@misc{PereraBoeckmann2019,
  author    = {Perera, Upeksha and B{\"o}ckmann, Christine},
  title     = {Solutions of direct and inverse even-order Sturm-Liouville problems using Magnus expansion},
  series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1336},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-47341},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-473414},
  pages     = {24},
  year      = {2019},
  abstract  = {In this paper 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 for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, 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. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.},
  language  = {en}
}
@misc{PornsawadSapsakulBoeckmann2019,
  author    = {Pornsawad, Pornsarp and Sapsakul, Nantawan and B{\"o}ckmann, Christine},
  title     = {A modified asymptotical regularization of nonlinear ill-posed problems},
  series = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Zweitver{\"o}ffentlichungen der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {1335},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-47343},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-473433},
  pages     = {19},
  year      = {2019},
  abstract  = {In this paper, we investigate the continuous version of modified iterative Runge-Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))-𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.},
  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{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}
}