Kaiserslautern - Fachbereich Mathematik
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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.
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.
The aim of this thesis is to introduce an equilibrium insurance market model and study its properties and possible applications in risk class management.
First, an insurance market model based on an equilibrium approach is developed. Depending on the premium, the insured will choose the amount of coverage they buy in order to maximize their expected utility. The behavior of the insurer in different market regimes is then compared. While the premiums in markets with perfect competition are calculated in order to make no profit at all, insurers try to maximize their margins in a monopolistic market.
In markets modeled in this way several phenomena become evident. Perhaps the most important one is the so-called push-out effect. When customers with different attributes are insured together, insurance might become so expensive for one type of customers that those agents are better off with buying no insurance at all. The push-out effect was already shown for theoretical examples in the literature. We present a comprehensive analysis of the equilibrium insurance market model and the push-out effect for different insurance products such as life, health and disability insurance contracts using real-life data from different sources. In a concluding chapter we formulate indicators when a push-out can be expected and when not.
Machine learning regression approaches such as neural networks have gained vast popularity in recent years. The exponential growth of computing power has enabled larger and more evolved networks that can perform increasingly complex tasks. In our feasibility study about the use of neural networks in the regression of equilibrium insurance premiums it is shown that this regression is quite robust and the risk of overfitting can almost be excluded -- as long as the regression is performed on at least a few thousand data points.
Grouping customers of different risk types into contracts is important for the stability and the robustness of an insurance market. This motivates the study of the optimal assignment of risk classes into contracts, also known as rating classes. We provide a theoretical framework that makes use of techniques from different mathematical fields such as non-linear optimization, convex analysis, herding theory, game theory and combinatorics. In addition, we are able to show that the market specifications have a large impact on the optimal allocation of risk classes to contracts by the insurer. However, there does not need to be an optimal risk class assignment for each of these specifications.
To address this issue, we present two different approaches, one more theoretical and another that can easily be implemented in practice. An extension of our model to markets with capacity constraints rounds off the topic and extends the applicability of our approach.
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.
The German energy mix, which provides an overview of the sources of electricity available in Germany, is changing as a result of the expansion of renewable energy sources. With this shift towards sustainable energy sources such as wind and solar power, the electricity market situation is also in flux. Whereas in the past there were few uncertainties in electricity generation and only demand was subject to stochastic uncertainties, generation is now subject to stochastic fluctuations as well, especially due to weather dependency. To provide a supportive framework for this different situation, the electricity market has introduced, among other things, the intraday market, products with half-hourly and quarter-hourly time slices, and a modified balancing energy market design. As a result, both electricity price forecasting and optimization issues remain topical.
In this thesis, we first address intraday market modeling and intraday index forecasting. To do so, we move to the level of individual bids in the intraday market and use them to model the limit order books of intraday products. Based on statistics of the modeled limit order books, we present a novel estimator for the intraday indices. Especially for less liquid products, the order book statistics contain relevant information that allows for significantly more accurate predictions in comparison to the benchmark estimator.
Unlike the intraday market, the day ahead market allows smaller companies without their own trading department to participate since it is operated as a market with daily auctions. We optimize the flexibility offer of such a small company in the day ahead market and model the prices with a stochastic multi-factor model already used in the industry. To make this model accessible for stochastic optimization, we discretize it in time and space using scenario trees. Here we present existing algorithms for scenario tree generation as well as our own extensions and adaptations. These are based on the nested distance, which measures the distance between two distributions of stochastic processes. Based on the resulting scenario trees, we apply the stochastic optimization methods of stochastic programming, dynamic programming, and reinforcement learning to illustrate in which context the methods are appropriate.
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.
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.
Estimation of Motion Vector Fields of Complex Microstructures by Time Series of Volume Images
(2023)
Mechanical tests form one of the pillars in development and assessment of modern materials. In a world that will be forced to handle its resources more carefully in the near future, development of materials that are favorable regarding for example weight or material consumption is inevitable. To guarantee that such materials can also be used in critical infrastructure, such as foamed materials in automotive industry or new types of concrete in civil engineering, mechanical properties like tensile or compressive strength have to be thoroughly described. One method to do so is by so called in situ tests, where the mechanical test is combined with an image acquisition technique such as Computed Tomography.
The resulting time series of volume images comprise the delicate and individual nature of each material. The objective of this thesis is to present and develop methods to unveil this behavior and make the motion accessible by algorithms. The estimation of motion has been tackled by many communities, and two of them have already made big effort to solve the problems we are facing. Digital Volume Correlation (DVC) on the one hand has been developed by material scientists and was applied in many different context in mechanical testing, but almost never produces displacement fields that allocate one vector per voxel. Medical Image Registration (MIR) on the other hand does produce voxel precise estimates, but is limited to very smooth motion estimates.
The unification of both families, DVC and MIR, under one roof, will therefore be illustrated in the first half of this thesis. Using the theory of inverse problems, we lay the mathematical foundations to explain why in our impression none of the families is sufficient to deal with all of the problems that come with motion estimation in in situ tests. We then proceed by presenting a third community in motion estimation, namely Optical flow, which is normally only applied in two dimensions. Nevertheless, within this community algorithms have been developed that meet many of our requirements. Strategies for large displacement exist as well as methods that resolve jumps, and on top the displacement is always calculated on pixel level. This thesis therefore proceeds by extending some of the most successful methods to 3D.
To ensure the competitiveness of our approach, the last part of this thesis deals with a detailed evaluation of proposed extensions. We focus on three types of materials, foam, fibre systems and concrete, and use simulated and real in situ tests to compare the Optical flow based methods to their competitors from DVC and MIR. By using synthetically generated and simulated displacement fields, we also assess the quality of the calculated displacement fields - a novelty in this area. We conclude this thesis by two specialized applications of our algorithm, which show how the voxel-precise displacement fields serve as useful information to engineers in investigating their materials.
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.