@phdthesis{MalemShinitski2023,
  author    = {Malem-Shinitski, Noa},
  title     = {Bayesian inference and modeling for point processes with applications from neuronal activity to scene viewing},
  doi       = {10.25932/publishup-61495},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-614952},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 129},
  year      = {2023},
  abstract  = {Point processes are a common methodology to model sets of events. From earthquakes to social media posts, from the arrival times of neuronal spikes to the timing of crimes, from stock prices to disease spreading -- these phenomena can be reduced to the occurrences of events concentrated in points. Often, these events happen one after the other defining a time--series. Models of point processes can be used to deepen our understanding of such events and for classification and prediction. Such models include an underlying random process that generates the events. This work uses Bayesian methodology to infer the underlying generative process from observed data. Our contribution is twofold -- we develop new models and new inference methods for these processes. We propose a model that extends the family of point processes where the occurrence of an event depends on the previous events. This family is known as Hawkes processes. Whereas in most existing models of such processes, past events are assumed to have only an excitatory effect on future events, we focus on the newly developed nonlinear Hawkes process, where past events could have excitatory and inhibitory effects. After defining the model, we present its inference method and apply it to data from different fields, among others, to neuronal activity. The second model described in the thesis concerns a specific instance of point processes --- the decision process underlying human gaze control. This process results in a series of fixated locations in an image. We developed a new model to describe this process, motivated by the known Exploration--Exploitation dilemma. Alongside the model, we present a Bayesian inference algorithm to infer the model parameters. Remaining in the realm of human scene viewing, we identify the lack of best practices for Bayesian inference in this field. We survey four popular algorithms and compare their performances for parameter inference in two scan path models. The novel models and inference algorithms presented in this dissertation enrich the understanding of point process data and allow us to uncover meaningful insights.},
  language  = {en}
}
@phdthesis{Schindler2023,
  author    = {Schindler, Daniel},
  title     = {Mathematical modeling and simulation of protrusion-driven cell dynamics},
  doi       = {10.25932/publishup-61327},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-613275},
  school      = {Universit{\"a}t Potsdam},
  pages     = {VI, 161},
  year      = {2023},
  abstract  = {Amoeboid cell motility takes place in a variety of biomedical processes such as cancer metastasis, embryonic morphogenesis, and wound healing. In contrast to other forms of cell motility, it is mainly driven by substantial cell shape changes. Based on the interplay of explorative membrane protrusions at the front and a slower-acting membrane retraction at the rear, the cell moves in a crawling kind of way. Underlying these protrusions and retractions are multiple physiological processes resulting in changes of the cytoskeleton, a meshwork of different multi-functional proteins. The complexity and versatility of amoeboid cell motility raise the need for novel computational models based on a profound theoretical framework to analyze and simulate the dynamics of the cell shape. The objective of this thesis is the development of (i) a mathematical framework to describe contour dynamics in time and space, (ii) a computational model to infer expansion and retraction characteristics of individual cell tracks and to produce realistic contour dynamics, (iii) and a complementing Open Science approach to make the above methods fully accessible and easy to use. In this work, we mainly used single-cell recordings of the model organism Dictyostelium discoideum. Based on stacks of segmented microscopy images, we apply a Bayesian approach to obtain smooth representations of the cell membrane, so-called cell contours. We introduce a one-parameter family of regularized contour flows to track reference points on the contour (virtual markers) in time and space. This way, we define a coordinate system to visualize local geometric and dynamic quantities of individual contour dynamics in so-called kymograph plots. In particular, we introduce the local marker dispersion as a measure to identify membrane protrusions and retractions in a fully automated way. This mathematical framework is the basis of a novel contour dynamics model, which consists of three biophysiologically motivated components: one stochastic term, accounting for membrane protrusions, and two deterministic terms to control the shape and area of the contour, which account for membrane retractions. Our model provides a fully automated approach to infer protrusion and retraction characteristics from experimental cell tracks while being also capable of simulating realistic and qualitatively different contour dynamics. Furthermore, the model is used to classify two different locomotion types: the amoeboid and a so-called fan-shaped type. With the complementing Open Science approach, we ensure a high standard regarding the usability of our methods and the reproducibility of our research. In this context, we introduce our software publication named AmoePy, an open-source Python package to segment, analyze, and simulate amoeboid cell motility. Furthermore, we describe measures to improve its usability and extensibility, e.g., by detailed run instructions and an automatically generated source code documentation, and to ensure its functionality and stability, e.g., by automatic software tests, data validation, and a hierarchical package structure. The mathematical approaches of this work provide substantial improvements regarding the modeling and analysis of amoeboid cell motility. We deem the above methods, due to their generalized nature, to be of greater value for other scientific applications, e.g., varying organisms and experimental setups or the transition from unicellular to multicellular movement. Furthermore, we enable other researchers from different fields, i.e., mathematics, biophysics, and medicine, to apply our mathematical methods. By following Open Science standards, this work is of greater value for the cell migration community and a potential role model for other Open Science contributions.},
  language  = {en}
}
@phdthesis{LopezValencia2023,
  author    = {Lopez Valencia, Diego Andres},
  title     = {The Milnor-Moore and Poincar{\´e}-Birkhoff-Witt theorems in the locality set up and the polar structure of Shintani zeta functions},
  doi       = {10.25932/publishup-59421},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-594213},
  school      = {Universit{\"a}t Potsdam},
  pages     = {147},
  year      = {2023},
  abstract  = {This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincar{\´e}-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions. The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincar{\´e}-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincar{\´e}-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts. The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.},
  language  = {en}
}
@phdthesis{Gehring2023,
  author    = {Gehring, Penelope},
  title     = {Non-local boundary conditions for the spin Dirac operator on spacetimes with timelike boundary},
  doi       = {10.25932/publishup-57775},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-577755},
  school      = {Universit{\"a}t Potsdam},
  pages     = {100},
  year      = {2023},
  abstract  = {Non-local boundary conditions - for example the Atiyah-Patodi-Singer (APS) conditions - for Dirac operators on Riemannian manifolds are rather well-understood, while not much is known for such operators on Lorentzian manifolds. Recently, B{\"a}r and Strohmaier [15] and Drago, Große, and Murro [27] introduced APS-like conditions for the spin Dirac operator on Lorentzian manifolds with spacelike and timelike boundary, respectively. While B{\"a}r and Strohmaier [15] showed the Fredholmness of the Dirac operator with these boundary conditions, Drago, Große, and Murro [27] proved the well-posedness of the corresponding initial boundary value problem under certain geometric assumptions. In this thesis, we will follow the footsteps of the latter authors and discuss whether the APS-like conditions for Dirac operators on Lorentzian manifolds with timelike boundary can be replaced by more general conditions such that the associated initial boundary value problems are still wellposed. We consider boundary conditions that are local in time and non-local in the spatial directions. More precisely, we use the spacetime foliation arising from the Cauchy temporal function and split the Dirac operator along this foliation. This gives rise to a family of elliptic operators each acting on spinors of the spin bundle over the corresponding timeslice. The theory of elliptic operators then ensures that we can find families of non-local boundary conditions with respect to this family of operators. Proceeding, we use such a family of boundary conditions to define a Lorentzian boundary condition on the whole timelike boundary. By analyzing the properties of the Lorentzian boundary conditions, we then find sufficient conditions on the family of non-local boundary conditions that lead to the well-posedness of the corresponding Cauchy problems. The well-posedness itself will then be proven by using classical tools including energy estimates and approximation by solutions of the regularized problems. Moreover, we use this theory to construct explicit boundary conditions for the Lorentzian Dirac operator. More precisely, we will discuss two examples of boundary conditions - the analogue of the Atiyah-Patodi-Singer and the chirality conditions, respectively, in our setting. For doing this, we will have a closer look at the theory of non-local boundary conditions for elliptic operators and analyze the requirements on the family of non-local boundary conditions for these specific examples.},
  language  = {en}
}
@phdthesis{Hain2022,
  author    = {Hain, Tobias Martin},
  title     = {Structure formation and identification in geometrically driven soft matter systems},
  doi       = {10.25932/publishup-55880},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-558808},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xviii, 171},
  year      = {2022},
  abstract  = {Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces. Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system's constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness. In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system "forced" to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate. We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase. The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy.},
  language  = {en}
}
@phdthesis{Schanner2022,
  author    = {Schanner, Maximilian Arthus},
  title     = {Correlation based modeling of the archeomagnetic field},
  doi       = {10.25932/publishup-55587},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-555875},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 146},
  year      = {2022},
  abstract  = {The geomagnetic main field is vital for live on Earth, as it shields our habitat against the solar wind and cosmic rays. It is generated by the geodynamo in the Earth's outer core and has a rich dynamic on various timescales. Global models of the field are used to study the interaction of the field and incoming charged particles, but also to infer core dynamics and to feed numerical simulations of the geodynamo. Modern satellite missions, such as the SWARM or the CHAMP mission, support high resolution reconstructions of the global field. From the 19 th century on, a global network of magnetic observatories has been established. It is growing ever since and global models can be constructed from the data it provides. Geomagnetic field models that extend further back in time rely on indirect observations of the field, i.e. thermoremanent records such as burnt clay or volcanic rocks and sediment records from lakes and seas. These indirect records come with (partially very large) uncertainties, introduced by the complex measurement methods and the dating procedure. Focusing on thermoremanent records only, the aim of this thesis is the development of a new modeling strategy for the global geomagnetic field during the Holocene, which takes the uncertainties into account and produces realistic estimates of the reliability of the model. This aim is approached by first considering snapshot models, in order to address the irregular spatial distribution of the records and the non-linear relation of the indirect observations to the field itself. In a Bayesian setting, a modeling algorithm based on Gaussian process regression is developed and applied to binned data. The modeling algorithm is then extended to the temporal domain and expanded to incorporate dating uncertainties. Finally, the algorithm is sequentialized to deal with numerical challenges arising from the size of the Holocene dataset. The central result of this thesis, including all of the aspects mentioned, is a new global geomagnetic field model. It covers the whole Holocene, back until 12000 BCE, and we call it ArchKalmag14k. When considering the uncertainties that are produced together with the model, it is evident that before 6000 BCE the thermoremanent database is not sufficient to support global models. For times more recent, ArchKalmag14k can be used to analyze features of the field under consideration of posterior uncertainties. The algorithm for generating ArchKalmag14k can be applied to different datasets and is provided to the community as an open source python package.},
  language  = {en}
}
@phdthesis{Fischer2022,
  author    = {Fischer, Jens Walter},
  title     = {Random dynamics in collective behavior - consensus, clustering \& extinction of populations},
  doi       = {10.25932/publishup-55372},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-553725},
  school      = {Universit{\"a}t Potsdam},
  pages     = {242},
  year      = {2022},
  abstract  = {The echo chamber model describes the development of groups in heterogeneous social networks. By heterogeneous social network we mean a set of individuals, each of whom represents exactly one opinion. The existing relationships between individuals can then be represented by a graph. The echo chamber model is a time-discrete model which, like a board game, is played in rounds. In each round, an existing relationship is randomly and uniformly selected from the network and the two connected individuals interact. If the opinions of the individuals involved are sufficiently similar, they continue to move closer together in their opinions, whereas in the case of opinions that are too far apart, they break off their relationship and one of the individuals seeks a new relationship. In this paper we examine the building blocks of this model. We start from the observation that changes in the structure of relationships in the network can be described by a system of interacting particles in a more abstract space. These reflections lead to the definition of a new abstract graph that encompasses all possible relational configurations of the social network. This provides us with the geometric understanding necessary to analyse the dynamic components of the echo chamber model in Part III. As a first step, in Part 7, we leave aside the opinions of the inidividuals and assume that the position of the edges changes with each move as described above, in order to obtain a basic understanding of the underlying dynamics. Using Markov chain theory, we find upper bounds on the speed of convergence of an associated Markov chain to its unique stationary distribution and show that there are mutually identifiable networks that are not apparent in the dynamics under analysis, in the sense that the stationary distribution of the associated Markov chain gives equal weight to these networks. In the reversible cases, we focus in particular on the explicit form of the stationary distribution as well as on the lower bounds of the Cheeger constant to describe the convergence speed. The final result of Section 8, based on absorbing Markov chains, shows that in a reduced version of the echo chamber model, a hierarchical structure of the number of conflicting relations can be identified. We can use this structure to determine an upper bound on the expected absorption time, using a quasi-stationary distribution. This hierarchy of structure also provides a bridge to classical theories of pure death processes. We conclude by showing how future research can exploit this link and by discussing the importance of the results as building blocks for a full theoretical understanding of the echo chamber model. Finally, Part IV presents a published paper on the birth-death process with partial catastrophe. The paper is based on the explicit calculation of the first moment of a catastrophe. This first part is entirely based on an analytical approach to second degree recurrences with linear coefficients. The convergence to 0 of the resulting sequence as well as the speed of convergence are proved. On the other hand, the determination of the upper bounds of the expected value of the population size as well as its variance and the difference between the determined upper bound and the actual value of the expected value. For these results we use almost exclusively the theory of ordinary nonlinear differential equations.},
  language  = {en}
}
@phdthesis{Hannes2022,
  author    = {Hannes, Sebastian},
  title     = {Boundary Value Problems for the Lorentzian Dirac Operator},
  doi       = {10.25932/publishup-54839},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-548391},
  school      = {Universit{\"a}t Potsdam},
  pages     = {67},
  year      = {2022},
  abstract  = {The index theorem for elliptic operators on a closed Riemannian manifold by Atiyah and Singer has many applications in analysis, geometry and topology, but it is not suitable for a generalization to a Lorentzian setting. In the case where a boundary is present Atiyah, Patodi and Singer provide an index theorem for compact Riemannian manifolds by introducing non-local boundary conditions obtained via the spectral decomposition of an induced boundary operator, so called APS boundary conditions. B{\"a}r and Strohmaier prove a Lorentzian version of this index theorem for the Dirac operator on a manifold with boundary by utilizing results from APS and the characterization of the spectral flow by Phillips. In their case the Lorentzian manifold is assumed to be globally hyperbolic and spatially compact, and the induced boundary operator is given by the Riemannian Dirac operator on a spacelike Cauchy hypersurface. Their results show that imposing APS boundary conditions for these boundary operator will yield a Fredholm operator with a smooth kernel and its index can be calculated by a formula similar to the Riemannian case. Back in the Riemannian setting, B{\"a}r and Ballmann provide an analysis of the most general kind of boundary conditions that can be imposed on a first order elliptic differential operator that will still yield regularity for solutions as well as Fredholm property for the resulting operator. These boundary conditions can be thought of as deformations to the graph of a suitable operator mapping APS boundary conditions to their orthogonal complement. This thesis aims at applying the boundary conditions found by B{\"a}r and Ballmann to a Lorentzian setting to understand more general types of boundary conditions for the Dirac operator, conserving Fredholm property as well as providing regularity results and relative index formulas for the resulting operators. As it turns out, there are some differences in applying these graph-type boundary conditions to the Lorentzian Dirac operator when compared to the Riemannian setting. It will be shown that in contrast to the Riemannian case, going from a Fredholm boundary condition to its orthogonal complement works out fine in the Lorentzian setting. On the other hand, in order to deduce Fredholm property and regularity of solutions for graph-type boundary conditions, additional assumptions for the deformation maps need to be made. The thesis is organized as follows. In chapter 1 basic facts about Lorentzian and Riemannian spin manifolds, their spinor bundles and the Dirac operator are listed. These will serve as a foundation to define the setting and prove the results of later chapters. Chapter 2 defines the general notion of boundary conditions for the Dirac operator used in this thesis and introduces the APS boundary conditions as well as their graph type deformations. Also the role of the wave evolution operator in finding Fredholm boundary conditions is analyzed and these boundary conditions are connected to notion of Fredholm pairs in a given Hilbert space. Chapter 3 focuses on the principal symbol calculation of the wave evolution operator and the results are used to proof Fredholm property as well as regularity of solutions for suitable graph-type boundary conditions. Also sufficient conditions are derived for (pseudo-)local boundary conditions imposed on the Dirac operator to yield a Fredholm operator with a smooth solution space. In the last chapter 4, a few examples of boundary conditions are calculated applying the results of previous chapters. Restricting to special geometries and/or boundary conditions, results can be obtained that are not covered by the more general statements, and it is shown that so-called transmission conditions behave very differently than in the Riemannian setting.},
  language  = {en}
}
@phdthesis{Mauerberger2022,
  author    = {Mauerberger, Stefan},
  title     = {Correlation based Bayesian modeling},
  doi       = {10.25932/publishup-53782},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-537827},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 128},
  year      = {2022},
  abstract  = {The motivation for this work was the question of reliability and robustness of seismic tomography. The problem is that many earth models exist which can describe the underlying ground motion records equally well. Most algorithms for reconstructing earth models provide a solution, but rarely quantify their variability. If there is no way to verify the imaged structures, an interpretation is hardly reliable. The initial idea was to explore the space of equivalent earth models using Bayesian inference. However, it quickly became apparent that the rigorous quantification of tomographic uncertainties could not be accomplished within the scope of a dissertation. In order to maintain the fundamental concept of statistical inference, less complex problems from the geosciences are treated instead. This dissertation aims to anchor Bayesian inference more deeply in the geosciences and to transfer knowledge from applied mathematics. The underlying idea is to use well-known methods and techniques from statistics to quantify the uncertainties of inverse problems in the geosciences. This work is divided into three parts: Part I introduces the necessary mathematics and should be understood as a kind of toolbox. With a physical application in mind, this section provides a compact summary of all methods and techniques used. The introduction of Bayesian inference makes the beginning. Then, as a special case, the focus is on regression with Gaussian processes under linear transformations. The chapters on the derivation of covariance functions and the approximation of non-linearities are discussed in more detail. Part II presents two proof of concept studies in the field of seismology. The aim is to present the conceptual application of the introduced methods and techniques with moderate complexity. The example about traveltime tomography applies the approximation of non-linear relationships. The derivation of a covariance function using the wave equation is shown in the example of a damped vibrating string. With these two synthetic applications, a consistent concept for the quantification of modeling uncertainties has been developed. Part III presents the reconstruction of the Earth's archeomagnetic field. This application uses the whole toolbox presented in Part I and is correspondingly complex. The modeling of the past 1000 years is based on real data and reliably quantifies the spatial modeling uncertainties. The statistical model presented is widely used and is under active development. The three applications mentioned are intentionally kept flexible to allow transferability to similar problems. The entire work focuses on the non-uniqueness of inverse problems in the geosciences. It is intended to be of relevance to those interested in the concepts of Bayesian inference.},
  language  = {en}
}
@phdthesis{Perera2021,
  author    = {Perera, Upeksha},
  title     = {Solutions of direct and inverse Sturm-Liouville problems},
  doi       = {10.25932/publishup-53006},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-530064},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 109},
  year      = {2021},
  abstract  = {Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm-Liouville Problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular and some singular SLPs of even orders (tested up to order eight), with a mix of boundary conditions (including non-separable and finite singular endpoints), accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. Next, a concrete implementation to the inverse Sturm-Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm-Liouville problems of order n=2,4 is verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides methods that can be adapted successfully for solving a direct (regular/singular) or inverse SLP of an arbitrary order with arbitrary boundary conditions.},
  language  = {en}
}
@phdthesis{Maier2021,
  author    = {Maier, Corinna},
  title     = {Bayesian data assimilation and reinforcement learning for model-informed precision dosing in oncology},
  doi       = {10.25932/publishup-51587},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-515870},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 138},
  year      = {2021},
  abstract  = {While patients are known to respond differently to drug therapies, current clinical practice often still follows a standardized dosage regimen for all patients. For drugs with a narrow range of both effective and safe concentrations, this approach may lead to a high incidence of adverse events or subtherapeutic dosing in the presence of high patient variability. Model-informedprecision dosing (MIPD) is a quantitative approach towards dose individualization based on mathematical modeling of dose-response relationships integrating therapeutic drug/biomarker monitoring (TDM) data. MIPD may considerably improve the efficacy and safety of many drug therapies. Current MIPD approaches, however, rely either on pre-calculated dosing tables or on simple point predictions of the therapy outcome. These approaches lack a quantification of uncertainties and the ability to account for effects that are delayed. In addition, the underlying models are not improved while applied to patient data. Therefore, current approaches are not well suited for informed clinical decision-making based on a differentiated understanding of the individually predicted therapy outcome. The objective of this thesis is to develop mathematical approaches for MIPD, which (i) provide efficient fully Bayesian forecasting of the individual therapy outcome including associated uncertainties, (ii) integrate Markov decision processes via reinforcement learning (RL) for a comprehensive decision framework for dose individualization, (iii) allow for continuous learning across patients and hospitals. Cytotoxic anticancer chemotherapy with its major dose-limiting toxicity, neutropenia, serves as a therapeutically relevant application example. For more comprehensive therapy forecasting, we apply Bayesian data assimilation (DA) approaches, integrating patient-specific TDM data into mathematical models of chemotherapy-induced neutropenia that build on prior population analyses. The value of uncertainty quantification is demonstrated as it allows reliable computation of the patient-specific probabilities of relevant clinical quantities, e.g., the neutropenia grade. In view of novel home monitoring devices that increase the amount of TDM data available, the data processing of sequential DA methods proves to be more efficient and facilitates handling of the variability between dosing events. By transferring concepts from DA and RL we develop novel approaches for MIPD. While DA-guided dosing integrates individualized uncertainties into dose selection, RL-guided dosing provides a framework to consider delayed effects of dose selections. The combined DA-RL approach takes into account both aspects simultaneously and thus represents a holistic approach towards MIPD. Additionally, we show that RL can be used to gain insights into important patient characteristics for dose selection. The novel dosing strategies substantially reduce the occurrence of both subtherapeutic and life-threatening neutropenia grades in a simulation study based on a recent clinical study (CEPAC-TDM trial) compared to currently used MIPD approaches. If MIPD is to be implemented in routine clinical practice, a certain model bias with respect to the underlying model is inevitable, as the models are typically based on data from comparably small clinical trials that reflect only to a limited extent the diversity in real-world patient populations. We propose a sequential hierarchical Bayesian inference framework that enables continuous cross-patient learning to learn the underlying model parameters of the target patient population. It is important to note that the approach only requires summary information of the individual patient data to update the model. This separation of the individual inference from population inference enables implementation across different centers of care. The proposed approaches substantially improve current MIPD approaches, taking into account new trends in health care and aspects of practical applicability. They enable progress towards more informed clinical decision-making, ultimately increasing patient benefits beyond the current practice.},
  language  = {en}
}
@phdthesis{Zass2021,
  author    = {Zass, Alexander},
  title     = {A multifaceted study of marked Gibbs point processes},
  doi       = {10.25932/publishup-51277},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-512775},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 104},
  year      = {2021},
  abstract  = {This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation. We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight. We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution. Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation. Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry.},
  language  = {en}
}
@phdthesis{Zadorozhnyi2021,
  author    = {Zadorozhnyi, Oleksandr},
  title     = {Contributions to the theoretical analysis of the algorithms with adversarial and dependent data},
  school      = {Universit{\"a}t Potsdam},
  pages     = {144},
  year      = {2021},
  abstract  = {In this work I present the concentration inequalities of Bernstein's type for the norms of Banach-valued random sums under a general functional weak-dependency assumption (the so-called \$\cC-\$mixing). The latter is then used to prove, in the asymptotic framework, excess risk upper bounds of the regularised Hilbert valued statistical learning rules under the τ-mixing assumption on the underlying training sample. These results (of the batch statistical setting) are then supplemented with the regret analysis over the classes of Sobolev balls of the type of kernel ridge regression algorithm in the setting of online nonparametric regression with arbitrary data sequences. Here, in particular, a question of robustness of the kernel-based forecaster is investigated. Afterwards, in the framework of sequential learning, the multi-armed bandit problem under \$\cC-\$mixing assumption on the arm's outputs is considered and the complete regret analysis of a version of Improved UCB algorithm is given. Lastly, probabilistic inequalities of the first part are extended to the case of deviations (both of Azuma-Hoeffding's and of Burkholder's type) to the partial sums of real-valued weakly dependent random fields (under the type of projective dependence condition).},
  language  = {en}
}
@phdthesis{Oancea2021,
  author    = {Oancea, Marius-Adrian},
  title     = {Spin Hall effects in general relativity},
  doi       = {10.25932/publishup-50229},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-502293},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 123},
  year      = {2021},
  abstract  = {The propagation of test fields, such as electromagnetic, Dirac or linearized gravity, on a fixed spacetime manifold is often studied by using the geometrical optics approximation. In the limit of infinitely high frequencies, the geometrical optics approximation provides a conceptual transition between the test field and an effective point-particle description. The corresponding point-particles, or wave rays, coincide with the geodesics of the underlying spacetime. For most astrophysical applications of interest, such as the observation of celestial bodies, gravitational lensing, or the observation of cosmic rays, the geometrical optics approximation and the effective point-particle description represent a satisfactory theoretical model. However, the geometrical optics approximation gradually breaks down as test fields of finite frequency are considered. In this thesis, we consider the propagation of test fields on spacetime, beyond the leading-order geometrical optics approximation. By performing a covariant Wentzel-Kramers-Brillouin analysis for test fields, we show how higher-order corrections to the geometrical optics approximation can be considered. The higher-order corrections are related to the dynamics of the spin internal degree of freedom of the considered test field. We obtain an effective point-particle description, which contains spin-dependent corrections to the geodesic motion obtained using geometrical optics. This represents a covariant generalization of the well-known spin Hall effect, usually encountered in condensed matter physics and in optics. Our analysis is applied to electromagnetic and massive Dirac test fields, but it can easily be extended to other fields, such as linearized gravity. In the electromagnetic case, we present several examples where the gravitational spin Hall effect of light plays an important role. These include the propagation of polarized light rays on black hole spacetimes and cosmological spacetimes, as well as polarization-dependent effects on the shape of black hole shadows. Furthermore, we show that our effective point-particle equations for polarized light rays reproduce well-known results, such as the spin Hall effect of light in an inhomogeneous medium, and the relativistic Hall effect of polarized electromagnetic wave packets encountered in Minkowski spacetime.},
  language  = {en}
}
@phdthesis{Friedrich2020,
  author    = {Friedrich, Alexander},
  title     = {Minimizers of generalized Willmore energies and applications in general relativity},
  doi       = {10.25932/publishup-48142},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-481423},
  school      = {Universit{\"a}t Potsdam},
  pages     = {100},
  year      = {2020},
  abstract  = {Das Willmore Funktional ist eine Funktion die jeder Fl{\"a}che in einer Riemannschen Mannigfaltigkeit, ihre totale mittlere Kr{\"u}mmung zuweist. Ein klassisches Problem der Differentialgeometrie ist es geschlossene (kompakt und ohne Rand) Fl{\"a}chen zu finden die das Willmore funktional minimieren, beziehungsweise die kritische Punkte des Willmore Funktionals sind. In dieser Doktorarbeit entwickeln wir ein Konzept von verallgemeinerten Willmore Funktionalen f{\"u}r Fl{\"a}chen in Riemannschen Mannigfaltigkeiten, wobei wir uns von physikalischen Modellen leiten lassen. Insbesondere ist hier die Hawking Energie der allgemeinen Relativit{\"a}tstheorie und die Biegungsenergie von d{\"u}nnen Membranen zu nennen. F{\"u}r dieses verallgemeinerten Willmore Funktionale beweisen wir die Existenz von Minimieren mit vorgeschriebenen Fl{\"a}cheninhalt, in einer geeigneten Klasse von verallgemeinerten Fl{\"a}chen. Insbesondere konstruieren wir Minimierer der oben erw{\"a}hnten Biegungsenergie mit vorgeschrieben Fl{\"a}cheninhalt und vorgeschriebenen, eingeschlossenem Volumen. Außerdem beweisen wir, dass kritische Punkte von verallgemeinerten Willmore Funktionalen mit vorgeschriebenen Fl{\"a}cheninhalt abseits endlich vieler Punkte glatt sind. Dabei st{\"u}tzen wir uns, wie auch im folgenden, auf die bestehende Theorie f{\"u}r das Willmore Funktional. An diese allgemeinen Resultate schließen wir eine detailliertere Analyse der Hawking Energie an. Im Kontext der allgemeinen Relativit{\"a}tstheorie beschreibt die Umgebungsmannigfaltigkeit den Raum zu einem Zeitpunkt. Daher sind wir an dem Wechselspiel zwischen der Hawking Energie und der umgebenden Mannigfaltigkeit interessiert. Wir charakterisieren Punkte in der umgebenden Mannigfaltigkeit f{\"u}r die es in jeder Umgebung eine kritische Fl{\"a}che mit vorgeschriebenem, kleinem Fl{\"a}cheninhalt gibt. Diese Punnkte werden als Konzentrationspunkte der Hawking Energie interpretiert. Außerdem berechnen wir eine Entwicklung der Hawking Energie auf kleinen, runden Sph{\"a}ren. Dadurch k{\"o}nnen wir eine Art Energiedichte der Hawking Energie identifizieren. Hierbei ist anzumerken, dass unsere Resultate im Kontrast zu Ergebnissen in der Literatur stehen. Dort wurde berechnet, dass die Entwicklung der Hawking Energie auf Sph{\"a}ren im Lichtkegel eines Punktes der umgebenden Mannigfaltigkeit in f{\"u}hrender Ordnung proportional zur der klassischen Energiedichte der allgemeinen Relativit{\"a}tstheorie ist. Zu diesem Zeitpunkt ist nicht klar wie diese Diskrepanz zu begr{\"u}nden ist. Ferner betrachten wir asymptotisch Schwarzschild Mannigfaltigkeiten. Sie sind ein Spezialfall von asymptotisch flachen Mannigfaltigkeiten, welche in der allgemeinen Relativit{\"a}tstheorie als Modelle f{\"u}r isolierte Systeme dienen. Die Schwarzschild Raumzeit selbst ist eine rotationssymmetrische Raumzeit die schwarzen Loch beschreibt. In diesen asymptotisch Schwarzschild Mannigfaltigkeiten konstruieren wir eine Bl{\"a}tterung des {\"a}ußeren Bereiches durch kritische Fl{\"a}chen der Hawking Energie mit vorgeschriebenen Fl{\"a}cheninhalt. Diese Bl{\"a}tterung kann in einem verallgemeinertem Sinne als Schwerpunkt des isolierten Systems betrachtet werden. Außerdem zeigen wir, dass die Hawking Energie entlang der Bl{\"a}tterung w{\"a}chst je gr{\"o}ßer die Fl{\"a}chen werden.},
  language  = {en}
}
@phdthesis{Reinhardt2020,
  author    = {Reinhardt, Maria},
  title     = {Hybrid filters and multi-scale models},
  doi       = {10.25932/publishup-47435},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-474356},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xiii, 102},
  year      = {2020},
  abstract  = {This thesis is concerned with Data Assimilation, the process of combining model predictions with observations. So called filters are of special interest. One is inter- ested in computing the probability distribution of the state of a physical process in the future, given (possibly) imperfect measurements. This is done using Bayes' rule. The first part focuses on hybrid filters, that bridge between the two main groups of filters: ensemble Kalman filters (EnKF) and particle filters. The first are a group of very stable and computationally cheap algorithms, but they request certain strong assumptions. Particle filters on the other hand are more generally applicable, but computationally expensive and as such not always suitable for high dimensional systems. Therefore it exists a need to combine both groups to benefit from the advantages of each. This can be achieved by splitting the likelihood function, when assimilating a new observation and treating one part of it with an EnKF and the other part with a particle filter. The second part of this thesis deals with the application of Data Assimilation to multi-scale models and the problems that arise from that. One of the main areas of application for Data Assimilation techniques is predicting the development of oceans and the atmosphere. These processes involve several scales and often balance rela- tions between the state variables. The use of Data Assimilation procedures most often violates relations of that kind, which leads to unrealistic and non-physical pre- dictions of the future development of the process eventually. This work discusses the inclusion of a post-processing step after each assimilation step, in which a minimi- sation problem is solved, which penalises the imbalance. This method is tested on four different models, two Hamiltonian systems and two spatially extended models, which adds even more difficulties.},
  language  = {en}
}
@phdthesis{Goebel2019,
  author    = {G{\"o}bel, Franziska},
  title     = {Contributions to multiscale statistical analysis on random geometric graphs},
  school      = {Universit{\"a}t Potsdam},
  pages     = {127},
  year      = {2019},
  language  = {en}
}
@phdthesis{Schwarz2020,
  author    = {Schwarz, Michael},
  title     = {Nodal domains and boundary representation for Dirichlet forms},
  school      = {Universit{\"a}t Potsdam},
  pages     = {164},
  year      = {2020},
  language  = {en}
}
@phdthesis{Muench2019,
  author    = {M{\"u}nch, Florentin},
  title     = {Discrete Ricci curvature, diameter bounds and rigidity},
  school      = {Universit{\"a}t Potsdam},
  pages     = {68},
  year      = {2019},
  language  = {en}
}
@phdthesis{Knoechel2019,
  author    = {Kn{\"o}chel, Jane},
  title     = {Model reduction of mechanism-based pharmacodynamic models and its link to classical drug effect models},
  doi       = {10.25932/publishup-44059},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-440598},
  school      = {Universit{\"a}t Potsdam},
  pages     = {vii, 147},
  year      = {2019},
  abstract  = {Continuous insight into biological processes has led to the development of large-scale, mechanistic systems biology models of pharmacologically relevant networks. While these models are typically designed to study the impact of diverse stimuli or perturbations on multiple system variables, the focus in pharmacological research is often on a specific input, e.g., the dose of a drug, and a specific output related to the drug effect or response in terms of some surrogate marker. To study a chosen input-output pair, the complexity of the interactions as well as the size of the models hinders easy access and understanding of the details of the input-output relationship. The objective of this thesis is the development of a mathematical approach, in specific a model reduction technique, that allows (i) to quantify the importance of the different state variables for a given input-output relationship, and (ii) to reduce the dynamics to its essential features -- allowing for a physiological interpretation of state variables as well as parameter estimation in the statistical analysis of clinical data. We develop a model reduction technique using a control theoretic setting by first defining a novel type of time-limited controllability and observability gramians for nonlinear systems. We then show the superiority of the time-limited generalised gramians for nonlinear systems in the context of balanced truncation for a benchmark system from control theory. The concept of time-limited controllability and observability gramians is subsequently used to introduce a state and time-dependent quantity called the input-response (ir) index that quantifies the importance of state variables for a given input-response relationship at a particular time. We subsequently link our approach to sensitivity analysis, thus, enabling for the first time the use of sensitivity coefficients for state space reduction. The sensitivity based ir-indices are given as a product of two sensitivity coefficients. This allows not only for a computational more efficient calculation but also for a clear distinction of the extent to which the input impacts a state variable and the extent to which a state variable impacts the output. The ir-indices give insight into the coordinated action of specific state variables for a chosen input-response relationship. Our developed model reduction technique results in reduced models that still allow for a mechanistic interpretation in terms of the quantities/state variables of the original system, which is a key requirement in the field of systems pharmacology and systems biology and distinguished the reduced models from so-called empirical drug effect models. The ir-indices are explicitly defined with respect to a reference trajectory and thereby dependent on the initial state (this is an important feature of the measure). This is demonstrated for an example from the field of systems pharmacology, showing that the reduced models are very informative in their ability to detect (genetic) deficiencies in certain physiological entities. Comparing our novel model reduction technique to the already existing techniques shows its superiority. The novel input-response index as a measure of the importance of state variables provides a powerful tool for understanding the complex dynamics of large-scale systems in the context of a specific drug-response relationship. Furthermore, the indices provide a means for a very efficient model order reduction and, thus, an important step towards translating insight from biological processes incorporated in detailed systems pharmacology models into the population analysis of clinical data.},
  language  = {en}
}
@phdthesis{Lewandowski2019,
  author    = {Lewandowski, Max},
  title     = {Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes},
  doi       = {10.25932/publishup-43938},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-439381},
  school      = {Universit{\"a}t Potsdam},
  pages     = {v, 69},
  year      = {2019},
  abstract  = {Quantenfeldtheorie auf gekr{\"u}mmten Raumzeiten ist eine semiklassische N{\"a}herung einer Quantentheorie der Gravitation, im Rahmen derer ein Quantenfeld unter dem Einfluss eines klassisch modellierten Gravitationsfeldes, also einer gekr{\"u}mmten Raumzeit, beschrieben wird. Eine der bemerkenswertesten Vorhersagen dieses Ansatzes ist die Erzeugung von Teilchen durch die gekr{\"u}mmte Raumzeit selbst, wie zum Beispiel durch Hawkings Verdampfen schwarzer L{\"o}cher und den Unruh Effekt. Andererseits deuten diese Aspekte bereits an, dass fundamentale Grundpfeiler der Theorie auf dem Minkowskiraum, insbesondere ein ausgezeichneter Vakuumzustand und damit verbunden der Teilchenbegriff, f{\"u}r allgemeine gekr{\"u}mmte Raumzeiten keine sinnvolle Entsprechung besitzen. Gleichermaßen ben{\"o}tigen wir eine alternative Implementierung von Kovarianz in die Theorie, da gekr{\"u}mmte Raumzeiten im Allgemeinen keine nicht-triviale globale Symmetrie aufweisen. Letztere Problematik konnte im Rahmen lokal-kovarianter Quantenfeldtheorie gel{\"o}st werden, wohingegen die Abwesenheit entsprechender Konzepte f{\"u}r Vakuum und Teilchen in diesem allgemeinen Fall inzwischen sogar in Form von no-go-Aussagen manifestiert wurde. Beim algebraischen Ansatz f{\"u}r eine Quantenfeldtheorie werden zun{\"a}chst Observablen eingef{\"u}hrt und erst anschließend Zust{\"a}nde via Zuordnung von Erwartungswerten. Obwohl die Observablen unter physikalischen Gesichtspunkten konstruiert werden, existiert dennoch eine große Anzahl von m{\"o}glichen Zust{\"a}nden, von denen viele, aus physikalischen Blickwinkeln betrachtet, nicht sinnvoll sind. Dieses Konzept von Zust{\"a}nden ist daher noch zu allgemein und bedarf weiterer physikalisch motivierter Einschr{\"a}nkungen. Beispielsweise ist es nat{\"u}rlich, sich im Falle freier Quantenfeldtheorien mit linearen Feldgleichungen auf quasifreie Zust{\"a}nde zu konzentrieren. Dar{\"u}ber hinaus ist die Renormierung von Erwartungswerten f{\"u}r Produkte von Feldern von zentraler Bedeutung. Dies betrifft insbesondere den Energie-Impuls-Tensor, dessen Erwartungswert durch distributionelle Bil{\"o}sungen der Feldgleichungen gegeben ist. Tats{\"a}chlich liefert J. Hadamard Theorie hyperbolischer Differentialgleichungen Bil{\"o}sungen mit festem singul{\"a}ren Anteil, so dass ein geeignetes Renormierungsverfahren definiert werden kann. Die sogenannte Hadamard-Bedingung an Bidistributionen steht f{\"u}r die Forderung einer solchen Singularit{\"a}tenstruktur und sie hat sich etabliert als nat{\"u}rliche Verallgemeinerung der f{\"u}r flache Raumzeiten formulierten Spektralbedingung. Seit Radzikowskis wegweisenden Resultaten l{\"a}sst sie sich außerdem lokal ausdr{\"u}cken, n{\"a}mlich als eine Bedingung an die Wellenfrontenmenge der Bil{\"o}sung. Diese Formulierung schl{\"a}gt eine Br{\"u}cke zu der von Duistermaat und H{\"o}rmander entwickelten mikrolokalen Analysis, die seitdem bei der {\"U}berpr{\"u}fung der Hadamard-Bedingung sowie der Konstruktion von Hadamard Zust{\"a}nden vielfach Verwendung findet und rasante Fortschritte auf diesem Gebiet ausgel{\"o}st hat. Obwohl unverzichtbar f{\"u}r die Analyse der Charakteristiken von Operatoren und ihrer Parametrizen sind die Methoden und Aussagen der mikrolokalen Analysis ungeeignet f{\"u}r die Analyse von nicht-singul{\"a}ren Strukturen und zentrale Aussagen sind typischerweise bis auf glatte Anteile formuliert. Beispielsweise lassen sich aus Radzikowskis Resultaten nahezu direkt Existenzaussagen und sogar ein konkretes Konstruktionsschema f{\"u}r Hadamard Zust{\"a}nde ableiten, die {\"u}brigen Eigenschaften (Bil{\"o}sung, Kausalit{\"a}t, Positivit{\"a}t) k{\"o}nnen jedoch auf diesem Wege nur modulo glatte Funktionen gezeigt werden. Es ist das Ziel dieser Dissertation, diesen Ansatz f{\"u}r lineare Wellenoperatoren auf Schnitten in Vektorb{\"u}ndeln {\"u}ber global-hyperbolischen Lorentz-Mannigfaltigkeiten zu vollenden und, ausgehend von einer lokalen Hadamard Reihe, Hadamard Zust{\"a}nde zu konstruieren. Beruhend auf Wightmans L{\"o}sung f{\"u}r die d'Alembert-Gleichung auf dem Minkowski-Raum und der Herleitung der avancierten und retardierten Fundamentall{\"o}sung konstruieren wir lokal Parametrizen in Form von Hadamard-Reihen und f{\"u}gen sie zu globalen Bil{\"o}sungen zusammen. Diese besitzen dann die Hadamard-Eigenschaft und wir zeigen anschließend, dass glatte Bischnitte existieren, die addiert werden k{\"o}nnen, so dass die verbleibenden Bedingungen erf{\"u}llt sind.},
  language  = {en}
}
@phdthesis{Angwenyi2019,
  author    = {Angwenyi, David},
  title     = {Time-continuous state and parameter estimation with application to hyperbolic SPDEs},
  doi       = {10.25932/publishup-43654},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-436542},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xi, 101},
  year      = {2019},
  abstract  = {Data assimilation has been an active area of research in recent years, owing to its wide utility. At the core of data assimilation are filtering, prediction, and smoothing procedures. Filtering entails incorporation of measurements' information into the model to gain more insight into a given state governed by a noisy state space model. Most natural laws are governed by time-continuous nonlinear models. For the most part, the knowledge available about a model is incomplete; and hence uncertainties are approximated by means of probabilities. Time-continuous filtering, therefore, holds promise for wider usefulness, for it offers a means of combining noisy measurements with imperfect model to provide more insight on a given state. The solution to time-continuous nonlinear Gaussian filtering problem is provided for by the Kushner-Stratonovich equation. Unfortunately, the Kushner-Stratonovich equation lacks a closed-form solution. Moreover, the numerical approximations based on Taylor expansion above third order are fraught with computational complications. For this reason, numerical methods based on Monte Carlo methods have been resorted to. Chief among these methods are sequential Monte-Carlo methods (or particle filters), for they allow for online assimilation of data. Particle filters are not without challenges: they suffer from particle degeneracy, sample impoverishment, and computational costs arising from resampling. The goal of this thesis is to:— i) Review the derivation of Kushner-Stratonovich equation from first principles and its extant numerical approximation methods, ii) Study the feedback particle filters as a way of avoiding resampling in particle filters, iii) Study joint state and parameter estimation in time-continuous settings, iv) Apply the notions studied to linear hyperbolic stochastic differential equations. The interconnection between It{\^o} integrals and stochastic partial differential equations and those of Stratonovich is introduced in anticipation of feedback particle filters. With these ideas and motivated by the variants of ensemble Kalman-Bucy filters founded on the structure of the innovation process, a feedback particle filter with randomly perturbed innovation is proposed. Moreover, feedback particle filters based on coupling of prediction and analysis measures are proposed. They register a better performance than the bootstrap particle filter at lower ensemble sizes. We study joint state and parameter estimation, both by means of extended state spaces and by use of dual filters. Feedback particle filters seem to perform well in both cases. Finally, we apply joint state and parameter estimation in the advection and wave equation, whose velocity is spatially varying. Two methods are employed: Metropolis Hastings with filter likelihood and a dual filter comprising of Kalman-Bucy filter and ensemble Kalman-Bucy filter. The former performs better than the latter.},
  language  = {en}
}
@phdthesis{Mahmoudi,
  author    = {Mahmoudi, Mahdi Hedayat},
  title     = {New applications of the edge calculus},
  school      = {Universit{\"a}t Potsdam},
  pages     = {126},
  language  = {en}
}
@phdthesis{Jakobs2019,
  author    = {Jakobs, Friedrich},
  title     = {Dubrovin-rings and their connection to Hughes-free skew fields of fractions},
  doi       = {10.25932/publishup-43556},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-435561},
  school      = {Universit{\"a}t Potsdam},
  pages     = {ix, 62},
  year      = {2019},
  abstract  = {One method of embedding groups into skew fields was introduced by A. I. Mal'tsev and B. H. Neumann (cf. [18, 19]). If G is an ordered group and F is a skew field, the set F((G)) of formal power series over F in G with well-ordered support forms a skew field into which the group ring F[G] can be embedded. Unfortunately it is not suficient that G is left-ordered since F((G)) is only an F-vector space in this case as there is no natural way to define a multiplication on F((G)). One way to extend the original idea onto left-ordered groups is to examine the endomorphism ring of F((G)) as explored by N. I. Dubrovin (cf. [5, 6]). It is possible to embed any crossed product ring F[G; η, σ] into the endomorphism ring of F((G)) such that each non-zero element of F[G; η, σ] defines an automorphism of F((G)) (cf. [5, 10]). Thus, the rational closure of F[G; η, σ] in the endomorphism ring of F((G)), which we will call the Dubrovin-ring of F[G; η, σ], is a potential candidate for a skew field of fractions of F[G; η, σ]. The methods of N. I. Dubrovin allowed to show that specific classes of groups can be embedded into a skew field. For example, N. I. Dubrovin contrived some special criteria, which are applicable on the universal covering group of SL(2, R). These methods have also been explored by J. Gr{\"a}ter and R. P. Sperner (cf. [10]) as well as N.H. Halimi and T. Ito (cf. [11]). Furthermore, it is of interest to know if skew fields of fractions are unique. For example, left and right Ore domains have unique skew fields of fractions (cf. [2]). This is not the general case as for example the free group with 2 generators can be embedded into non-isomorphic skew fields of fractions (cf. [12]). It seems likely that Ore domains are the most general case for which unique skew fields of fractions exist. One approach to gain uniqueness is to restrict the search to skew fields of fractions with additional properties. I. Hughes has defined skew fields of fractions of crossed product rings F[G; η, σ] with locally indicable G which fulfill a special condition. These are called Hughes-free skew fields of fractions and I. Hughes has proven that they are unique if they exist [13, 14]. This thesis will connect the ideas of N. I. Dubrovin and I. Hughes. The first chapter contains the basic terminology and concepts used in this thesis. We present methods provided by N. I. Dubrovin such as the complexity of elements in rational closures and special properties of endomorphisms of the vector space of formal power series F((G)). To combine the ideas of N.I. Dubrovin and I. Hughes we introduce Conradian left-ordered groups of maximal rank and examine their connection to locally indicable groups. Furthermore we provide notations for crossed product rings, skew fields of fractions as well as Dubrovin-rings and prove some technical statements which are used in later parts. The second chapter focuses on Hughes-free skew fields of fractions and their connection to Dubrovin-rings. For that purpose we introduce series representations to interpret elements of Hughes-free skew fields of fractions as skew formal Laurent series. This 1 Introduction allows us to prove that for Conradian left-ordered groups G of maximal rank the statement "F[G; η, σ] has a Hughes-free skew field of fractions" implies "The Dubrovin ring of F [G; η, σ] is a skew field". We will also prove the reverse and apply the results to give a new prove of Theorem 1 in [13]. Furthermore we will show how to extend injective ring homomorphisms of some crossed product rings onto their Hughes-free skew fields of fractions. At last we will be able to answer the open question whether Hughes--free skew fields are strongly Hughes-free (cf. [17, page 53]).},
  language  = {en}
}
@phdthesis{Fiedler2019,
  author    = {Fiedler, Bernhard},
  title     = {Change-point detection for seismicity parameters},
  school      = {Universit{\"a}t Potsdam},
  pages     = {129},
  year      = {2019},
  language  = {en}
}
@phdthesis{Tinpun2019,
  author    = {Tinpun, Kittisak},
  title     = {Relative rank of infinite full transformation semigroups with restricted range},
  school      = {Universit{\"a}t Potsdam},
  year      = {2019},
  language  = {en}
}
@phdthesis{Jende2018,
  author    = {Jende, Alexander},
  title     = {On the characterization of particular orthogroups by disjunctions of identities},
  school      = {Universit{\"a}t Potsdam},
  pages     = {112},
  year      = {2018},
  abstract  = {In this thesis, we discuss the characterization of orthogroups by so-called disjunctions of identities. The orthogroups are a subclass of the class of completely regular semigroups, a generalization of the concept of a group. Thus there is for all elements of an orthogroup some kind of an inverse element such that both elements commute. Based on a fundamental result by A.H. Clifford, every completely regular semigroup is a semilattice of completely simple semigroups. This allows the description the gross structure of such semigroup. In particular every orthogroup is a semilattice of rectangular groups which are isomorphic to direct products of rectangular bands and groups. Semilattices of rectangular groups coming from various classes are characterized using the concept of an alternative variety, a generalization of the classical idea of a variety by Birkhoff. After starting with some fundamental definitions and results concerning semigroups, we introduce the concept of disjunctions of identities and summarize some necessary properties. In particular we present some disjunction of identities which is sufficient for a semigroup for being completely regular. Furthermore we derive from this identity some statements concerning Rees matrix semigroups, a possible representation of completely simple semigroups. A main result of this thesis is the general description of disjunctions of identities such that a completely regular semigroup satisfying the described identity is a semilattice of left groups (right groups / groups). In this case the completely regular semigroup is an orthogroup. Furthermore we define various classes of rectangular groups such that there is an exponent taken from a set of pairwise coprime positive integers. An important result is the characterization of the class of all semilattices of particular rectangular groups (taken from the classes defined before) using a set-theoretic minimal set of disjunctions of identities. Additionally we investigate semilattices of groups (so-called Clifford semigroups). For this purpose we consider abelian groups of particular exponents and prove some well-known results from the theory of Clifford semigroups in an alternative way applying the concept of disjunctions of identities. As a practical application of the results concerning semilattices of left zero semigroups and right zero semigroups we identify a particular transformation semigroup. For more detailed information about the product of two arbitrary elements of a semilattice of semigroups we introduce the concept of strong semilattices of semigroups. It is well-known that a semilattice of groups is a strong semilattice of groups. So we can characterize a strong semilattice of groups of particular pairwise coprime exponents by disjunctions of identities. Additionally we describe the class of all strong semilattices of left zero semigroups and right zero semigroups with the help of such kind of identity, and we relate this statement to the theory of normal bands. A possible extension of the already described semilattices of rectangular groups can be achieved by an auxiliary total order (in terms of chains of semigroups). To this end we present a corresponding characterization due to disjunctions of identities which is obviously minimal. A list of open questions which have arisen during the research for this thesis, but left crude, is attached.},
  language  = {en}
}
@phdthesis{Khalil2018,
  author    = {Khalil, Sara},
  title     = {Boundary Value Problems on Manifolds with Singularities},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-419018},
  school      = {Universit{\"a}t Potsdam},
  pages     = {10, 160},
  year      = {2018},
  abstract  = {In the thesis there are constructed new quantizations for pseudo-differential boundary value problems (BVPs) on manifolds with edge. The shape of operators comes from Boutet de Monvel's calculus which exists on smooth manifolds with boundary. The singular case, here with edge and boundary, is much more complicated. The present approach simplifies the operator-valued symbolic structures by using suitable Mellin quantizations on infinite stretched model cones of wedges with boundary. The Mellin symbols themselves are, modulo smoothing ones, with asymptotics, holomorphic in the complex Mellin covariable. One of the main results is the construction of parametrices of elliptic elements in the corresponding operator algebra, including elliptic edge conditions.},
  language  = {en}
}
@phdthesis{LindbladPetersen2017,
  author    = {Lindblad Petersen, Oliver},
  title     = {The Cauchy problem for the linearised Einstein equation and the Goursat problem for wave equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-410216},
  school      = {Universit{\"a}t Potsdam},
  pages     = {108},
  year      = {2017},
  abstract  = {In this thesis, we study two initial value problems arising in general relativity. The first is the Cauchy problem for the linearised Einstein equation on general globally hyperbolic spacetimes, with smooth and distributional initial data. We extend well-known results by showing that given a solution to the linearised constraint equations of arbitrary real Sobolev regularity, there is a globally defined solution, which is unique up to addition of gauge solutions. Two solutions are considered equivalent if they differ by a gauge solution. Our main result is that the equivalence class of solutions depends continuously on the corre- sponding equivalence class of initial data. We also solve the linearised constraint equations in certain cases and show that there exist arbitrarily irregular (non-gauge) solutions to the linearised Einstein equation on Minkowski spacetime and Kasner spacetime. In the second part, we study the Goursat problem (the characteristic Cauchy problem) for wave equations. We specify initial data on a smooth compact Cauchy horizon, which is a lightlike hypersurface. This problem has not been studied much, since it is an initial value problem on a non-globally hyperbolic spacetime. Our main result is that given a smooth function on a non-empty, smooth, compact, totally geodesic and non-degenerate Cauchy horizon and a so called admissible linear wave equation, there exists a unique solution that is defined on the globally hyperbolic region and restricts to the given function on the Cauchy horizon. Moreover, the solution depends continuously on the initial data. A linear wave equation is called admissible if the first order part satisfies a certain condition on the Cauchy horizon, for example if it vanishes. Interestingly, both existence of solution and uniqueness are false for general wave equations, as examples show. If we drop the non-degeneracy assumption, examples show that existence of solution fails even for the simplest wave equation. The proof requires precise energy estimates for the wave equation close to the Cauchy horizon. In case the Ricci curvature vanishes on the Cauchy horizon, we show that the energy estimates are strong enough to prove local existence and uniqueness for a class of non-linear wave equations. Our results apply in particular to the Taub-NUT spacetime and the Misner spacetime. It has recently been shown that compact Cauchy horizons in spacetimes satisfying the null energy condition are necessarily smooth and totally geodesic. Our results therefore apply if the spacetime satisfies the null energy condition and the Cauchy horizon is compact and non-degenerate.},
  language  = {en}
}
@phdthesis{Santilli2017,
  author    = {Santilli, Mario},
  title     = {Higher order rectifiability in Euclidean space},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-403632},
  school      = {Universit{\"a}t Potsdam},
  pages     = {45},
  year      = {2017},
  abstract  = {The first main goal of this thesis is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset A of the Euclidean space and for every integer k ≥ 2 we introduce the approximate differential of order k of A and we prove it is a Borel map whose domain is a (possibly empty) Borel set. This concept could be helpful to deal with higher order rectifiable sets in applications. The other goal is to extend to general closed sets a well known theorem of Alberti on the second order rectifiability properties of the boundary of convex bodies. The Alberti theorem provides a stratification of second order rectifiable subsets of the boundary of a convex body based on the dimension of the (convex) normal cone. Considering a suitable generalization of this normal cone for general closed subsets of the Euclidean space and employing some results from the first part we can prove that the same stratification exists for every closed set.},
  language  = {en}
}
@phdthesis{Pedeches2017,
  author    = {P{\´e}d{\`e}ches, Laure},
  title     = {Stochastic models for collective motions of populations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-405491},
  school      = {Universit{\"a}t Potsdam},
  pages     = {187},
  year      = {2017},
  abstract  = {Stochastisches Modell f{\"u}r kollektive Bewegung von Populationen In dieser Doktorarbeit befassen wir uns mit stochastischen Systemen, die eines der mysteri{\"o}sesten biologischen Ph{\"a}nomene als Modell darstellen: die kollektive Bewegung von Gemeinschaften. Diese werden bei V{\"o}gel- und Fischschw{\"a}rmen, aber auch bei manchen Bakterien, Viehherden oder gar bei Menschen beobachtet. Dieser Verhaltenstyp spielt ebenfalls in anderen Bereichen wie Finanzwesen, Linguistik oder auch Robotik eine Rolle. Wir nehmen uns der Dynamik einer Gruppe von N Individuen, insbesondere zweier asymptotischen Verhaltenstypen an. Einerseits befassen wir uns mit den Eigenschaften der Ergodizit{\"a}t in Langzeit: Existenz einer invarianten Wahrscheinlichkeitsverteilung durch Ljapunow-Funktionen, und Konvergenzrate der {\"U}bergangshalbgruppe gegen diese Wahrscheinlichkeit. Eine ebenfalls zentrale Thematik unserer Forschung ist der Begriff Flocking: es wird damit definiert, dass eine Gruppe von Individuen einen dynamischen Konsens ohne hierarchische Struktur erreichen kann; mathematisch gesehen entspricht dies der Aneinanderreihung der Geschwindigkeiten und dem Zusammenkommen des Schwarmes. Andererseits gehen wir das Ph{\"a}nomen der "Propagation of Chaos" an, wenn die Anzahl N der Teilchen ins Unendliche tendiert: die Bewegungen der jeweiligen Individuen werden asymptotisch unabh{\"a}ngig. Unser Ausgangspunkt ist das Cucker-Smale-Modell, ein deterministisches kinetisches Molekular-Modell f{\"u}r eine Gruppe ohne hierarchische Struktur. Die Wechselwirkung zwischen zwei Teilchen variiert gem{\"a}ß deren "Kommunikationsrate", die wiederum von deren relativen Entfernung abh{\"a}ngt und polynomisch abnimmt. Im ersten Kapitel adressieren wir das asymptotische Verhalten eines Cucker-Smale-Modells mit Rauschst{\"o}rung und dessen Varianten. Kapitel 2 stellt mehrere Definitionen des Flockings in einem Zufallsrahmen dar: diverse stochastische Systeme, die verschiedenen Rauschformen entsprechen (die eine gest{\"o}rte Umgebung, den "freien Willen" des jeweiligen Individuums oder eine unterbrochene {\"U}bertragung suggerieren) werden im Zusammenhang mit diesen Begriffen unter die Lupe genommen. Das dritte Kapitel basiert auf der "Cluster Expansion"-Methode aus der statistischen Mechanik. Wir beweisen die exponentielle Ergodizit{\"a}t von gewissen nicht-Markow-Prozessen mit nicht-glattem Drift und wenden diese Ergebnisse auf St{\"o}rungen des Ornstein-Uhlenbeck-Prozesses an. Im letzten Teil, nehmen wir uns der zweidimensionalen parabolisch-elliptischen Gleichung von Keller-Segel an. Wir beweisen die Existenz einer L{\"o}sung, welche in gewisser Hinsicht einzig ist, indem wir, mittels Vergleich mit Bessel-Prozessen und der Dirichlet Formtheorie, m{\"o}gliche Stoßtypen zwischen den Teilchen ermitteln.},
  language  = {en}
}
@phdthesis{Muecke2017,
  author    = {M{\"u}cke, Nicole},
  title     = {Direct and inverse problems in machine learning},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-403479},
  school      = {Universit{\"a}t Potsdam},
  pages     = {159},
  year      = {2017},
  abstract  = {We analyze an inverse noisy regression model under random design with the aim of estimating the unknown target function based on a given set of data, drawn according to some unknown probability distribution. Our estimators are all constructed by kernel methods, which depend on a Reproducing Kernel Hilbert Space structure using spectral regularization methods. A first main result establishes upper and lower bounds for the rate of convergence under a given source condition assumption, restricting the class of admissible distributions. But since kernel methods scale poorly when massive datasets are involved, we study one example for saving computation time and memory requirements in more detail. We show that Parallelizing spectral algorithms also leads to minimax optimal rates of convergence provided the number of machines is chosen appropriately. We emphasize that so far all estimators depend on the assumed a-priori smoothness of the target function and on the eigenvalue decay of the kernel covariance operator, which are in general unknown. To obtain good purely data driven estimators constitutes the problem of adaptivity which we handle for the single machine problem via a version of the Lepskii principle.},
  language  = {en}
}
@phdthesis{Mera2017,
  author    = {Mera, Azal Jaafar Musa},
  title     = {The Navier-Stokes equations for elliptic quasicomplexes},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-398495},
  school      = {Universit{\"a}t Potsdam},
  pages     = {101},
  year      = {2017},
  abstract  = {The classical Navier-Stokes equations of hydrodynamics are usually written in terms of vector analysis. More promising is the formulation of these equations in the language of differential forms of degree one. In this way the study of Navier-Stokes equations includes the analysis of the de Rham complex. In particular, the Hodge theory for the de Rham complex enables one to eliminate the pressure from the equations. The Navier-Stokes equations constitute a parabolic system with a nonlinear term which makes sense only for one-forms. A simpler model of dynamics of incompressible viscous fluid is given by Burgers' equation. This work is aimed at the study of invariant structure of the Navier-Stokes equations which is closely related to the algebraic structure of the de Rham complex at step 1. To this end we introduce Navier-Stokes equations related to any elliptic quasicomplex of first order differential operators. These equations are quite similar to the classical Navier-Stokes equations including generalised velocity and pressure vectors. Elimination of the pressure from the generalised Navier-Stokes equations gives a good motivation for the study of the Neumann problem after Spencer for elliptic quasicomplexes. Such a study is also included in the work.We start this work by discussion of Lam{\´e} equations within the context of elliptic quasicomplexes on compact manifolds with boundary. The non-stationary Lam{\´e} equations form a hyperbolic system. However, the study of the first mixed problem for them gives a good experience to attack the linearised Navier-Stokes equations. On this base we describe a class of non-linear perturbations of the Navier-Stokes equations, for which the solvability results still hold.},
  language  = {en}
}
@phdthesis{Cheng2016,
  author    = {Cheng, Yuan},
  title     = {Recursive state estimation in dynamical systems},
  school      = {Universit{\"a}t Potsdam},
  pages     = {84},
  year      = {2016},
  language  = {en}
}
@phdthesis{Solms2017,
  author    = {Solms, Alexander Maximilian},
  title     = {Integrating nonlinear mixed effects and physiologically-based modeling approaches for the analysis of repeated measurement studies},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-397070},
  school      = {Universit{\"a}t Potsdam},
  pages     = {x, 141},
  year      = {2017},
  abstract  = {During the drug discovery \& development process, several phases encompassing a number of preclinical and clinical studies have to be successfully passed to demonstrate safety and efficacy of a new drug candidate. As part of these studies, the characterization of the drug's pharmacokinetics (PK) is an important aspect, since the PK is assumed to strongly impact safety and efficacy. To this end, drug concentrations are measured repeatedly over time in a study population. The objectives of such studies are to describe the typical PK time-course and the associated variability between subjects. Furthermore, underlying sources significantly contributing to this variability, e.g. the use of comedication, should be identified. The most commonly used statistical framework to analyse repeated measurement data is the nonlinear mixed effect (NLME) approach. At the same time, ample knowledge about the drug's properties already exists and has been accumulating during the discovery \& development process: Before any drug is tested in humans, detailed knowledge about the PK in different animal species has to be collected. This drug-specific knowledge and general knowledge about the species' physiology is exploited in mechanistic physiological based PK (PBPK) modeling approaches -it is, however, ignored in the classical NLME modeling approach. Mechanistic physiological based models aim to incorporate relevant and known physiological processes which contribute to the overlying process of interest. In comparison to data--driven models they are usually more complex from a mathematical perspective. For example, in many situations, the number of model parameters outrange the number of measurements and thus reliable parameter estimation becomes more complex and partly impossible. As a consequence, the integration of powerful mathematical estimation approaches like the NLME modeling approach -which is widely used in data-driven modeling -and the mechanistic modeling approach is not well established; the observed data is rather used as a confirming instead of a model informing and building input. Another aggravating circumstance of an integrated approach is the inaccessibility to the details of the NLME methodology so that these approaches can be adapted to the specifics and needs of mechanistic modeling. Despite the fact that the NLME modeling approach exists for several decades, details of the mathematical methodology is scattered around a wide range of literature and a comprehensive, rigorous derivation is lacking. Available literature usually only covers selected parts of the mathematical methodology. Sometimes, important steps are not described or are only heuristically motivated, e.g. the iterative algorithm to finally determine the parameter estimates. Thus, in the present thesis the mathematical methodology of NLME modeling is systemically described and complemented to a comprehensive description, comprising the common theme from ideas and motivation to the final parameter estimation. Therein, new insights for the interpretation of different approximation methods used in the context of the NLME modeling approach are given and illustrated; furthermore, similarities and differences between them are outlined. Based on these findings, an expectation-maximization (EM) algorithm to determine estimates of a NLME model is described. Using the EM algorithm and the lumping methodology by Pilari2010, a new approach on how PBPK and NLME modeling can be combined is presented and exemplified for the antibiotic levofloxacin. Therein, the lumping identifies which processes are informed by the available data and the respective model reduction improves the robustness in parameter estimation. Furthermore, it is shown how apriori known factors influencing the variability and apriori known unexplained variability is incorporated to further mechanistically drive the model development. Concludingly, correlation between parameters and between covariates is automatically accounted for due to the mechanistic derivation of the lumping and the covariate relationships. A useful feature of PBPK models compared to classical data-driven PK models is in the possibility to predict drug concentration within all organs and tissue in the body. Thus, the resulting PBPK model for levofloxacin is used to predict drug concentrations and their variability within soft tissues which are the site of action for levofloxacin. These predictions are compared with data of muscle and adipose tissue obtained by microdialysis, which is an invasive technique to measure a proportion of drug in the tissue, allowing to approximate the concentrations in the interstitial fluid of tissues. Because, so far, comparing human in vivo tissue PK and PBPK predictions are not established, a new conceptual framework is derived. The comparison of PBPK model predictions and microdialysis measurements shows an adequate agreement and reveals further strengths of the presented new approach. We demonstrated how mechanistic PBPK models, which are usually developed in the early stage of drug development, can be used as basis for model building in the analysis of later stages, i.e. in clinical studies. As a consequence, the extensively collected and accumulated knowledge about species and drug are utilized and updated with specific volunteer or patient data. The NLME approach combined with mechanistic modeling reveals new insights for the mechanistic model, for example identification and quantification of variability in mechanistic processes. This represents a further contribution to the learn \& confirm paradigm across different stages of drug development. Finally, the applicability of mechanism--driven model development is demonstrated on an example from the field of Quantitative Psycholinguistics to analyse repeated eye movement data. Our approach gives new insight into the interpretation of these experiments and the processes behind.},
  language  = {en}
}
@phdthesis{Samaras2016,
  author    = {Samaras, Stefanos},
  title     = {Microphysical retrieval of non-spherical aerosol particles using regularized inversion of multi-wavelength lidar data},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-396528},
  school      = {Universit{\"a}t Potsdam},
  pages     = {xiv, 190},
  year      = {2016},
  abstract  = {Numerous reports of relatively rapid climate changes over the past century make a clear case of the impact of aerosols and clouds, identified as sources of largest uncertainty in climate projections. Earth's radiation balance is altered by aerosols depending on their size, morphology and chemical composition. Competing effects in the atmosphere can be further studied by investigating the evolution of aerosol microphysical properties, which are the focus of the present work. The aerosol size distribution, the refractive index, and the single scattering albedo are commonly used such properties linked to aerosol type, and radiative forcing. Highly advanced lidars (light detection and ranging) have reduced aerosol monitoring and optical profiling into a routine process. Lidar data have been widely used to retrieve the size distribution through the inversion of the so-called Lorenz-Mie model (LMM). This model offers a reasonable treatment for spherically approximated particles, it no longer provides, though, a viable description for other naturally occurring arbitrarily shaped particles, such as dust particles. On the other hand, non-spherical geometries as simple as spheroids reproduce certain optical properties with enhanced accuracy. Motivated by this, we adapt the LMM to accommodate the spheroid-particle approximation introducing the notion of a two-dimensional (2D) shape-size distribution. Inverting only a few optical data points to retrieve the shape-size distribution is classified as a non-linear ill-posed problem. A brief mathematical analysis is presented which reveals the inherent tendency towards highly oscillatory solutions, explores the available options for a generalized solution through regularization methods and quantifies the ill-posedness. The latter will improve our understanding on the main cause fomenting instability in the produced solution spaces. The new approach facilitates the exploitation of additional lidar data points from depolarization measurements, associated with particle non-sphericity. However, the generalization of LMM vastly increases the complexity of the problem. The underlying theory for the calculation of the involved optical cross sections (T-matrix theory) is computationally so costly, that would limit a retrieval analysis to an unpractical point. Moreover the discretization of the model equation by a 2D collocation method, proposed in this work, involves double integrations which are further time consuming. We overcome these difficulties by using precalculated databases and a sophisticated retrieval software (SphInX: Spheroidal Inversion eXperiments) especially developed for our purposes, capable of performing multiple-dataset inversions and producing a wide range of microphysical retrieval outputs. Hybrid regularization in conjunction with minimization processes is used as a basis for our algorithms. Synthetic data retrievals are performed simulating various atmospheric scenarios in order to test the efficiency of different regularization methods. The gap in contemporary literature in providing full sets of uncertainties in a wide variety of numerical instances is of major concern here. For this, the most appropriate methods are identified through a thorough analysis on an overall-behavior basis regarding accuracy and stability. The general trend of the initial size distributions is captured in our numerical experiments and the reconstruction quality depends on data error level. Moreover, the need for more or less depolarization points is explored for the first time from the point of view of the microphysical retrieval. Finally, our approach is tested in various measurement cases giving further insight for future algorithm improvements.},
  language  = {en}
}
@phdthesis{Habal2013,
  author    = {Habal, Nadia},
  title     = {Operators on singular manifolds},
  address   = {Potsdam},
  pages     = {203 S.},
  year      = {2013},
  language  = {en}
}
@phdthesis{Susanti2013,
  author    = {Susanti, Yeni},
  title     = {On Particular n-Clones},
  address   = {Potsdam},
  pages     = {108 S.},
  year      = {2013},
  language  = {en}
}
@phdthesis{Rattana2013,
  author    = {Rattana, Amornrat},
  title     = {Direct and inverse sturm-liouville problems of order four},
  address   = {Potsdam},
  pages     = {121 S.},
  year      = {2013},
  language  = {en}
}
@phdthesis{Rungrottheera2013,
  author    = {Rungrottheera, Wannarut},
  title     = {Corner pseudo-differential operators},
  address   = {Potsdam},
  pages     = {139 S.},
  year      = {2013},
  language  = {en}
}
@phdthesis{Keller2012,
  author    = {Keller, Peter},
  title     = {Mathematical modeling of molecular motors},
  address   = {Potsdam},
  pages     = {116 S.},
  year      = {2012},
  language  = {en}
}
@phdthesis{Schachtschneider2011,
  author    = {Schachtschneider, Reyko},
  title     = {Error distribution in regional inversions of potential fields from satellite data},
  address   = {Potsdam},
  pages     = {118 S.},
  year      = {2011},
  language  = {en}
}
@phdthesis{Sarasit2011,
  author    = {Sarasit, Napaporn},
  title     = {Algebraic properties of sets of terms},
  address   = {Potsdam},
  pages     = {91 S.},
  year      = {2011},
  language  = {en}
}
@phdthesis{Pornsawad2010,
  author    = {Pornsawad, Pornsarp},
  title     = {Solution of nonlinear inverse ill-posed problems via Runge-Kutta methods},
  address   = {Potsdam},
  pages     = {104 S.},
  year      = {2010},
  language  = {en}
}
@phdthesis{Ramadan2010,
  author    = {Ramadan, Ayad},
  title     = {Statistical model for categorical data},
  address   = {Potsdam},
  pages     = {iv, 100 S. : graph. Darst.},
  year      = {2010},
  language  = {en}
}
@phdthesis{Hayn2010,
  author    = {Hayn, Michael},
  title     = {Wavelet analysis and spline modeling of geophysical data on the sphere},
  address   = {Potsdam},
  pages     = {95 S. : graph. Darst.},
  year      = {2010},
  language  = {en}
}
@phdthesis{Kammanee2010,
  author    = {Kammanee, Athassawat},
  title     = {Some inverse potential problems},
  address   = {Potsdam},
  pages     = {XIV, 128 S.},
  year      = {2010},
  language  = {en}
}
@phdthesis{Ly2009,
  author    = {Ly, Ibrahim},
  title     = {Asymptotic solutions of the Cauchy problem for nonlinear elliptic differential equations},
  address   = {Potsdam},
  pages     = {VI, 100 S.},
  year      = {2009},
  language  = {en}
}
@phdthesis{Krainer2009,
  author    = {Krainer, Thomas},
  title     = {Elliptic boundary value problems on manifolds with corners},
  pages     = {VI, 269 S.},
  year      = {2009},
  language  = {en}
}
@phdthesis{Lekkoksung2010,
  author    = {Lekkoksung, Somsak},
  title     = {Hyperidentities and hypersubstitutions of many-sorted algebras},
  address   = {Potsdam},
  pages     = {X, 98 S.},
  year      = {2010},
  language  = {en}
}