models are considered in the class of generalised Newtonian fluids, where viscosity depends on the pressure and modulo of the strain-rate tensor. We showed the hyperbolic nature for the evolutionary model in 1D and ill-posed behaviour for 2D and 3D. The steady state equations are always hyperbolic. In the 2D problem we derived a prototype nonlinear backward parabolic equation for the velocity and the similar equation for the shear-rate. Analysis of derived PDE showed the finite blow up time. Blow up time depends on the initial condition. Full 2D and antiplane 3D model were investigated numerically with finite element method. For 2D model we showed the presence of boundary layers. Antiplane 3D model was investigated with the Runge Kutta Discontinuous Galerkin method with mesh addoption. Numerical results confirmed that such a numerical method can be a good choice for the simulations of the slow dense granular flow.
The thesis investigates the phenomenon of hypocoercivity for Langevin-type equations on manifolds via a powerful abstract Hilbert space method. In applications, hypocoercivity experienced by the semigroup can be used to find optimal parameters for the production of nonwoven fleeces. Furthermore, the last chapter introduces a new scaling limit technique: Employing the concept of so-called stratifolds we can show Kuwae-Shioya-Mosco convergence of anisotropic 3D fibre lay-down models to an isotropic 2D model.
Following the ideas presented in Dahlhaus (2000) and Dahlhaus and Sahm (2000) for time series, we build a Whittle-type approximation of the Gaussian likelihood for locally stationary random fields. To achieve this goal, we extend a Szegö-type formula, for the multidimensional and local stationary case and secondly we derived a set of matrix approximations using elements of the spectral theory of stochastic processes. The minimization of the Whittle likelihood leads to the so-called Whittle estimator \(\widehat{\theta}_{T}\). For the sake of simplicity we assume known mean (without loss of generality zero mean), and hence \(\widehat{\theta}_{T}\) estimates the parameter vector of the covariance matrix \(\Sigma_{\theta}\).
We investigate the asymptotic properties of the Whittle estimate, in particular uniform convergence of the likelihoods, and consistency and Gaussianity of the estimator. A main point is a detailed analysis of the asymptotic bias which is considerably more difficult for random fields than for time series. Furthemore, we prove in case of model misspecification that the minimum of our Whittle likelihood still converges, where the limit is the minimum of the Kullback-Leibler information divergence.
Finally, we evaluate the performance of the Whittle estimator through computational simulations and estimation of conditional autoregressive models, and a real data application.
Mrázek et al. [14] proposed a unified approach to curve estimation which combines
localization and regularization. In this thesis we will use their approach to study
some asymptotic properties of local smoothers with regularization. In Particular, we
shall discuss the regularized local least squares (RLLS) estimate with correlated errors
(more precisely with stationary time series errors), and then based on this approach
we will discuss the case when the kernel function is dirac function and compare our
smoother with the spline smoother. Finally, we will do some simulation study.
Mrázek et al. [25] proposed a unified approach to curve estimation which combines localization and regularization. Franke et al. [10] used that approach to discuss the case of the regularized local least-squares (RLLS) estimate. In this thesis we will use the unified approach of Mrázek et al. to study some asymptotic properties of local smoothers with regularization. In particular, we shall discuss the Huber M-estimate and its limiting cases towards the L2 and the L1 cases. For the regularization part, we will use quadratic regularization. Then, we will define a more general class of regularization functions. Finally, we will do a Monte Carlo simulation study to compare different types of estimates.
We investigate the long-term behaviour of diffusions on the non-negative real numbers under killing at some random time. Killing can occur at zero as well as in the interior of the state space. The diffusion follows a stochastic differential equation driven by a Brownian motion. The diffusions we are working with will almost surely be killed. In large parts of this thesis we only assume the drift coefficient to be continuous. Further, we suppose that zero is regular and that infinity is natural. We condition the diffusion on survival up to time t and let t tend to infinity looking for a limiting behaviour.
Linear algebra, together with polynomial arithmetic, is the foundation of computer algebra. The algorithms have improved over the last 20 years, and the current state of the art algorithms for matrix inverse, solution of a linear system and determinants have a theoretical sub-cubic complexity. This thesis presents fast and practical algorithms for some classical problems in linear algebra over number fields and polynomial rings. Here, a number field is a finite extension of the field of rational numbers, and the polynomial rings we considered in this thesis are over finite fields.
One of the key problems of symbolic computation is intermediate coefficient swell: the bit length of intermediate results can grow during the computation compared to those in the input and output. The standard strategy to overcome this is not to compute the number directly but to compute it modulo some other numbers, using either the Chinese remainder theorem (CRT) or a variation of Newton-Hensel lifting. Often, the final step of these algorithms is combined with reconstruction methods such as rational reconstruction to convert the integral result into the rational solution. Here, we present reconstruction methods over number fields with a fast and simple vector-reconstruction algorithm.
The state of the art method for computing the determinant over integers is due to Storjohann. When generalizing his method over number field, we encountered the problem that modules generated by the rows of a matrix over number fields are in general not free, thus Strojohann's method cannot be used directly. Therefore, we have used the theory of pseudo-matrices to overcome this problem. As a sub-problem of this application, we generalized a unimodular certification method for pseudo-matrices: similar to the integer case, we check whether the determinant of the given pseudo matrix is a unit by testing the integrality of the corresponding dual module using higher-order lifting.
One of the main algorithms in linear algebra is the Dixon solver for linear system solving due to Dixon. Traditionally this algorithm is used only for square systems having a unique solution. Here we generalized Dixon algorithm for non-square linear system solving. As the solution is not unique, we have used a basis of the kernel to normalize the solution. The implementation is accompanied by a fast kernel computation algorithm that also extends to compute the reduced-row-echelon form of a matrix over integers and number fields.
The fast implementations for computing the characteristic polynomial and minimal polynomial over number fields use the CRT-based modular approach. Finally, we extended Storjohann's determinant computation algorithm over polynomial ring over finite fields, with its sub-algorithms for reconstructions and unimodular certification. In this case, we face the problem of intermediate degree swell. To avoid this phenomenon, we used higher-order lifting techniques in the unimodular certification algorithm. We have successfully used the half-gcd approach to optimize the rational polynomial reconstruction.
Life insurance companies are asked by the Solvency II regime to retain capital requirements against economically adverse developments. This ensures that they are continuously able to meet their payment obligations towards the policyholders. When relying on an internal model approach, an insurer's solvency capital requirement is defined as the 99.5% value-at-risk of its full loss probability distribution over the coming year. In the introductory part of this thesis, we provide the actuarial modeling tools and risk aggregation methods by which the companies can accomplish the derivations of these forecasts. Since the industry still lacks the computational capacities to fully simulate these distributions, the insurers have to refer to suitable approximation techniques such as the least-squares Monte Carlo (LSMC) method. The key idea of LSMC is to run only a few wisely selected simulations and to process their output further to obtain a risk-dependent proxy function of the loss. We dedicate the first part of this thesis to establishing a theoretical framework of the LSMC method. We start with how LSMC for calculating capital requirements is related to its original use in American option pricing. Then we decompose LSMC into four steps. In the first one, the Monte Carlo simulation setting is defined. The second and third steps serve the calibration and validation of the proxy function, and the fourth step yields the loss distribution forecast by evaluating the proxy model. When guiding through the steps, we address practical challenges and propose an adaptive calibration algorithm. We complete with a slightly disguised real-world application. The second part builds upon the first one by taking up the LSMC framework and diving deeper into its calibration step. After a literature review and a basic recapitulation, various adaptive machine learning approaches relying on least-squares regression and model selection criteria are presented as solutions to the proxy modeling task. The studied approaches range from ordinary and generalized least-squares regression variants over GLM and GAM methods to MARS and kernel regression routines. We justify the combinability of the regression ingredients mathematically and compare their approximation quality in slightly altered real-world experiments. Thereby, we perform sensitivity analyses, discuss numerical stability and run comprehensive out-of-sample tests. The scope of the analyzed regression variants extends to other high-dimensional variable selection applications. Life insurance contracts with early exercise features can be priced by LSMC as well due to their analogies to American options. In the third part of this thesis, equity-linked contracts with American-style surrender options and minimum interest rate guarantees payable upon contract termination are valued. We allow randomness and jumps in the movements of the interest rate, stochastic volatility, stock market and mortality. For the simultaneous valuation of numerous insurance contracts, a hybrid probability measure and an additional regression function are introduced. Furthermore, an efficient seed-related simulation procedure accounting for the forward discretization bias and a validation concept are proposed. An extensive numerical example rounds off the last part.
In this thesis, a new concept to prove Mosco convergence of gradient-type Dirichlet forms within the \(L^2\)-framework of K.~Kuwae and T.~Shioya for varying reference measures is developed.
The goal is, to impose as little additional conditions as possible on the sequence of reference measure \({(\mu_N)}_{N\in \mathbb N}\), apart from weak convergence of measures.
Our approach combines the method of Finite Elements from numerical analysis with the topic of Mosco convergence.
We tackle the problem first on a finite-dimensional substructure of the \(L^2\)-framework, which is induced by finitely many basis functions on the state space \(\mathbb R^d\).
These are shifted and rescaled versions of the archetype tent function \(\chi^{(d)}\).
For \(d=1\) the archetype tent function is given by
\[\chi^{(1)}(x):=\big((-x+1)\land(x+1)\big)\lor 0,\quad x\in\mathbb R.\]
For \(d\geq 2\) we define a natural generalization of \(\chi^{(1)}\) as
\[\chi^{(d)}(x):=\Big(\min_{i,j\in\{1,\dots,d\}}\big(\big\{1+x_i-x_j,1+x_i,1-x_i\big\}\big)\Big)_+,\quad x\in\mathbb R^d.\]
Our strategy to obtain Mosco convergence of
\(\mathcal E^N(u,v)=\int_{\mathbb R^d}\langle\nabla u,\nabla v\rangle_\text{euc}d\mu_N\) towards \(\mathcal E(u,v)=\int_{\mathbb R^d}\langle\nabla u,\nabla v\rangle_\text{euc}d\mu\) for \(N\to\infty\)
involves as a preliminary step to restrict those bilinear forms to arguments \(u,v\) from the vector space spanned by the finite family \(\{\chi^{(d)}(\frac{\,\cdot\,}{r}-\alpha)\) \(|\alpha\in Z\}\) for
a finite index set \(Z\subset\mathbb Z^d\) and a scaling parameter \(r\in(0,\infty)\).
In a diagonal procedure, we consider a zero-sequence of scaling parameters and a sequence of index sets exhausting \(\mathbb Z^d\).
The original problem of Mosco convergence, \(\mathcal E^N\) towards \(\mathcal E\) w.r.t.~arguments \(u,v\) form the respective minimal closed form domains extending the pre-domain \(C_b^1(\mathbb R^d)\), can be solved
by such a diagonal procedure if we ask for some additional conditions on the Radon-Nikodym derivatives \(\rho_N(x)=\frac{d\mu_N(x)}{d x}\), \(N\in\mathbb N\). The essential requirement reads
\[\frac{1}{(2r)^d}\int_{[-r,r]^d}|\rho_N(x)- \rho_N(x+y)|d y \quad \overset{r\to 0}{\longrightarrow} \quad 0 \quad \text{in } L^1(d x),\,
\text{uniformly in } N\in\mathbb N.\]
As an intermediate step towards a setting with an infinite-dimensional state space, we let $E$ be a Suslin space and analyse the Mosco convergence of
\(\mathcal E^N(u,v)=\int_E\int_{\mathbb R^d}\langle\nabla_x u(z,x),\nabla_x v(z,x)\rangle_\text{euc}d\mu_N(z,x)\) with reference measure \(\mu_N\) on \(E\times\mathbb R^d\) for \(N\in\mathbb N\).
The form \(\mathcal E^N\) can be seen as a superposition of gradient-type forms on \(\mathbb R^d\).
Subsequently, we derive an abstract result on Mosco convergence for classical gradient-type Dirichlet forms
\(\mathcal E^N(u,v)=\int_E\langle \nabla u,\nabla v\rangle_Hd\mu_N\) with reference measure \(\mu_N\) on a Suslin space $E$ and a tangential Hilbert space \(H\subseteq E\).
The preceding analysis of superposed gradient-type forms can be used on the component forms \(\mathcal E^{N}_k\), which provide the decomposition
\(\mathcal E^{N}=\sum_k\mathcal E^{N}_k\). The index of the component \(k\) runs over a suitable orthonormal basis of admissible elements in \(H\).
For the asymptotic form \(\mathcal E\) and its component forms \(\mathcal E^k\), we have to assume \(D(\mathcal E)=\bigcap_kD(\mathcal E^k)\) regarding their domains, which is equivalent to the Markov uniqueness of \(\mathcal E\).
The abstract results are tested on an example from statistical mechanics.
Under a scaling limit, tightness of the family of laws for a microscopic dynamical stochastic interface model over \((0,1)^d\) is shown and its asymptotic Dirichlet form identified.
The considered model is based on a sequence of weakly converging Gaussian measures \({(\mu_N)}_{N\in\mathbb N}\) on \(L^2((0,1)^d)\), which are
perturbed by a class of physically relevant non-log-concave densities.
This thesis deals with the simulation of large insurance portfolios. On the one hand, we need to model the contracts' development and the insured collective's structure and dynamics. On the other hand, an important task is the forward projection of the given balance sheet. Questions that are interesting in this context, such as the question of the default probability up to a certain time or the question of whether interest rate promises can be kept in the long term, cannot be answered analytically without strong simplifications. Reasons for this are high dependencies between the insurer's assets and liabilities, interactions between existing and new contracts due to claims on a collective reserve, potential policy features such as a guaranteed interest rate, and individual surrender options of the insured. As a consequence, we need numerical calculations, and especially the volatile financial markets require stochastic simulations. Despite the fact that advances in technology with increasing computing capacities allow for faster computations, a contract-specific simulation of all policies is often an impossible task. This is due to the size and heterogeneity of insurance portfolios, long time horizons, and the number of necessary Monte Carlo simulations. Instead, suitable approximation techniques are required.
In this thesis, we therefore develop compression methods, where the insured collective is grouped into cohorts based on selected contract-related criteria and then only an enormously reduced number of representative contracts needs to be simulated. We also show how to efficiently integrate new contracts into the existing insurance portfolio. Our grouping schemes are flexible, can be applied to any insurance portfolio, and maintain the existing structure of the insured collective. Furthermore, we investigate the efficiency of the compression methods and their quality in approximating the real life insurance portfolio.
For the simulation of the insurance business, we introduce a stochastic asset-liability management (ALM) model. Starting with an initial insurance portfolio, our aim is the forward projection of a given balance sheet structure. We investigate conditions for a long-term stability or stationarity corresponding to the idea of a solid and healthy insurance company. Furthermore, a main result is the proof that our model satisfies the fundamental balance sheet equation at the end of every period, which is in line with the principle of double-entry bookkeeping. We analyze several strategies for investing in the capital market and for financing the due obligations. Motivated by observed weaknesses, we develop new, more sophisticated strategies. In extensive simulation studies, we illustrate the short- and long-term behavior of our ALM model and show impacts of different business forms, the predicted new business, and possible capital market crashes on the profitability and stability of a life insurer.
This work aims at including nonlinear elastic shell models in a multibody framework. We focus our attention to Kirchhoff-Love shells and explore the benefits of an isogeometric approach, the latest development in finite element methods, within a multibody system. Isogeometric analysis extends isoparametric finite elements to more general functions such as B-Splines and Non-Uniform Rational B-Splines (NURBS) and works on exact geometry representations even at the coarsest level of discretizations. Using NURBS as basis functions, high regularity requirements of the shell model, which are difficult to achieve with standard finite elements, are easily fulfilled. A particular advantage is the promise of simplifying the mesh generation step, and mesh refinement is easily performed by eliminating the need for communication with the geometry representation in a Computer-Aided Design (CAD) tool.
Quite often the domain consists of several patches where each patch is parametrized by means of NURBS, and these patches are then glued together by means of continuity conditions. Although the techniques known from domain decomposition can be carried over to this situation, the analysis of shell structures is substantially more involved as additional angle preservation constraints between the patches might arise. In this work, we address this issue in the stationary and transient case and make use of the analogy to constrained mechanical systems with joints and springs as interconnection elements. Starting point of our work is the bending strip method which is a penalty approach that adds extra stiffness to the interface between adjacent patches and which is found to lead to a so-called stiff mechanical system that might suffer from ill-conditioning and severe stepsize restrictions during time integration. As a remedy, an alternative formulation is developed that improves the condition number of the system and removes the penalty parameter dependence. Moreover, we study another alternative formulation with continuity constraints applied to triples of control points at the interface. The approach presented here to tackle stiff systems is quite general and can be applied to all penalty problems fulfilling some regularity requirements.
The numerical examples demonstrate an impressive convergence behavior of the isogeometric approach even for a coarse mesh, while offering substantial savings with respect to the number of degrees of freedom. We show a comparison between the different multipatch approaches and observe that the alternative formulations are well conditioned, independent of any penalty parameter and give the correct results. We also present a technique to couple the isogeometric shells with multibody systems using a pointwise interaction.
In this thesis we develop a shape optimization framework for isogeometric analysis in the optimize first–discretize then setting. For the discretization we use
isogeometric analysis (iga) to solve the state equation, and search optimal designs in a space of admissible b-spline or nurbs combinations. Thus a quite
general class of functions for representing optimal shapes is available. For the
gradient-descent method, the shape derivatives indicate both stopping criteria and search directions and are determined isogeometrically. The numerical treatment requires solvers for partial differential equations and optimization methods, which introduces numerical errors. The tight connection between iga and geometry representation offers new ways of refining the geometry and analysis discretization by the same means. Therefore, our main concern is to develop the optimize first framework for isogeometric shape optimization as ground work for both implementation and an error analysis. Numerical examples show that this ansatz is practical and case studies indicate that it allows local refinement.
In this thesis we present a new method for nonlinear frequency response analysis of mechanical vibrations.
For an efficient spatial discretization of nonlinear partial differential equations of continuum mechanics we employ the concept of isogeometric analysis. Isogeometric finite element methods have already been shown to possess advantages over classical finite element discretizations in terms of exact geometry representation and higher accuracy of numerical approximations using spline functions.
For computing nonlinear frequency response to periodic external excitations, we rely on the well-established harmonic balance method. It expands the solution of the nonlinear ordinary differential equation system resulting from spatial discretization as a truncated Fourier series in the frequency domain.
A fundamental aspect for enabling large-scale and industrial application of the method is model order reduction of the spatial discretization of the equation of motion. Therefore we propose the utilization of a modal projection method enhanced with modal derivatives, providing second-order information. We investigate the concept of modal derivatives theoretically and using computational examples we demonstrate the applicability and accuracy of the reduction method for nonlinear static computations and vibration analysis.
Furthermore, we extend nonlinear vibration analysis to incompressible elasticity using isogeometric mixed finite element methods.
In this thesis, we present the basic concepts of isogeometric analysis (IGA) and we consider Poisson's equation as model problem. Since in IGA the physical domain is parametrized via a geometry function that goes from a parameter domain, e.g. the unit square or unit cube, to the physical one, we present a class of parametrizations that can be viewed as a generalization of polar coordinates, known as the scaled boundary parametrizations (SB-parametrizations). These are easy to construct and are particularly attractive when only the boundary of a domain is available. We then present an IGA approach based on these parametrizations, that we call scaled boundary isogeometric analysis (SB-IGA). The SB-IGA derives the weak form of partial differential equations in a different way from the standard IGA. For the discretization projection
on a finite-dimensional space, we choose in both cases Galerkin's method. Thanks to this technique, we state an equivalence theorem for linear elliptic boundary value problems between the standard IGA, when it makes use of an SB-parametrization,
and the SB-IGA. We solve Poisson's equation with Dirichlet boundary conditions on different geometries and with different SB-parametrizations.
This thesis is devoted to the computational aspects of intersection theory and enumerative geometry. The first results are a Sage package Schubert3 and a Singular library schubert.lib which both provide the key functionality necessary for computations in intersection theory and enumerative geometry. In particular, we describe an alternative method for computations in Schubert calculus via equivariant intersection theory. More concretely, we propose an explicit formula for computing the degree of Fano schemes of linear subspaces on hypersurfaces. As a special case, we also obtain an explicit formula for computing the number of linear subspaces on a general hypersurface when this number is finite. This leads to a much better performance than classical Schubert calculus.
Another result of this thesis is related to the computation of Gromov-Witten invariants. The most powerful method for computing Gromov-Witten invariants is the localization of moduli spaces of stable maps. This method was introduced by Kontsevich in 1995. It allows us to compute Gromov-Witten invariants via Bott's formula. As an insightful application, we computed the numbers of rational curves on general complete intersection Calabi-Yau threefolds in projective spaces up to degree six. The results are all in agreement with predictions made from mirror symmetry.
Intersection Theory on Tropical Toric Varieties and Compactifications of Tropical Parameter Spaces
(2011)
We study toric varieties over the tropical semifield. We define tropical cycles inside these toric varieties and extend the stable intersection of tropical cycles in R^n to these toric varieties. In particular, we show that every tropical cycle can be degenerated into a sum of torus-invariant cycles. This allows us to tropicalize algebraic cycles of toric varieties over an algebraically closed field with non-Archimedean valuation. We see that the tropicalization map is a homomorphism on cycles and an isomorphism on cycle classes. Furthermore, we can use projective toric varieties to compactify known tropical varieties and study their combinatorics. We do this for the tropical Grassmannian in the Plücker embedding and compactify the tropical parameter space of rational degree d curves in tropical projective space using Chow quotients of the tropical Grassmannian.
This thesis is concerned with interest rate modeling by means of the potential approach. The contribution of this work is twofold. First, by making use of the potential approach and the theory of affine Markov processes, we develop a general class of rational models to the term structure of interest rates which we refer to as "the affine rational potential model". These models feature positive interest rates and analytical pricing formulae for zero-coupon bonds, caps, swaptions, and European currency options. We present some concrete models to illustrate the scope of the affine rational potential model and calibrate a model specification to real-world market data. Second, we develop a general family of "multi-curve potential models" for post-crisis interest rates. Our models feature positive stochastic basis spreads, positive term structures, and analytic pricing formulae for interest rate derivatives. This modeling framework is also flexible enough to accommodate negative interest rates and positive basis spreads.
Since the early days of representation theory of finite groups in the 19th century, it was known that complex linear representations of finite groups live over number fields, that is, over finite extensions of the field of rational numbers.
While the related question of integrality of representations was answered negatively by the work of Cliff, Ritter and Weiss as well as by Serre and Feit, it was not known how to decide integrality of a given representation.
In this thesis we show that there exists an algorithm that given a representation of a finite group over a number field decides whether this representation can be made integral.
Moreover, we provide theoretical and numerical evidence for a conjecture, which predicts the existence of splitting fields of irreducible characters with integrality properties.
In the first part, we describe two algorithms for the pseudo-Hermite normal form, which is crucial when handling modules over ring of integers.
Using a newly developed computational model for ideal and element arithmetic in number fields, we show that our pseudo-Hermite normal form algorithms have polynomial running time.
Furthermore, we address a range of algorithmic questions related to orders and lattices over Dedekind domains, including computation of genera, testing local isomorphism, computation of various homomorphism rings and computation of Solomon zeta functions.
In the second part we turn to the integrality of representations of finite groups and show that an important ingredient is a thorough understanding of the reduction of lattices at almost all prime ideals.
By employing class field theory and tools from representation theory we solve this problem and eventually describe an algorithm for testing integrality.
After running the algorithm on a large set of examples we are led to a conjecture on the existence of integral and nonintegral splitting fields of characters.
By extending techniques of Serre we prove the conjecture for characters with rational character field and Schur index two.
Diese Dissertation besteht aus zwei aktuellen Themen im Bereich Finanzmathematik, die voneinander unabhängig sind.
Beim ersten Thema, "Flexible Algorithmen zur Bewertung komplexer Optionen mit mehreren Eigenschaften mittels der funktionalen Programmiersprache Haskell", handelt es sich um ein interdisziplinäres Projekt, in dem eine wissenschaftliche Brücke zwischen der Optionsbewertung und der funktionalen Programmierung geschlagen wurde.
Im diesem Projekt wurde eine funktionale Bibliothek zur Konstruktion von Optionen
entworfen, in dem es eine Reihe von grundlegenden Konstruktoren gibt, mit denen
man verschiedene Optionen kombinieren kann. Im Rahmen der funktionalen Bibliothek
wurde ein allgemeiner Algorithmus entwickelt, durch den die aus den Konstruktoren
kombinierten Optionen bewertet werden können.
Der mathematische Aspekt des Projekts besteht in der Entwicklung eines neuen Konzeptes zur Bewertung der Optionen. Dieses Konzept basiert auf dem Binomialmodell, welches in den letzten Jahren eine weite Verbreitung im Forschungsgebiet der Optionsbewertung fand. Der kerne Algorithmus des Konzeptes ist eine Kombination von mehreren
sorgfältig ausgewählten numerischen Methoden in Bezug auf den Binomialbaum. Diese
Kombination ist nicht trivial, sondern entwikelt sich nach bestimmten Regeln und ist eng mit den grundlegenden Konstruktoren verknüpft.
Ein wichtiger Charakterzug des Projekts ist die funktionale Denkweise. D. h. der Algorithmus ließ sich mithilfe einer funktionalen Programmiersprache formulieren. In unserem Projekt wurde Haskell verwendet.
Das zweite Thema, Monte-Carlo-Simulation des Deltas und (Cross-)Gammas von
Bermuda-Swaptions im LIBOR-Marktmodell, bezieht sich auf ein zentrales Problem der
Finanzmathematik, nämlich die Bestimmung der Risikoparameter komplexer Zinsderivate.
In dieser Arbeit wurde die numerische Berechnung des Delta-Vektors einer Bermuda-
Swaption ausführlich untersucht und die neue Herausforderung, die Gamma-Matrix einer Bermuda-Swaption exakt simulieren, erfolgreich gemeistert. Die beiden Risikoparameter spielen bei Handelsstrategien in Form des Delta-Hedgings und Gamma-Hedgings eine entscheidende Rolle. Das zugrunde liegende Zinsstrukturmodell ist das LIBORMarktmodell, welches in den letzten Jahren eine auffällige Entwicklung in der Finanzmathematik gemacht hat. Bei der Simulation und Anwendung des LIBOR-Marktmodells fällt die Monte-Carlo-Simulation ins Gewicht.
Für die Berechung des Delta-Vektors einer Bermuda-Swaption wurden drei klassische und drei von uns entwickelte numerische Methoden vorgestellt und gegenübergestellt, welche fast alle vorhandenen Arten der Monte-Carlo-Simulation zur Berechnung des Delta-Vektors einer Bermuda-Swaption enthalten.
Darüber hinaus gibt es in der Arbeit noch zwei neu entwickelte Methoden, um die Gamma-Matrix einer Bermuda-Swaption exakt zu berechnen, was völlig neu im Forschungsgebiet der Computational-Finance ist. Eine ist die modifizierte Finite-Differenzen-Methode. Die andere ist die reine Pathwise-Methode, die auf pfadweiser Differentialrechnung basiert und einem robusten und erwartungstreuen Simulationsverfahren entspricht.
In this thesis diverse problems concerning inflation-linked products are dealt with. To start with, two models for inflation are presented, including a geometric Brownian motion for consumer price index itself and an extended Vasicek model for inflation rate. For both suggested models the pricing formulas of inflation-linked products are derived using the risk-neutral valuation techniques. As a result Black and Scholes type closed form solutions for a call option on inflation index for a Brownian motion model and inflation evolution for an extended Vasicek model as well as for an inflation-linked bond are calculated. These results have been already presented in Korn and Kruse (2004) [17]. In addition to these inflation-linked products, for the both inflation models the pricing formulas of a European put option on inflation, an inflation cap and floor, an inflation swap and an inflation swaption are derived. Consequently, basing on the derived pricing formulas and assuming the geometric Brownian motion process for an inflation index, different continuous-time portfolio problems as well as hedging problems are studied using the martingale techniques as well as stochastic optimal control methods. These utility optimization problems are continuous-time portfolio problems in different financial market setups and in addition with a positive lower bound constraint on the final wealth of the investor. When one summarizes all the optimization problems studied in this work, one will have the complete picture of the inflation-linked market and both counterparts of market-participants, sellers as well as buyers of inflation-linked financial products. One of the interesting results worth mentioning here is naturally the fact that a regular risk-averse investor would like to sell and not buy inflation-linked products due to the high price of inflation-linked bonds for example and an underperformance of inflation-linked bonds compared to the conventional risk-free bonds. The relevance of this observation is proved by investigating a simple optimization problem for the extended Vasicek process, where as a result we still have an underperforming inflation-linked bond compared to the conventional bond. This situation does not change, when one switches to an optimization of expected utility from the purchasing power, because in its nature it is only a change of measure, where we have a different deflator. The negativity of the optimal portfolio process for a normal investor is in itself an interesting aspect, but it does not affect the optimality of handling inflation-linked products compared to the situation not including these products into investment portfolio. In the following, hedging problems are considered as a modeling of the other half of inflation market that is inflation-linked products buyers. Natural buyers of these inflation-linked products are obviously institutions that have payment obligations in the future that are inflation connected. That is why we consider problems of hedging inflation-indexed payment obligations with different financial assets. The role of inflation-linked products in the hedging portfolio is shown to be very important by analyzing two alternative optimal hedging strategies, where in the first one an investor is allowed to trade as inflation-linked bond and in the second one he is not allowed to include an inflation-linked bond into his hedging portfolio. Technically this is done by restricting our original financial market, which is made of a conventional bond, inflation index and a stock correlated with inflation index, to the one, where an inflation index is excluded. As a whole, this thesis presents a wide view on inflation-linked products: inflation modeling, pricing aspects of inflation-linked products, various continuous-time portfolio problems with inflation-linked products as well as hedging of inflation-related payment obligations.
Index Insurance for Farmers
(2021)
In this thesis we focus on weather index insurance for agriculture risk. Even though such an index insurance is easily applicable and reduces information asymmetries, the demand for it is quite low. This is in particular due to the basis risk and the lack of knowledge about it’s effectiveness. The basis risk is the difference between the index insurance payout and the actual loss of the insured. We evaluate the performance of weather index insurance in different contexts, because proper knowledge about index insurance will help to use it as a successful alternative for traditional crop insurance. In addition to that, we also propose and discuss methods to reduce the basis risk.
We also analyze the performance of an agriculture loan which is interlinked with a weather index insurance. We show that an index insurance with actuarial fair or subsidized premium helps to reduce the loan default probability. While we first consider an index insurance with a commonly used linear payout function for this analysis, we later design an index insurance payout function which maximizes the expected utility of the insured. Then we show that, an index insurance with that optimal payout function is more appropriate for bundling with an agriculture loan. The optimal payout function also helps to reduce the basis risk. In addition, we show that a lender who issues agriculture loans can be better off by purchasing a weather index insurance in some circumstances.
We investigate the market equilibrium for weather index insurance by assuming risk averse farmers and a risk averse insurer. When we consider two groups of farmers with different risks, we show that the low risk group subsidizes the high risk group when both should pay the same premium for the index insurance. Further, according to the analysis of an index insurance in an informal risk sharing environment, we observe that the demand of the index insurance can be increased by selling it to a group of farmers who informally share the risk based on the insurance payout, because it reduces the adverse effect of the basis risk. Besides of that we analyze the combination of an index insurance with a gap insurance. Such a combination can increase the demand and reduce the basis risk of the index insurance if we choose the correct levels of premium and of gap insurance cover. Moreover our work shows that index insurance can be a good alternative to proportional and excess loss reinsurance when it is issued at a low enough price.
Laser-induced interstitial thermotherapy (LITT) is a minimally invasive procedure to destroy liver
tumors through thermal ablation. Mathematical models are the basis for computer simulations
of LITT, which support the practitioner in planning and monitoring the therapy.
In this thesis, we propose three potential extensions of an established mathematical model of
LITT, which is based on two nonlinearly coupled partial differential equations (PDEs) modeling
the distribution of the temperature and the laser radiation in the liver.
First, we introduce the Cattaneo–LITT model for delayed heat transfer in this context, prove its
well-posedness and study the effect of an inherent delay parameter numerically.
Second, we model the influence of large blood vessels in the heat-transfer model by means
of a spatially varying blood-perfusion rate. This parameter is unknown at the beginning of
each therapy because it depends on the individual patient and the placement of the LITT
applicator relative to the liver. We propose a PDE-constrained optimal-control problem for the
identification of the blood-perfusion rate, prove the existence of an optimal control and prove
necessary first-order optimality conditions. Furthermore, we introduce a numerical example
based on which we demonstrate the algorithmic solution of this problem.
Third, we propose a reformulation of the well-known PN model hierarchy with Marshak
boundary conditions as a coupled system of second-order PDEs to approximate the radiative-transfer
equation. The new model hierarchy is derived in a general context and is applicable
to a wide range of applications other than LITT. It can be generated in an automated way by
means of algebraic transformations and allows the solution with standard finite-element tools.
We validate our formulation in a general context by means of various numerical experiments.
Finally, we investigate the coupling of this new model hierarchy with the LITT model numerically.
Emission trading systems (ETS) represent a widely used instrument to control greenhouse
gas emissions, while minimizing reduction costs. In an ETS, the desired amount of emissions in
a predefined time period is fixed in advance; corresponding to this amount, tradeable allowances
are handed out or auctioned to companies which underlie the system. Emissions which are not
covered by an allowance are subject to a penalty at the end of the time period.
Emissions depend on non-deterministic parameters such as weather and the state of the
economy. Therefore, it is natural to view emissions as a stochastic quantity. This introduces a
challenge for the companies involved: In planning their abatement actions, they need to avoid
penalty payments without knowing their total amount of emissions. We consider a stochastic control approach to address this problem: In a continuous-time model, we use the rate of
emission abatement as a control in minimizing the costs that arise from penalty payments and
abatement costs. In a simplified variant of this model, the resulting Hamilton-Jacobi-Bellman
(HJB) equation can be solved analytically.
Taking the viewpoint of a regulator of an ETS, our main interest is to determine the resulting
emissions and to evaluate their compliance with the given emission target. Additionally, as an
incentive for investments in low-emission technologies, a high allowance price with low variability
is desirable. Both the resulting emissions and the allowance price are not directly given by the
solution to the stochastic control problem. Instead we need to solve a stochastic differential
equation (SDE), where the abatement rate enters as the drift term. Due to the nature of the
penalty function, the abatement rate is not continuous. This means that classical results on
existence and uniqueness of a solution as well as convergence of numerical methods, such as the
Euler-Maruyama scheme, do not apply. Therefore, we prove similar results under assumptions
suitable for our case. By applying a standard verification theorem, we show that the stochastic
control approach delivers an optimal abatement rate.
We extend the model by considering several consecutive time periods. This enables us to
model the transfer of unused allowances to the subsequent time period. In formulating the
multi-period model, we pursue two different approaches: In the first, we assume the value that
the company anticipates for an unused allowance to be constant throughout one time period.
We proceed similarly to the one-period model and again obtain an analytical solution. In the
second approach, we introduce an additional stochastic process to simulate the evolution of the
anticipated price for an unused allowance.
The model so far assumes that allowances are allocated for free. Therefore, we construct
another model extension to incorporate the auctioning of allowances. Then, additionally the
problem of choosing the optimal demand at the auction needs to be solved. We find that
the auction price equals the allowance price at the beginning of the respective time period.
Furthermore, we show that the resulting emissions as well as the allowance price are unaffected
by the introduction of auctioning in the setting of our model.
To perform numerical simulations, we first solve the characteristic partial differential equation
derived from the HJB equation by applying the method of lines. Then we apply the Euler-
Maruyama scheme to solve the SDE, delivering realizations of the resulting emissions and the
allowance price paths.
Simulation results indicate that, under realistic settings, the probability of non-compliance
with the emission target is quite large. It can be reduced for instance by an increase of the
penalty. In the multi-period model, we observe that by allowing the transfer of allowances to the
subsequent time period, the probability of non-compliance decreases considerably.
In many industrial applications fast and accurate solutions of linear elliptic partial differential equations are needed as one of the building blocks of more complex problems. The domains are often highly complex and meshing turns out to be expensive and difficult to obtain with a sufficient quality. In such cases methods with a regular, not boundary adapted grid offer an attractive alternative. The Explicit Jump Immersed Interface Method is one of these algorithms. The main interest of this work lies in solving the linear elasticity equations. For this purpose the existing EJIIM algorithm has been extended to three dimensions. The Poisson equation is always considered in parallel as the most typical representative of elliptic PDEs. During the work it became clear that EJIIM can have very high computational memory requirements. To overcome this problem an improvement, Reduced EJIIM is proposed. The main theoretical result in this work is the proof of the smoothing property of inverses of elliptic finite difference operators in two and three space dimensions. It is an often observed phenomena that the local truncation error is allowed to be of lower order along some lower dimensional manifold without influencing the global convergence order of the solution.
This dissertation is intended to give a systematic treatment of hypersurface singularities in arbitrary characteristic which provides the necessary tools, theoretically and computationally, for the purpose of classification. This thesis consists of five chapters: In chapter 1, we introduce the background on isolated hypersurface singularities needed for our work. In chapter 2, we formalize the notions of piecewise-homogeneous grading and we discuss thoroughly non-degeneracy in arbitrary characteristic. Chapter 3 is devoted to determinacy and normal forms of isolated hypersurface singularities. In the first part, we give finite determinacy theorems in arbitrary characteristic with respect to right respectively contact equivalence. Furthermore, we show that "isolated" and finite determinacy properties are equivalent. In the second part, we formalize Arnol'd's key ideas for the computation of normal forms an define the conditions (AA) and (AAC). The last part of Chapter 3 is devoted to the study of normal forms in the general setting of hypersurface singularities imposing neither condition (A) nor Newton-Nondegeneracy. In Chapter 4, we present algorithms which we implement in Singular for the purpose of explicit computation of regular bases and normal forms. In chapter 5, we transfer some classical results on invariants over the field C of complex numbers to algebraically closed fields of characteristic zero known as Lefschetz principle.
The dissertation deals with the application of Hub Location models in public transport planning. The author proposes new mathematical models along with different solution approaches to solve the instances. Moreover, a novel multi-period formulation is proposed as an extension to the general model. Due to its high complexity heuristic approaches are formulated to find a good solution within a reasonable amount of time.
This work aims to study textile structures in the frame of linear elasticity to understand how
the structure and material parameters influence the macroscopic homogenized model. More
precisely, we are interested in how the textile design parameters, such as the ratio between
fibers’ distance and cross-section width, the strength of the contact sliding between yarns,
and the partial clamp on the textile boundaries determine the phenomena that one can see in
shear experiments with textiles. Among others, when the warp and weft yarns change their
in-plane angles first and, after reaching some critical shear angle, the textile plate comes out
of the plane, and its folding starts.
The textile structure under consideration is a woven square, partially clamped on the left
and bottom boundary, made of long thin fibers that cross each other in a periodic pattern.
The fibers cannot penetrate each other, and in-plane sliding is allowed. This last assumption,
together with the partial clamp, adds new levels of complexity to the problem due to
the anisotropy in the yarn’s behavior in the unclamped subdomains of the textile.
The limiting behavior and macroscopic strain fields are found by passing to the limit with
respect to the yarn’s thickness r and the distance between them e, parameters that are asymptotically
related. The homogenization and dimension reduction are done via the unfolding
method, which separates the macroscopic scale from the periodicity cell. In addition to the
homogenization, a dimension reduction from a 3D to a 2D problem is applied. Adapting
the classical unfolding results to both the anisotropic context and to lattice grids (which are
constructed starting from the center lines of the rods crossing each other) are the main tools
we developed to tackle this type of model. They represent the first part of the thesis and are
published in Falconi, Griso, and Orlik, 2022b and Falconi, Griso, and Orlik, 2022a.
Given the parameters mentioned above, we then proceed to classify different textile problems,
incorporating the results from other works on the topic and thoroughly investigating
some others. After the study is conducted, we draw conclusions and give a mathematical
explanation concerning the expected approximation of the displacements, the expected solvability
of the limit problems, and the phenomena mentioned above. The results can be found
in “Asymptotic behavior for textiles with loose contact”, which has been recently submitted.
The thesis consists of two parts. In the first part we consider the stable Auslander--Reiten quiver of a block \(B\) of a Hecke algebra of the symmetric group at a root of unity in characteristic zero. The main theorem states that if the ground field is algebraically closed and \(B\) is of wild representation type, then the tree class of every connected component of the stable Auslander--Reiten quiver \(\Gamma_{s}(B)\) of \(B\) is \(A_{\infty}\). The main ingredient of the proof is a skew group algebra construction over a quantum complete intersection. Also, for these algebras the stable Auslander--Reiten quiver is computed in the case where the defining parameters are roots of unity. As a result, the tree class of every connected component of the stable Auslander--Reiten quiver is \(A_{\infty}\).\[\]
In the second part of the thesis we are concerned with branching rules for Hecke algebras of the symmetric group at a root of unity. We give a detailed survey of the theory initiated by I. Grojnowski and A. Kleshchev, describing the Lie-theoretic structure that the Grothendieck group of finite-dimensional modules over a cyclotomic Hecke algebra carries. A decisive role in this approach is played by various functors that give branching rules for cyclotomic Hecke algebras that are independent of the underlying field. We give a thorough definition of divided power functors that will enable us to reformulate the Scopes equivalence of a Scopes pair of blocks of Hecke algebras of the symmetric group. As a consequence we prove that two indecomposable modules that correspond under this equivalence have a common vertex. In particular, we verify the Dipper--Du Conjecture in the case where the blocks under consideration have finite representation type.
This thesis is separated into three main parts: Development of Gaussian and White Noise Analysis, Hamiltonian Path Integrals as White Noise Distributions, Numerical methods for polymers driven by fractional Brownian motion.
Throughout this thesis the Donsker's delta function plays a key role. We investigate this generalized function also in Chapter 2. Moreover we show by giving a counterexample, that the general definition for complex kernels is not true.
In Chapter 3 we take a closer look to generalized Gauss kernels and generalize these concepts to the case of vector-valued White Noise. These results are the basis for Hamiltonian path integrals of quadratic type. The core result of this chapter gives conditions under which pointwise products of generalized Gauss kernels and certain Hida distributions have a mathematical rigorous meaning as distributions in the Hida space.
In Chapter 4 we discuss operators which are related to applications for Feynman Integrals as differential operators, scaling, translation and projection. We show the relation of these operators to differential operators, which leads to the well-known notion of so called convolution operators. We generalize the central homomorphy theorem to regular generalized functions.
We generalize the concept of complex scaling to scaling with bounded operators and discuss the relation to generalized Radon-Nikodym derivatives. With the help of this we consider products of generalized functions in chapter 5. We show that the projection operator from the Wick formula for products with Donsker's deltais not closable on the square-integrable functions..
In Chapter 5 we discuss products of generalized functions. Moreover the Wick formula is revisited. We investigate under which conditions and on which spaces the Wick formula can be generalized to. At the end of the chapter we consider the products of Donsker's delta function with a generalized function with help of a measure transformation. Here also problems as measurability are concerned.
In Chapter 6 we characterize Hamiltonian path integrands for the free particle, the harmonic oscillator and the charged particle in a constant magnetic field as Hida distributions. This is done in terms of the T-transform and with the help of the results from chapter 3. For the free particle and the harmonic oscillator we also investigate the momentum space propagators. At the same time, the $T$-transform of the constructed Feynman integrands provides us with their generating functional. In Chapter 7, we can show that the generalized expectation (generating functional at zero) gives the Greens function to the corresponding Schrödinger equation.
Moreover, with help of the generating functional we can show that the canonical commutation relations for the free particle and the harmonic oscillator in phase space are fulfilled. This confirms on a mathematical rigorous level the heuristics developed by Feynman and Hibbs.
In Chapter 8 we give an outlook, how the scaling approach which is successfully applied in the Feynman integral setting can be transferred to the phase space setting. We give a mathematical rigorous meaning to an analogue construction to the scaled Feynman-Kac kernel. It is open if the expression solves the Schrödinger equation. At least for quadratic potentials we can get the right physics.
In the last chapter, we focus on the numerical analysis of polymer chains driven by fractional Brownian motion. Instead of complicated lattice algorithms, our discretization is based on the correlation matrix. Using fBm one can achieve a long-range dependence of the interaction of the monomers inside a polymer chain. Here a Metropolis algorithm is used to create the paths of a polymer driven by fBm taking the excluded volume effect in account.
Gröbner bases are one of the most powerful tools in computer algebra and commutative algebra, with applications in algebraic geometry and singularity theory. From the theoretical point of view, these bases can be computed over any field using Buchberger's algorithm. In practice, however, the computational efficiency depends on the arithmetic of the coefficient field.
In this thesis, we consider Gröbner bases computations over two types of coefficient fields. First, consider a simple extension \(K=\mathbb{Q}(\alpha)\) of \(\mathbb{Q}\), where \(\alpha\) is an algebraic number, and let \(f\in \mathbb{Q}[t]\) be the minimal polynomial of \(\alpha\). Second, let \(K'\) be the algebraic function field over \(\mathbb{Q}\) with transcendental parameters \(t_1,\ldots,t_m\), that is, \(K' = \mathbb{Q}(t_1,\ldots,t_m)\). In particular, we present efficient algorithms for computing Gröbner bases over \(K\) and \(K'\). Moreover, we present an efficient method for computing syzygy modules over \(K\).
To compute Gröbner bases over \(K\), starting from the ideas of Noro [35], we proceed by joining \(f\) to the ideal to be considered, adding \(t\) as an extra variable. But instead of avoiding superfluous S-pair reductions by inverting algebraic numbers, we achieve the same goal by applying modular methods as in [2,4,27], that is, by inferring information in characteristic zero from information in characteristic \(p > 0\). For suitable primes \(p\), the minimal polynomial \(f\) is reducible over \(\mathbb{F}_p\). This allows us to apply modular methods once again, on a second level, with respect to the
modular factors of \(f\). The algorithm thus resembles a divide and conquer strategy and
is in particular easily parallelizable. Moreover, using a similar approach, we present an algorithm for computing syzygy modules over \(K\).
On the other hand, to compute Gröbner bases over \(K'\), our new algorithm first specializes the parameters \(t_1,\ldots,t_m\) to reduce the problem from \(K'[x_1,\ldots,x_n]\) to \(\mathbb{Q}[x_1,\ldots,x_n]\). The algorithm then computes a set of Gröbner bases of specialized ideals. From this set of Gröbner bases with coefficients in \(\mathbb{Q}\), it obtains a Gröbner basis of the input ideal using sparse multivariate rational interpolation.
At current state, these algorithms are probabilistic in the sense that, as for other modular Gröbner basis computations, an effective final verification test is only known for homogeneous ideals or for local monomial orderings. The presented timings show that for most examples, our algorithms, which have been implemented in SINGULAR [17], are considerably faster than other known methods.