Kaiserslautern - Fachbereich Mathematik
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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.
Solving probabilistic-robust optimization problems using methods from semi-infinite optimization
(2023)
Optimization under uncertainty is one field of mathematics which is strongly inspired by real world problems. To handle uncertainties several models have arisen. One of these is the probust model where a combination of probabilistic and worst-case uncertainty is considered. So far, just problem instances with a special structure can be dealt with. In this thesis, we introduce solving techniques applicable for any probust optimization problem. On the one hand, we create upper bounds for the solution value by solving a sequence of chance constrained optimization problems. These bounds are based on discretization schemes which are inspired by semi-infinite optimization. On the other hand, we create lower bounds by solving a sequence of set-approximation problems. Here, we substitute the original event set by an appropriate family of sets. We examine the performance of the corresponding algorithms on simple packing problems where we can provide the probust solution analytically. Afterwards, we solve a water reservoir and a distillation problem and compare the probust solutions with solutions arising from other uncertainty models.
This thesis is primarily motivated by a project with Deutsche Bahn about offer preparation in rail freight transport. At its core, a customer should be offered three train paths to choose from in response to a freight train request. As part of this cooperation with DB Netz AG, we investigated how to compute these train paths efficiently. They should be all "good" but also "as different as possible". We solved this practical problem using combinatorial optimization techniques.
At the beginning of this thesis, we describe the practical aspects of our research collaboration. The more theoretical problems, which we consider afterwards, are divided into two parts.
In Part I, we deal with a dual pair of problems on directed graphs with two designated end-vertices. The Almost Disjoint Paths (ADP) problem asks for a maximum number of paths between the end-vertices any two of which have at most one arc in common. In comparison, for the Separating by Forbidden Pairs (SFP) problem we have to select as few arc pairs as possible such that every path between the end-vertices contains both arcs of a chosen pair. The main results of this more theoretical part are the classifications of ADP as an NP-complete and SFP as a Sigma-2-P-complete problem.
In Part II, we address a simplified version of the practical project: the Fastest Path with Time Profiles and Waiting (FPTPW) problem. In a directed acyclic graph with durations on the arcs and time windows at the vertices, we search for a fastest path from a source to a target vertex. We are only allowed to be at a vertex within its time windows, and we are only allowed to wait at specified vertices. After introducing departure-duration functions we develop solution algorithms based on these. We consider special cases that significantly reduce the complexity or are of practical relevance. Furthermore, we show that already this simplified problem is in general NP-hard and investigate the complexity status more closely.
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.
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.
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.
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 dissertation presents a generalization of the generalized grey Brownian motion with componentwise independence, called a vector-valued generalized grey Brownian motion (vggBm), and builds a framework of mathematical analysis around this process with the aim of solving stochastic differential equations with respect to this process. Similar to that of the one-dimensional case, the construction of vggBm starts with selecting the appropriate nuclear triple, and construct the corresponding probability measure on the co-nuclear space. Since independence of components are essential in constructing vggBm, a natural way to achieve this is to use the nuclear triple of product spaces: \[ \mathcal{S}_d(\mathbb{R}) \subset L^2_d(\mathbb{R}) \subset \mathcal{S}_d'(\mathbb{R}), \]
where \( L^2_d(\mathbb{R}) \) is the real separable Hilbert space of \( \mathbb{R}^d \)-valued square integrable functions on \( \mathbb{R} \) with respect to the Lebesgue measure, \( \mathcal{S}_d(\mathbb{R}) \) is the external direct sum of \(d\) copies of the nuclear space \(\mathcal{S}(\mathbb{R})\) of Schwartz test functions, and \(\mathcal{S}_d'(\mathbb{R})\) is the dual space of \(\mathcal{S}_d(\mathbb{R})\).
The probability measure used is the the \(d\)-fold product measure of the Mittag-Leffler measure, denoted by \(\mu_{\beta}^{\otimes d}\), whose characteristic function is given by \[ \int_{\mathcal{S}_d'(\mathbb{R})} e^{i\langle\omega,\varphi\rangle}\,\text{d}\mu_{\beta}^{\otimes d}(\omega) = \prod_{k=1}^{d}E_\beta\left(-\frac{1}{2}\langle\varphi_k,\varphi_k\rangle\right),\qquad \varphi\in \mathcal{S}_d(\mathbb{R}), \]
where \( \beta\in(0,1] \), and \( E_\beta \) is the Mittag-Leffler function. Vector-valued generalized grey Brownian motion, denoted by \( B^{\beta,\alpha}_{d}:=(B^{\beta,\alpha}_{d,t})_{t\geq 0}\), is then defined as a process taking values in \( L^2(\mu_{\beta}^{\otimes d};\mathbb{R}^d) \) given by
\[ B^{\beta,\alpha}_{d,t}(\omega) := (\langle\omega_1,M^{\alpha/2}_{-}1\!\!1_{[0,t)}\rangle,\dots,\langle\omega_d,M^{\alpha/2}_{-}1\!\!1_{[0,t)}\rangle),\quad \omega\in\mathcal{S}_d'(\mathbb{R}), \]
where \( M^{\alpha/2} \) is an appropriate fractional operator indexed by \( \alpha\in(0,2) \) and \( 1\!\!1_{[0,t)} \) is the indicator function on the interval \( [0,t) \). This process is, in general, not the aforementioned \(d\)-dimensional analogues of ggBm for \(d\geq 2\), since componentwise independence of the latter process holds only in the Gaussian case.
The study of analysis around vggBm starts with accessibility to Appell systems, so that characterizations and tools for the analysis of the corresponding distribution spaces are established. Then, explicit examples of the use of these characterizations and tools are given: the construction of Donsker's delta function, the existence of local times and self-intersection local times of vggBm, the existence of the derivative of vggBm in the sense of distributions, and the existence of solutions to linear stochastic differential equations with respect to vggBm.
This thesis concerns itself with the long-term behavior of generalized Langevin dynamics with multiplicative noise,
i.e. the solutions to a class of two-component stochastic differential equations in \( \mathbb{R}^{d_1}\times\mathbb{R}^{d_2} \)
subject to outer influence induced by potentials \( \Phi \) and \( \Psi \),
where the stochastic term is only present in the second component, on which it is dependent.
In particular, convergence to an equilibrium defined by an invariant initial distribution \( \mu \) is shown
for weak solutions to the generalized Langevin equation obtained via generalized Dirichlet forms,
and the convergence rate is estimated by applying hypocoercivity methods relying on weak or classical Poincaré inequalities.
As a prerequisite, the space of compactly supported smooth functions is proven to be a domain of essential m-dissipativity
for the associated Kolmogorov backward operator on \(L^2(\mu)\).
In the second part of the thesis, similar Langevin dynamics are considered, however defined on a product of infinite-dimensional separable Hilbert spaces.
The set of finitely based smooth bounded functions is shown to be a domain of essential m-dissipativity for the corresponding Kolmogorov operator \( L \) on \( L^2(\mu) \)
for a Gaussian measure \( \mu \), by applying the previous finite-dimensional result to appropriate restrictions of \( L \).
Under further bounding conditions on the diffusion coefficient relative to the covariance operators of \( \mu \),
hypocoercivity of the generated semigroup is proved, as well as the existence of an associated weakly continuous Markov process
which analytically weakly provides a weak solution to the considered Langevin equation.