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Sun, 06 Sep 2015 15:50:38 +0200Sun, 06 Sep 2015 15:50:38 +0200Coercive functions from a topological viewpoint and properties of minimizing sets of convex functions appearing in image restoration
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4100
Many tasks in image processing can be tackled by modeling an appropriate data fidelity term \(\Phi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}\) and then solve one of the regularized minimization problems \begin{align*}
&{}(P_{1,\tau}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \big\{ \Phi(x) \;{\rm s.t.}\; \Psi(x) \leq \tau \big\} \\ &{}(P_{2,\lambda}) \qquad \mathop{\rm argmin}_{x \in \mathbb R^n} \{ \Phi(x) + \lambda \Psi(x) \}, \; \lambda > 0 \end{align*} with some function \(\Psi: \mathbb{R}^n \rightarrow \mathbb{R} \cup \{+\infty\}\) and a good choice of the parameter(s). Two tasks arise naturally here: \begin{align*} {}& \text{1. Study the solver sets \({\rm SOL}(P_{1,\tau})\) and
\({\rm SOL}(P_{2,\lambda})\) of the minimization problems.} \\ {}& \text{2. Ensure that the minimization problems have solutions.} \end{align*} This thesis provides contributions to both tasks: Regarding the first task for a more special setting we prove that there are intervals \((0,c)\) and \((0,d)\) such that the setvalued curves \begin{align*}
\tau \mapsto {}& {\rm SOL}(P_{1,\tau}), \; \tau \in (0,c) \\ {} \lambda \mapsto {}& {\rm SOL}(P_{2,\lambda}), \; \lambda \in (0,d) \end{align*} are the same, besides an order reversing parameter change \(g: (0,c) \rightarrow (0,d)\). Moreover we show that the solver sets are changing all the time while \(\tau\) runs from \(0\) to \(c\) and \(\lambda\) runs from \(d\) to \(0\).
In the presence of lower semicontinuity the second task is done if we have additionally coercivity. We regard lower semicontinuity and coercivity from a topological point of view and develop a new technique for proving lower semicontinuity plus coercivity.
Dropping any lower semicontinuity assumption we also prove a theorem on the coercivity of a sum of functions.René Ciakdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4100Tue, 09 Jun 2015 15:50:38 +0200Upscaling Approaches for Nonlinear Processes in Lithium-Ion Batteries
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4086
Lithium-ion batteries are broadly used nowadays in all kinds of portable electronics, such as laptops, cell phones, tablets, e-book readers, digital cameras, etc. They are preferred to other types of rechargeable batteries due to their superior characteristics, such as light weight and high energy density, no memory effect, and a big number of charge/discharge cycles. The high demand and applicability of Li-ion batteries naturally give rise to the unceasing necessity of developing better batteries in terms of performance and lifetime. The aim of the mathematical modelling of Li-ion batteries is to help engineers test different battery configurations and electrode materials faster and cheaper. Lithium-ion batteries are multiscale systems. A typical Li-ion battery consists of multiple connected electrochemical battery cells. Each cell has two electrodes - anode and cathode, as well as a separator between them that prevents a short circuit.
Both electrodes have porous structure composed of two phases - solid and electrolyte. We call macroscale the lengthscale of the whole electrode and microscale - the lengthscale at which we can distinguish the complex porous structure of the electrodes. We start from a Li-ion battery model derived on the microscale. The model is based on nonlinear diffusion type of equations for the transport of Lithium ions and charges in the electrolyte and in the active material. Electrochemical reactions on the solid-electrolyte interface couple the two phases. The interface kinetics is modelled by the highly nonlinear Butler-Volmer interface conditions. Direct numerical simulations with standard methods, such as the Finite Element Method or Finite Volume Method, lead to ill-conditioned problems with a huge number of degrees of freedom which are difficult to solve. Therefore, the aim of this work is to derive upscaled models on the lengthscale of the whole electrode so that we do not have to resolve all the small-scale features of the porous microstructure thus reducing the computational time and cost. We do this by applying two different upscaling techniques - the Asymptotic Homogenization Method and the Multiscale Finite Element Method (MsFEM). We consider the electrolyte and the solid as two self-complementary perforated domains and we exploit this idea with both upscaling methods. The first method is restricted only to periodic media and periodically oscillating solutions while the second method can be applied to randomly oscillating solutions and is based on the Finite Element Method framework. We apply the Asymptotic Homogenization Method to derive a coupled macro-micro upscaled model under the assumption of periodic electrode microstructure. A crucial step in the homogenization procedure is the upscaling of the Butler-Volmer interface conditions. We rigorously determine the asymptotic order of the interface exchange current densities and we perform a comprehensive numerical study in order to validate the derived homogenized Li-ion battery model. In order to upscale the microscale battery problem in the case of random electrode microstructure we apply the MsFEM, extended to problems in perforated domains with Neumann boundary conditions on the holes. We conduct a detailed numerical investigation of the proposed algorithm and we show numerical convergence of the method that we design. We also apply the developed technique to a simplified two-dimensional Li-ion battery problem and we show numerical convergence of the solution obtained with the MsFEM to the reference microscale one. Vasilena Taralovadoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4086Thu, 28 May 2015 09:01:35 +0200Simulation of Degradation Processes in Lithium-Ion Batteries
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4085
Lithium-ion batteries are increasingly becoming an ubiquitous part of our everyday life - they are present in mobile phones, laptops, tools, cars, etc. However, there are still many concerns about their longevity and their safety. In this work we focus on the simulation of several degradation mechanisms on the microscopic scale, where one can resolve the active materials inside the electrodes of the lithium-ion batteries as porous structures. We mainly study two aspects - heat generation and mechanical stress. For the former we consider an electrochemical non-isothermal model on the spatially resolved porous scale to observe the temperature increase inside a battery cell, as well as to observe the individual heat sources to assess their contributions to the total heat generation. As a result from our experiments, we determined that the temperature has very small spatial variance for our test cases and thus allows for an ODE formulation of the heat equation.
The second aspect that we consider is the generation of mechanical stress as a result of the insertion of lithium ions in the electrode materials. We study two approaches - using small strain models and finite strain models. For the small strain models, the initial geometry and the current geometry coincide. The model considers a diffusion equation for the lithium ions and equilibrium equation for the mechanical stress. First, we test a single perforated cylindrical particle using different boundary conditions for the displacement and with Neumann boundary conditions for the diffusion equation. We also test for cylindrical particles, but with boundary conditions for the diffusion equation in the electrodes coming from an isothermal electrochemical model for the whole battery cell. For the finite strain models we take in consideration the deformation of the initial geometry as a result of the intercalation and the mechanical stress. We compare two elastic models to study the sensitivity of the predicted elastic behavior on the specific model used. We also consider a softening of the active material dependent on the concentration of the lithium ions and using data for silicon electrodes. We recover the general behavior of the stress from known physical experiments.
Some models, like the mechanical models we use, depend on the local values of the concentration to predict the mechanical stress. In that sense we perform a short comparative study between the Finite Element Method with tetrahedral elements and the Finite Volume Method with voxel volumes for an isothermal electrochemical model.
The spatial discretizations of the PDEs are done using the Finite Element Method. For some models we have discontinuous quantities where we adapt the FEM accordingly. The time derivatives are discretized using the implicit Backward Euler method. The nonlinear systems are linearized using the Newton method. All of the discretized models are implemented in a C++ framework developed during the thesis. Maxim Taralovdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4085Thu, 28 May 2015 08:47:34 +0200Isogeometric Finite Element Analysis of Nonlinear Structural Vibrations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4079
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.
Oliver Weegerdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4079Wed, 20 May 2015 11:46:03 +0200Isogeometric Shell Discretizations for Flexible Multibody Dynamics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4076
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. Anmol Goyaldoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4076Tue, 19 May 2015 09:55:55 +0200Portfolio Optimization and Stochastic Control under Transaction Costs
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4073
This thesis is concerned with stochastic control problems under transaction costs. In particular, we consider a generalized menu cost problem with partially controlled regime switching, general multidimensional running cost problems and the maximization of long-term growth rates in incomplete markets. The first two problems are considered under a general cost structure that includes a fixed cost component, whereas the latter is analyzed under proportional and Morton-Pliska
transaction costs.
For the menu cost problem and the running cost problem we provide an equivalent characterization of the value function by means of a generalized version of the Ito-Dynkin formula instead of the more restrictive, traditional approach via the use of quasi-variational inequalities (QVIs). Based on the finite element method and weak solutions of QVIs in suitable Sobolev spaces, the value function is constructed iteratively. In addition to the analytical results, we study a novel application of the menu cost problem in management science. We consider a company that aims to implement an optimal investment and marketing strategy and must decide when to issue a new version of a product and when and how much
to invest into marketing.
For the long-term growth rate problem we provide a rigorous asymptotic analysis under both proportional and Morton-Pliska transaction costs in a general incomplete market that includes, for instance, the Heston stochastic volatility model and the Kim-Omberg stochastic excess return model as special cases. By means of a dynamic programming approach leading-order optimal strategies are constructed
and the leading-order coefficients in the expansions of the long-term growth rates are determined. Moreover, we analyze the asymptotic performance of Morton-Pliska strategies in settings with proportional transaction costs. Finally, pathwise optimality of the constructed strategies is established.Yaroslav Melnykdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4073Mon, 18 May 2015 10:01:57 +0200Robustness for regression models with asymmetric error distribution
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4046
In this work we focus on the regression models with asymmetrical error distribution,
more precisely, with extreme value error distributions. This thesis arises in the framework
of the project "Robust Risk Estimation". Starting from July 2011, this project won
three years funding by the Volkswagen foundation in the call "Extreme Events: Modelling,
Analysis, and Prediction" within the initiative "New Conceptual Approaches to
Modelling and Simulation of Complex Systems". The project involves applications in
Financial Mathematics (Operational and Liquidity Risk), Medicine (length of stay and
cost), and Hydrology (river discharge data). These applications are bridged by the
common use of robustness and extreme value statistics.
Within the project, in each of these applications arise issues, which can be dealt with by
means of Extreme Value Theory adding extra information in the form of the regression
models. The particular challenge in this context concerns asymmetric error distributions,
which significantly complicate the computations and make desired robustification
extremely difficult. To this end, this thesis makes a contribution.
This work consists of three main parts. The first part is focused on the basic notions
and it gives an overview of the existing results in the Robust Statistics and Extreme
Value Theory. We also provide some diagnostics, which is an important achievement of
our project work. The second part of the thesis presents deeper analysis of the basic
models and tools, used to achieve the main results of the research.
The second part is the most important part of the thesis, which contains our personal
contributions. First, in Chapter 5, we develop robust procedures for the risk management
of complex systems in the presence of extreme events. Mentioned applications use time
structure (e.g. hydrology), therefore we provide extreme value theory methods with time
dynamics. To this end, in the framework of the project we considered two strategies. In
the first one, we capture dynamic with the state-space model and apply extreme value
theory to the residuals, and in the second one, we integrate the dynamics by means of
autoregressive models, where the regressors are described by generalized linear models.
More precisely, since the classical procedures are not appropriate to the case of outlier
presence, for the first strategy we rework classical Kalman smoother and extended
Kalman procedures in a robust way for different types of outliers and illustrate the performance
of the new procedures in a GPS application and a stylized outlier situation.
To apply approach to shrinking neighborhoods we need some smoothness, therefore for
the second strategy, we derive smoothness of the generalized linear model in terms of
L2 differentiability and create sufficient conditions for it in the cases of stochastic and
deterministic regressors. Moreover, we set the time dependence in these models by
linking the distribution parameters to the own past observations. The advantage of
our approach is its applicability to the error distributions with the higher dimensional
parameter and case of regressors of possibly different length for each parameter. Further,
we apply our results to the models with generalized Pareto and generalized extreme value
error distributions.
Finally, we create the exemplary implementation of the fixed point iteration algorithm
for the computation of the optimally robust in
uence curve in R. Here we do not aim to
provide the most
exible implementation, but rather sketch how it should be done and
retain points of particular importance. In the third part of the thesis we discuss three applications,
operational risk, hospitalization times and hydrological river discharge data,
and apply our code to the real data set taken from Jena university hospital ICU and
provide reader with the various illustrations and detailed conclusions.Daria Pupashenkodoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4046Thu, 16 Apr 2015 13:53:08 +0200Worst-Case Portfolio Optimization: Transaction Costs and Bubbles
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4045
In this thesis we extend the worst-case modeling approach as first introduced by Hua and Wilmott (1997) (option pricing in discrete time) and Korn and Wilmott (2002) (portfolio optimization in continuous time) in various directions.
In the continuous-time worst-case portfolio optimization model (as first introduced by Korn and Wilmott (2002)), the financial market is assumed to be under the threat of a crash in the sense that the stock price may crash by an unknown fraction at an unknown time. It is assumed that only an upper bound on the size of the crash is known and that the investor prepares for the worst-possible crash scenario. That is, the investor aims to find the strategy maximizing her objective function in the worst-case crash scenario.
In the first part of this thesis, we consider the model of Korn and Wilmott (2002) in the presence of proportional transaction costs. First, we treat the problem without crashes and show that the value function is the unique viscosity solution of a dynamic programming equation (DPE) and then construct the optimal strategies. We then consider the problem in the presence of crash threats, derive the corresponding DPE and characterize the value function as the unique viscosity solution of this DPE.
In the last part, we consider the worst-case problem with a random number of crashes by proposing a regime switching model in which each state corresponds to a different crash regime. We interpret each of the crash-threatened regimes of the market as states in which a financial bubble has formed which may lead to a crash. In this model, we prove that the value function is a classical solution of a system of DPEs and derive the optimal strategies.
Christoph Belakdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4045Tue, 07 Apr 2015 10:17:10 +0200Modeling and design optimization of textile-like materials via homogenization and one-dimensional models of elasticity
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4019
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.
Vladimir Shiryaevdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4019Mon, 09 Mar 2015 14:42:08 +0100Modeling and Simulation of a Moving Rigid Body in a Rarefied Gas
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4012
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. Samir Shresthadoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4012Wed, 04 Mar 2015 11:43:53 +0100Testrig optimization by block loads: Remodelling of damage as Gaussian functions and their clustering method
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4003
In automotive testrigs we apply load time series to components such that the outcome is as close as possible to some reference data. The testing procedure should in general be less expensive and at the same time take less time for testing. In my thesis, I propose a testrig damage optimization problem (WSDP). This approach improves upon the testrig stress optimization problem (TSOP) used as a state of the art by industry experts.
In both (TSOP) and (WSDP), we optimize the load time series for a given testrig configuration. As the name suggests, in (TSOP) the reference data is the stress time series. The detailed behaviour of the stresses as functions of time are sometimes not the most important topic. Instead the damage potential of the stress signals are considered. Since damage is not part of the objectives in the (TSOP) the total damage computed from the optimized load time series is not optimal with respect to the reference damage. Additionally, the load time series obtained is as long as the reference stress time series and the total damage computation needs cycle counting algorithms and Goodmann corrections. The use of cycle counting algorithms makes the computation of damage from load time series non-differentiable.
To overcome the issues discussed in the previous paragraph this thesis uses block loads for the load time series. Using of block loads makes the damage differentiable with respect to the load time series. Additionally, in some special cases it is shown that damage is convex when block loads are used and no cycle counting algorithms are required. Using load time series with block loads enables us to use damage in the objective function of the (WSDP).
During every iteration of the (WSDP), we have to find the maximum total damage over all plane angles. The first attempt at solving the (WSDP) uses discretization of the interval for plane angle to find the maximum total damage at each iteration. This is shown to give unreliable results and makes maximum total damage function non-differentiable with respect to the plane angle. To overcome this, damage function for a given surface stress tensor due to a block load is remodelled by Gaussian functions. The parameters for the new model are derived.
When we model the damage by Gaussian function, the total damage is computed as a sum of Gaussian functions. The plane with the maximum damage is similar to the modes of the Gaussian Mixture Models (GMM), the difference being that the Gaussian functions used in GMM are probability density functions which is not the case in the damage approximation presented in this work. We derive conditions for a single maximum for Gaussian functions, similar to the ones given for the unimodality of GMM by Aprausheva et al. in [1].
By using the conditions for a single maximum we give a clustering algorithm that merges the Gaussian functions in the sum as clusters. Each cluster obtained through clustering is such that it has a single maximum in the absence of other Gaussian functions of the sum. The approximate point of the maximum of each cluster is used as the starting point for a fixed point equation on the original damage function to get the actual maximum total damage at each iteration.
We implement the method for the (TSOP) and the two methods (with discretization and with clustering) for (WSDP) on two example problems. The results obtained from the (WSDP) using discretization is shown to be better than the results obtained from the (TSOP). Furthermore we show that, (WSDP) using clustering approach to finding the maximum total damage, takes less number of iterations and is more reliable than using discretization.Chhitiz Buchasiadoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4003Tue, 24 Feb 2015 11:08:29 +0100Combinations of Boolean Groebner Bases and SAT Solvers
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3958
In this thesis, we combine Groebner basis with SAT Solver in different manners.
Both SAT solvers and Groebner basis techniques have their own strength and weakness.
Combining them could fix their weakness.
The first combination is using Groebner techniques to learn additional binary clauses for SAT solver from a selection of clauses. This combination is first proposed by Zengler and Kuechlin.
However, in our experiments, about 80 percent Groebner basis computations give no new binary clauses.
By selecting smaller and more compact input for Groebner basis computations, we can significantly
reduce the number of inefficient Groebner basis computations, learn much more binary clauses. In addition,
the new strategy can reduce the solving time of a SAT Solver in general, especially for large and hard problems.
The second combination is using all-solution SAT solver and interpolation to compute Boolean Groebner bases of Boolean elimination ideals of a given ideal. Computing Boolean Groebner basis of the given ideal is an inefficient method in case we want to eliminate most of the variables from a big system of Boolean polynomials.
Therefore, we propose a more efficient approach to handle such cases.
In this approach, the given ideal is translated to the CNF formula. Then an all-solution SAT Solver is used to find the projection of all solutions of the given ideal. Finally, an algorithm, e.g. Buchberger-Moeller Algorithm, is used to associate the reduced Groebner basis to the projection.
We also optimize the Buchberger-Moeller Algorithm for lexicographical ordering and compare it with Brickenstein's interpolation algorithm.
Finally, we combine Groebner basis and abstraction techniques to the verification of some digital designs that contain complicated data paths.
For a given design, we construct an abstract model.
Then, we reformulate it as a system of polynomials in the ring \({\mathbb Z}_{2^k}[x_1,\dots,x_n]\).
The variables are ordered in a way such that the system has already been a Groebner basis w.r.t lexicographical monomial ordering.
Finally, the normal form is employed to prove the desired properties.
To evaluate our approach, we verify the global property of a multiplier and a FIR filter using the computer algebra system Singular. The result shows that our approach is much faster than the commercial verification tool from Onespin on these benchmarks.Thanh Hung Nguyendoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3958Thu, 18 Dec 2014 14:11:19 +0100Multilevel Constructions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3942
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.Anton Kostiukdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3942Wed, 10 Dec 2014 08:29:03 +0100Zinsoptimiertes Schuldenmanagement
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3931
Das zinsoptimierte Schuldenmanagement hat zum Ziel, eine möglichst effiziente Abwägung zwischen den erwarteten Finanzierungskosten einerseits und den Risiken für den Staatshaushalt andererseits zu finden. Um sich diesem Spannungsfeld zu nähern, schlagen wir erstmals die Brücke zwischen den Problemstellungen des Schuldenmanagements und den Methoden der zeitkontinuierlichen, dynamischen Portfoliooptimierung.
Das Schlüsselelement ist dabei eine neue Metrik zur Messung der Finanzierungskosten, die Perpetualkosten. Diese spiegeln die durchschnittlichen zukünftigen Finanzierungskosten wider und beinhalten sowohl die bereits bekannten Zinszahlungen als auch die noch unbekannten Kosten für notwendige Anschlussfinanzierungen. Daher repräsentiert die Volatilität der Perpetualkosten auch das Risiko einer bestimmten Strategie; je langfristiger eine Finanzierung ist, desto kleiner ist die Schwankungsbreite der Perpetualkosten.
Die Perpetualkosten ergeben sich als Produkt aus dem Barwert eines Schuldenportfolios und aus der vom Portfolio unabhängigen Perpetualrate. Für die Modellierung des Barwertes greifen wir auf das aus der dynamischen Portfoliooptimierung bekannte Konzept eines selbstfinanzierenden Bondportfolios zurück, das hier auf einem mehrdimensionalen affin-linearen Zinsmodell basiert. Das Wachstum des Schuldenportfolios wird dabei durch die Einbeziehung des Primärüberschusses des Staates gebremst bzw. verhindert, indem wir diesen als externen Zufluss in das selbstfinanzierende Modell aufnehmen.
Wegen der Vielfältigkeit möglicher Finanzierungsinstrumente wählen wir nicht deren Wertanteile als Kontrollvariable, sondern kontrollieren die Sensitivitäten des Portfolios gegenüber verschiedenen Zinsbewegungen. Aus optimalen Sensitivitäten können in einem nachgelagerten Schritt dann optimale Wertanteile für verschiedenste Finanzierungsinstrumente abgeleitet werden. Beispielhaft demonstrieren wir dies mittels Rolling-Horizon-Bonds unterschiedlicher Laufzeit.
Schließlich lösen wir zwei Optimierungsprobleme mit Methoden der stochastischen Kontrolltheorie. Dabei wird stets der erwartete Nutzen der Perpetualkosten maximiert. Die Nutzenfunktionen sind jeweils an das Schuldenmanagement angepasst und zeichnen sich insbesondere dadurch aus, dass höhere Kosten mit einem niedrigeren Nutzen einhergehen. Im ersten Problem betrachten wir eine Potenznutzenfunktion mit konstanter relativer Risikoaversion, im zweiten wählen wir eine Nutzenfunktion, welche die Einhaltung einer vorgegebenen Schulden- bzw. Kostenobergrenze garantiert.Christoph Petersdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3931Mon, 24 Nov 2014 09:09:39 +0100Variance Reduction Procedures for Market Risk Estimation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3885
Monte Carlo simulation is one of the commonly used methods for risk estimation on financial markets, especially for option portfolios, where any analytical approximation is usually too inaccurate. However, the usually high computational effort for complex portfolios with a large number of underlying assets motivates the application of variance reduction procedures. Variance reduction for estimating the probability of high portfolio losses has been extensively studied by Glasserman et al. A great variance reduction is achieved by applying an exponential twisting importance sampling algorithm together with stratification. The popular and much faster Delta-Gamma approximation replaces the portfolio loss function in order to guide the choice of the importance sampling density and it plays the role of the stratification variable. The main disadvantage of the proposed algorithm is that it is derived only in the case of Gaussian and some heavy-tailed changes in risk factors.
Hence, our main goal is to keep the main advantage of the Monte Carlo simulation, namely its ability to perform a simulation under alternative assumptions on the distribution of the changes in risk factors, also in the variance reduction algorithms. Step by step, we construct new variance reduction techniques for estimating the probability of high portfolio losses. They are based on the idea of the Cross-Entropy importance sampling procedure. More precisely, the importance sampling density is chosen as the closest one to the optimal importance sampling density (zero variance estimator) out of some parametric family of densities with respect to Kullback - Leibler cross-entropy. Our algorithms are based on the special choices of the parametric family and can now use any approximation of the portfolio loss function. A special stratification is developed, so that any approximation of the portfolio loss function under any assumption of the distribution of the risk factors can be used. The constructed algorithms can easily be applied for any distribution of risk factors, no matter if light- or heavy-tailed. The numerical study exhibits a greater variance reduction than of the algorithm from Glasserman et al. The use of a better approximation may improve the performance of our algorithms significantly, as it is shown in the numerical study.
The literature on the estimation of the popular market risk measures, namely VaR and CVaR, often refers to the algorithms for estimating the probability of high portfolio losses, describing the corresponding transition process only briefly. Hence, we give a consecutive discussion of this problem. Results necessary to construct confidence intervals for both measures under the mentioned variance reduction procedures are also given.
Mykhailo Pupashenkodoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3885Wed, 01 Oct 2014 09:47:40 +0200New aspects of optimal investment in continuous time
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3867
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.Nhat Thu Trandoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3867Tue, 09 Sep 2014 12:50:40 +0200Edgeworth Expansions for Binomial Trees
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3861
In the theory of option pricing one is usually concerned with evaluating expectations under the risk-neutral measure in a continuous-time model.
However, very often these values cannot be calculated explicitly and numerical methods need to be applied to approximate the desired quantity. Monte Carlo simulations, numerical methods for PDEs and the lattice approach are the methods typically employed. In this thesis we consider the latter approach, with the main focus on binomial trees.
The binomial method is based on the concept of weak convergence. The discrete-time model is constructed so as to ensure convergence in distribution to the continuous process. This means that the expectations calculated in the binomial tree can be used as approximations of the option prices in the continuous model. The binomial method is easy to implement and can be adapted to options with different types of payout structures, including American options. This makes the approach very appealing. However, the problem is that in many cases, the convergence of the method is slow and highly irregular, and even a fine discretization does not guarantee accurate price approximations. Therefore, ways of improving the convergence properties are required.
We apply Edgeworth expansions to study the convergence behavior of the lattice approach. We propose a general framework, that allows to obtain asymptotic expansion for both multinomial and multidimensional trees. This information is then used to construct advanced models with superior convergence properties.
In binomial models we usually deal with triangular arrays of lattice random vectors. In this case the available results on Edgeworth expansions for lattices are not directly applicable. Therefore, we first present Edgeworth expansions, which are also valid for the binomial tree setting. We then apply these result to the one-dimensional and multidimensional Black-Scholes models. We obtain third order expansions
for general binomial and trinomial trees in the 1D setting, and construct advanced models for digital, vanilla and barrier options. Second order expansion are provided for the standard 2D binomial trees and advanced models are constructed for the two-asset digital and the two-asset correlation options. We also present advanced binomial models for a multidimensional setting.Alona Bockdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3861Tue, 02 Sep 2014 09:07:50 +0200Portfoliooptimierung im Binomialmodell
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3849
Die Dissertation "Portfoliooptimierung im Binomialmodell" befasst sich mit der Frage, inwieweit
das Problem der optimalen Portfolioauswahl im Binomialmodell lösbar ist bzw. inwieweit
die Ergebnisse auf das stetige Modell übertragbar sind. Dabei werden neben dem
klassischen Modell ohne Kosten und ohne Veränderung der Marktsituation auch Modellerweiterungen
untersucht.Henriette Krönerdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3849Thu, 14 Aug 2014 08:15:18 +0200Algorithms in SINGULAR: Parallelization, Syzygies, and Singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3840
This thesis, whose subject is located in the field of algorithmic commutative algebra and algebraic geometry, consists of three parts.
The first part is devoted to parallelization, a technique which allows us to take advantage of the computational power of modern multicore processors. First, we present parallel algorithms for the normalization of a reduced affine algebra A over a perfect field. Starting from the algorithm of Greuel, Laplagne, and Seelisch, we propose two approaches. For the local-to-global approach, we stratify the singular locus Sing(A) of A, compute the normalization locally at each stratum and finally reconstruct the normalization of A from the local results. For the second approach, we apply modular methods to both the global and the local-to-global normalization algorithm.
Second, we propose a parallel version of the algorithm of Gianni, Trager, and Zacharias for primary decomposition. For the parallelization of this algorithm, we use modular methods for the computationally hardest steps, such as for the computation of the associated prime ideals in the zero-dimensional case and for the standard bases computations. We then apply an innovative fast method to verify that the result is indeed a primary decomposition of the input ideal. This allows us to skip the verification step at each of the intermediate modular computations.
The proposed parallel algorithms are implemented in the open-source computer algebra system SINGULAR. The implementation is based on SINGULAR's new parallel framework which has been developed as part of this thesis and which is specifically designed for applications in mathematical research.
In the second part, we propose new algorithms for the computation of syzygies, based on an in-depth analysis of Schreyer's algorithm. Here, the main ideas are that we may leave out so-called "lower order terms" which do not contribute to the result of the algorithm, that we do not need to order the terms of certain module elements which occur at intermediate steps, and that some partial results can be cached and reused.
Finally, the third part deals with the algorithmic classification of singularities over the real numbers. First, we present a real version of the Splitting Lemma and, based on the classification theorems of Arnold, algorithms for the classification of the simple real singularities. In addition to the algorithms, we also provide insights into how real and complex singularities are related geometrically. Second, we explicitly describe the structure of the equivalence classes of the unimodal real singularities of corank 2. We prove that the equivalences are given by automorphisms of a certain shape. Based on this theorem, we explain in detail how the structure of the equivalence classes can be computed using SINGULAR and present the results in concise form. The probably most surprising outcome is that the real singularity type \(J_{10}^-\) is actually redundant.Andreas Steenpassdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3840Wed, 30 Jul 2014 10:37:00 +0200Algorithmic aspects of tropical intersection theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3823
In the first part of this thesis we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus is the study of moduli spaces, where the underlying combinatorics of the varieties involved allow a much more efficient way of computing certain tropical cycles. The algorithms discussed here have been implemented in an extension for polymake, a software for polyhedral computations.
In the second part we apply the algorithmic toolkit developed in the first part to the study of tropical double Hurwitz cycles. Hurwitz cycles are a higher-dimensional generalization of Hurwitz numbers, which count covers of \(\mathbb{P}^1\) by smooth curves of a given genus with a certain fixed ramification behaviour. Double Hurwitz numbers provide a strong connection between various mathematical disciplines, including algebraic geometry, representation theory and combinatorics. The tropical cycles have a rather complex combinatorial nature, so it is very difficult to study them purely "by hand". Being able to compute examples has been very helpful
in coming up with theoretical results. Our main result states that all marked and unmarked Hurwitz cycles are connected in codimension one and that for a generic choice of simple ramification points the marked cycle is a multiple of an irreducible cycle. In addition we provide computational examples to show that this is the strongest possible statement.Simon Hampedoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3823Thu, 03 Jul 2014 09:26:06 +0200Efficient Algorithms for Flow Simulation related to Nuclear Reactor Safety
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3826
Safety analysis is of ultimate importance for operating Nuclear Power Plants (NPP). The overall
modeling and simulation of physical and chemical processes occuring in the course of an accident
is an interdisciplinary problem and has origins in fluid dynamics, numerical analysis, reactor tech-
nology and computer programming. The aim of the study is therefore to create the foundations
of a multi-dimensional non-isothermal fluid model for a NPP containment and software tool based
on it. The numerical simulations allow to analyze and predict the behavior of NPP systems under
different working and accident conditions, and to develop proper action plans for minimizing the
risks of accidents, and/or minimizing the consequences of possible accidents. A very large number
of scenarios have to be simulated, and at the same time acceptable accuracy for the critical param-
eters, such as radioactive pollution, temperature, etc., have to be achieved. The existing software
tools are either too slow, or not accurate enough. This thesis deals with developing customized al-
gorithm and software tools for simulation of isothermal and non-isothermal flows in a containment
pool of NPP. Requirements to such a software are formulated, and proper algorithms are presented.
The goal of the work is to achieve a balance between accuracy and speed of calculation, and to
develop customized algorithm for this special case. Different discretization and solution approaches
are studied and those which correspond best to the formulated goal are selected, adjusted, and when
possible, analysed. Fast directional splitting algorithm for Navier-Stokes equations in complicated
geometries, in presence of solid and porous obstales, is in the core of the algorithm. Developing
suitable pre-processor and customized domain decomposition algorithms are essential part of the
overall algorithm amd software. Results from numerical simulations in test geometries and in real
geometries are presented and discussed.Tatiana Gornakdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3826Thu, 03 Jul 2014 08:29:14 +0200Efficient algorithms for Asymmetric Flow Field Flow Fractionation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3811
This thesis is devoted to the modeling and simulation of Asymmetric Flow Field Flow Fractionation, which is a technique for separating particles of submicron scale. This process is a part of large family of Field Flow Fractionation techniques and has a very broad range of industrial applications, e. g. in microbiology, chemistry, pharmaceutics, environmental analysis.
Mathematical modeling is crucial for this process, as due to the own nature of the process, lab ex- periments are difficult and expensive to perform. On the other hand, there are several challenges for the mathematical modeling: huge dominance (up to 106 times) of the flow over the diffusion, highly stretched geometry of the device. This work is devoted to developing fast and efficient algorithms, which take into the account the challenges, posed by the application, and provide reliable approximations for the quantities of interest.
We present a new Multilevel Monte Carlo method for estimating the distribution functions on a compact interval, which are of the main interest for Asymmetric Flow Field Flow Fractionation. Error estimates for this method in terms of computational cost are also derived.
We optimize the flow control at the Focusing stage under the given constraints on the flow and present an important ingredients for the further optimization, such as two-grid Reduced Basis method, specially adapted for the Finite Volume discretization approach.Tigran Nagapetyandoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3811Wed, 04 Jun 2014 09:42:28 +0200Pedestrian Flow Models
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3803
There have been many crowd disasters because of poor planning of the events. Pedestrian models are useful in analysing the behavior of pedestrians in advance to the events so that no pedestrians will be harmed during the event. This thesis deals with pedestrian flow models on microscopic, hydrodynamic and scalar scales. By following the Hughes' approach, who describes the crowd as a thinking fluid, we use the solution of the Eikonal equation to compute the optimal path for pedestrians. We start with the microscopic model for pedestrian flow and then derive the hydrodynamic and scalar models from it. We use particle methods to solve the governing equations. Moreover, we have coupled a mesh free particle method to the fixed grid for solving the Eikonal equation. We consider an example with a large number of pedestrians to investigate our models for different settings of obstacles and for different parameters. We also consider the pedestrian flow in a straight corridor and through T-junction and compare our numerical results with the experiments. A part of this work is devoted for finding a mesh free method to solve the Eikonal equation. Most of the available methods to solve the Eikonal equation are restricted to either cartesian grid or triangulated grid. In this context, we propose a mesh free method to solve the Eikonal equation, which can be applicable to any arbitrary grid and useful for the complex geometries. Raghavender Etikyaladoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3803Fri, 16 May 2014 12:02:59 +0200Intersection theory with applications to the computation of Gromov-Witten invariants
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3750
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.
Hiep Dangdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3750Fri, 14 Mar 2014 08:54:11 +0100On the distribution of eigenspaces in classical groups over finite rings and the Cohen-Lenstra heuristic
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3732
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.Michael Adamdoctoralthesishttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3732Tue, 18 Feb 2014 13:17:02 +0100