TY - JOUR A1 - Lewandowski, Max T1 - Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes JF - Journal of mathematical physics N2 - According to Radzikowski’s celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF−G˜F+G˜A−G˜R) in the sense of Duistermaat and Hörmander [Acta Math. 128, 183–269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by Bär, Ginoux, and Pfäffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Zürich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF−G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied. Y1 - 2022 U6 - https://doi.org/10.1063/5.0055753 SN - 0022-2488 SN - 1089-7658 VL - 63 IS - 1 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Zöller, Gert T1 - A note on the estimation of the maximum possible earthquake magnitude based on extreme value theory for the Groningen Gas Field JF - The bulletin of the Seismological Society of America : BSSA N2 - Extreme value statistics is a popular and frequently used tool to model the occurrence of large earthquakes. The problem of poor statistics arising from rare events is addressed by taking advantage of the validity of general statistical properties in asymptotic regimes. In this note, I argue that the use of extreme value statistics for the purpose of practically modeling the tail of the frequency-magnitude distribution of earthquakes can produce biased and thus misleading results because it is unknown to what degree the tail of the true distribution is sampled by data. Using synthetic data allows to quantify this bias in detail. The implicit assumption that the true M-max is close to the maximum observed magnitude M-max,M-observed restricts the class of the potential models a priori to those with M-max = M-max,M-observed + Delta M with an increment Delta M approximate to 0.5... 1.2. This corresponds to the simple heuristic method suggested by Wheeler (2009) and labeled :M-max equals M-obs plus an increment." The incomplete consideration of the entire model family for the frequency-magnitude distribution neglects, however, the scenario of a large so far unobserved earthquake. Y1 - 2022 U6 - https://doi.org/10.1785/0120210307 SN - 0037-1106 SN - 1943-3573 VL - 112 IS - 4 SP - 1825 EP - 1831 PB - Seismological Society of America CY - El Cerito, Calif. ER - TY - JOUR A1 - Kaya, Adem A1 - Freitag, Melina A. T1 - Conditioning analysis for discrete Helmholtz problems JF - Computers and mathematics with applications : an international journal N2 - In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions. KW - Helmholtz problem KW - Condition number KW - Ill-conditioning KW - Indefinite KW - matrices Y1 - 2022 U6 - https://doi.org/10.1016/j.camwa.2022.05.016 SN - 0898-1221 SN - 1873-7668 VL - 118 SP - 171 EP - 182 PB - Elsevier Science CY - Amsterdam ER - TY - JOUR A1 - Houdebert, Pierre A1 - Zass, Alexander T1 - An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions JF - Journal of applied probability / Applied Probability Trust N2 - We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature beta. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction. KW - Gibbs point process KW - DLR equations KW - uniqueness KW - Dobrushin criterion; KW - cluster expansion KW - disagreement percolation Y1 - 2022 U6 - https://doi.org/10.1017/jpr.2021.70 SN - 0021-9002 SN - 1475-6072 VL - 59 IS - 2 SP - 541 EP - 555 PB - Cambridge Univ. Press CY - Cambridge ER - TY - JOUR A1 - Evans, Myfanwy E. A1 - Hyde, Stephen T. T1 - Symmetric Tangling of Honeycomb Networks JF - Symmetry N2 - Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox. KW - tangles KW - knots KW - networks KW - periodic entanglement KW - molecular weaving KW - graphs Y1 - 2022 U6 - https://doi.org/10.3390/sym14091805 SN - 2073-8994 VL - 14 SP - 1 EP - 13 PB - MDPI CY - Basel, Schweiz ET - 9 ER - TY - JOUR A1 - Hyde, Stephen T. A1 - Evans, Myfanwy E. T1 - Symmetric tangled Platonic polyhedra JF - Proceedings of the National Academy of Sciences of the United States of America N2 - Conventional embeddings of the edge-graphs of Platonic polyhedra, {f,z}, where f,z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schonflies symbols I-h, O-h, and T-d). Tangled Platonic {f,z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the "theta(z)" polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo. KW - regular polyhedra KW - compound polyhedra KW - helicates KW - metal-organic KW - frameworks KW - clathrin Y1 - 2022 U6 - https://doi.org/10.1073/pnas.2110345118 SN - 0027-8424 SN - 1091-6490 VL - 119 IS - 1 PB - National Acad. of Sciences CY - Washington ER - TY - THES A1 - Hain, Tobias Martin T1 - Structure formation and identification in geometrically driven soft matter systems T1 - Strukturbildung und Identifikation in geometrisch getriebenen weiche Materie-Systemen N2 - 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. N2 - Die Unterteilung eines Raums durch Grenzflächen führt zu Raumaufteilungen, die für die Selbstorganisation weicher Materie relevant sind. Bekannte Beispiele sind zelluläre Medien, wie z.B. Seifenschaum, der aus Luftblasen besteht, getrennt durch Wände aus Wasser und Seife, und komplexere Partitionen, wie sie z.B. durch bikontinuierliche Minimalflächen erzeugt werden. In dieser Arbeit werden mit Hilfe von Computersimulationen Systeme weicher Materie in Bezug auf den Zusammenhang zwischen dem im System vorherrschenden, physikalischen Kräften und der Struktur der resultierenden Grenzflächen oder Partitionen untersucht. Der Schwerpunkt liegt hierbei auf zwei Systemen, eine Copolymerschmelze und das sogenannte Quantizer Problem, bei denen der treibende Faktor der Strukturbildung, nämlich die Minimierung der freien Energie, aus einem Zusammenspiel der Minimierung der Oberfläche der Grenzflächen und der gleichzeitigen Minimierung der Elastizitätsenergie besteht. Unter diesen Gegebenheiten bevorzugen solche Systeme Zellen gleichmäßiger Größe. Im ersten Teil der Arbeit befassen wir uns mit der Bildung von scharfen Grenzflächen in Systemen von Copolymeren. Wir analysieren die zylindrische Phase eines Copolymersystems, das gezwungen wird, sich auf einer kugelförmigen Oberfläche zu organisieren. Die Topologie dieser Oberfläche erlaubt es der optimalen Konfiguration, dem Sechseckgitter, nicht, sich zu bilden. Für dreiarmige Copolymere wird die entstehende Struktur durch Lösungen des sogenannten Thomson Problems beschrieben. Letzteres sucht nach der Konfigurationen von abstoßenden Ladungen auf einer Kugeloberfläche mit minimaler Energie. Auf einem Substrat haben wir eine Kombination aus drei ineinandergreifende Lösungen des Thomson Problems gefunden, wobei der Typ der Lösungen statistisch von dem Radius des Substrates abhängt. Anschließend untersuchen wir die Bildung von amorphen und kristallinen Strukturen im Quantizersystem, einem teilchenbasierenden Modell, dessen Energiefunktional keine Oberflächenspannung enthält und möglichst kugelförmige Zellen gleicher Größe begünstigt. Wird das System quasistatisch im thermodynamischen Gleichgewicht abgekühlt, kristallisiert das Quantizersystem in den geordneten BCC Grundzustand. Wird das System allerdings zu schnell abgekühlt, sodass es sich nicht mehr im thermodynamischen Gleichgewicht befindet, bildet sich eine amorphe, annähernd hyperuniforme Struktur aus. Wir konnten zeigen, dass diese Struktur bemerkenswert unabhängig von den Ausganszuständen, sowie der Art der Energieminimierung zu sein scheint. Im Ausblick erweitern wir das Quantizersystem, indem wir Oberflächenspannung einführen. Unsere Ergebnisse deutet darauf hin, dass dieses so erweiterte Modell Mizellenphasen in Polymersystem modellieren kann. Wir beobachten einen Phasenübergang von einer ungeordneten, flüssigen Phase hin zu einer festen Frank-Caspar-Phase. Der zweite Teil der Arbeit behandelt bikontinuierliche Grenzflächen, die den Raum in zwei netzwerkartige Domänen aufteilen. Wir führen eine Open-Source Software ein, das die Identifizierung von Strukturen anhand derer Mikroskopaufnahmen ermöglicht. Hierzu erweitern und verbessern wir eine Methode, die durch den Abgleich experimentell zugänglicher Projektionen in Mikroskopaufnahmen mit berechneten Projektionen potenzieller Strukturen basiert. Dieses Verfahren wurde erstmal von Deng und Mieczkowski (1998) eingeführt. Die simulierten Strukturen basieren auf einer Nodalflächenmodellierung von dreifach-periodischen Flächen konstanter mittlerer Krümmung. Wir führen eine Fallstudie an Zellmembranen von Etioplasten, den Vorläufern von Chloroplasten, durch. Wir konnten die Struktur dieser Etioplasten als die Diamond-Struktur identifizieren. Als Ausblick automatisieren wir den Identifizierungsproyess mit Hilfe von Deep-Learning-Methoden. Erste Ergebnisse zeigen, dass mit diesem Ansatz die Identifizierung von Strukturen mit ausgezeichneter Genauigkeit gelingt. KW - soft matter KW - geometry KW - self-assembly KW - structure formation KW - quantizer KW - polymer KW - Geometrie KW - Polymere KW - Quantizer KW - Selbstassemblierung KW - weiche Materie KW - Strukturbildung Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-558808 N1 - Parts of this publication are reproduced with permission from the Royal Society of Chemistry and AIP Publishing. ER - TY - JOUR A1 - Maier, Corinna Sabrina A1 - Wiljes, Jana de A1 - Hartung, Niklas A1 - Kloft, Charlotte A1 - Huisinga, Wilhelm T1 - A continued learning approach for model-informed precision dosing BT - Updating models in clinical practice JF - CPT: pharmacometrics & systems pharmacology N2 - Model-informed precision dosing (MIPD) is a quantitative dosing framework that combines prior knowledge on the drug-disease-patient system with patient data from therapeutic drug/ biomarker monitoring (TDM) to support individualized dosing in ongoing treatment. Structural models and prior parameter distributions used in MIPD approaches typically build on prior clinical trials that involve only a limited number of patients selected according to some exclusion/inclusion criteria. Compared to the prior clinical trial population, the patient population in clinical practice can be expected to also include altered behavior and/or increased interindividual variability, the extent of which, however, is typically unknown. Here, we address the question of how to adapt and refine models on the level of the model parameters to better reflect this real-world diversity. We propose an approach for continued learning across patients during MIPD using a sequential hierarchical Bayesian framework. The approach builds on two stages to separate the update of the individual patient parameters from updating the population parameters. Consequently, it enables continued learning across hospitals or study centers, because only summary patient data (on the level of model parameters) need to be shared, but no individual TDM data. We illustrate this continued learning approach with neutrophil-guided dosing of paclitaxel. The present study constitutes an important step toward building confidence in MIPD and eventually establishing MIPD increasingly in everyday therapeutic use. Y1 - 2021 U6 - https://doi.org/10.1002/psp4.12745 SN - 2163-8306 VL - 11 IS - 2 SP - 185 EP - 198 PB - London CY - Nature Publ. Group ER - TY - THES A1 - Fischer, Jens Walter T1 - Random dynamics in collective behavior - consensus, clustering & extinction of populations T1 - Stochastische Dynamiken in kollektivem Verhalten: Konsens, Gruppenbildung, Aussterben von Populationen N2 - 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. N2 - Beziehungen und damit Interaktion sowie Diskussion, aber auch Konflikt und Opposition bilden die Grundbausteine einer jeden Gesellschaft. Häufig wird Kommunikation als der übergreigende Begriff zur Beschreibung interner Strukturen einer Gesellschaft identifiziert. Dabei muss es sich aber nicht um eine Gesellschaft im Sinne von Nationen handeln, sondern kann auch schlicht eine Gruppe von Menschen umfassen, die miteinander strukturiert interagieren, beispielsweise, eine Gruppe von Angestellten, die an einem gemeinsamen Projekt arbeiten, oder die Mitglieder eines sozialen Netzwerks. In dieser Arbeit befassen wir uns mit der mathematischen Beschreibung solcher Prozesse innerhalb von Gruppen und Gesellschaften und legen dabei unseren Fokus auf die Bildung eines Konsens durch Interaktion aber auch die Konsequenzen von Konflikt und das potentielle Aussterben einer Population. Dabei werden zwei Modelle im Fokus des Interesses stehen: Das Echokammer Model sowie eine Erweiterung des Geburts-Todes Prozesses, die die Möglichkeit eines radikalen Abfalls der Populationsgr öße miteinschließt. Wir beginnen mit einer Einführung in Part I und teilen die verbleibende Arbeit in drei Teile auf, wobei sich die ersten beiden technischen Abschnitte, Part II und III, mit einer ausführlichen Analyse der Bausteine des Echokammer Models befassen und im dritten Abschnitt, in Part IV, der erweiterte Geburts- Todes Prozess untersucht wird. Dieser wird im Folgenden als Geburts-Todes Prozess mit teilweiser Katastrophe bezeichnet werden. Das Echokammer Model beschreibt die Entwicklung von Gruppen in zunächst heterogenen sozialen Netzwerken. Unter einem heterogenen sozialen Netzwerk verstehen wir dabei eine Menge von Individuen, von denen jedes exakt eine Meinungen vertritt. Meinungen werden vereinfacht durch Werte in [0, 1] modelliert. Bestehende Beziehungen unter den Individuen können dann durch einen Graphen dargestellt werden. Es handelt sich bei dem Echokammer Modell um ein zeit-diskretes Modell, das entsprechend, ähnlich einem Brettspiel, in Zügen abläuft. In jedem Zug wird zufällig gleichverteilt eine bestehende Beziehung aus dem Netzwerk ausgewählt und die beiden verbundenen Individuen interagieren. Dabei kann es zu zwei verschiedenen Interaktionen kommen. Sind die Meinungen der betroffenen Individuen hinreichend ähnlich, so nähern sie sich weiter in ihren Meinungen an, während sie im Fall von Meinungen, die zu weit von einander liegen, ihre Beziehung auflösen und sich eines der Individuen eine neue Beziehung sucht. 8 In dieser Arbeit untersuchen wir theoretisch die Bausteine dieses Modells. Dabei legen wir die Beobachtung zu Grunde, dass die Veränderungen der Beziehungsstruktur im Netzwerk durch einen System von interagierenden Partikeln auf einem abstrakteren Raum beschrieben werden kann. Dies erlaubt es insbesondere graphentheoretische überlegungen in die Analyse einfließen zu lassen. Diese überlegungen werden ausührlich in Part II diskutiert und führen zur Definition eines neuen, abstrahierten Graphens, der alle möglichen Beziehungskonfigurationen des sozialen Netzwerks umfasst. Dies erlaubt es uns einen ähnlichkeitsbegriff für Beziehungskonfigurationen auf Basis der benachbarten Knoten in besagtem Graphen zu definieren. Dies liefert uns das notwendige geometrische Verständnis um in Part III die dynamischen Komponenten des Echokammer models zu analysieren. Insbesondere fokusieren wir uns dabei auf die Dynamik der Kanten, für die bisher in der Literatur noch keine Ergebnisse existieren. Wir lassen zunächst in Abschnitt 7 die Meinungen der Individuen beiseite und nehmen an, dass die Position der Kanten sich in jedem Zug wie zuvor beschrieben ändert, um eine grundlegendes Verständnis der unterliegenden Dynamik zu erhalten. Unter der Verwendung der Theorie von Markovketten finden wir obere Schranken an die Konvergenzgeschwindigkeit einer assoziierten Markovkette gegen ihre eindeutige stationäre Verteilung und zeigen, dass es Netzwerke gibt, die miteinander identifizierbar und unter der analysierten Dynamik daheingehend ununterscheinbar sind, dass die stationäre Verteilung der assozierten Markovkette diesen Netzwerken dasselbe Gewicht zuordnet. Anschließend beweisen wir eine Reihe von quantitativen Resultaten, die sich insbesondere in Fällen, in denen die assozierte Markovkette reversibel ist, als berechenbar herausstellen. Insbesondere die explizite Form der stationären Verteilung sowie untere Schranken an die Cheeger Konstante zur Beschreibung der Konvergenzgeschwindigkeit stehen dabei im Fokus und werden ausführlich diskutiert. Nach dieser vertieften Analyse des reduzierten Modells, fügen wir die Meinungen unserer Betrachtung wieder hinzu. Das abschließende Result in Abschnitt 8, basierend auf absorbierenden Markovketten, liefert dann, dass in einer reduzierte Version des Echokammer Modells, in dem sich Individuen ähnlicher Meinung nicht annähern, eine hierarchische Struktur der Anzahl der konfliktreichen Beziehung identifiziert werden kann. Dies können wir ausnutzen, um eine obere Schranke an die erwartete Absorptionszeit, unter Zuhilfenahme einer quasi-stationären Verteilung, zu bestimmen. Diese hierarchische Struktur bildet außerdem eine Brücke zu klassischen Theorien von Geburts-Todes und, insbesondere, reinen Todes-Prozessen, für die eine reiche Literatur existiert. Wir zeigen abschließend auf, wie künftige Forschung diese Verbindung ausnutzen kann und diskutieren die Wichtigkeit der Ergbenisse als Bausteine eines vollständigen theoretischen Verständnisses des Echokammer Modells. Part IV stellt abschließend einen veröffentlichten Artikel vor, der sich dem Geburts- Todes Prozess mit teilweiser Katastrophe widmet. Besagter Artikel steht dabei auf zwei Säulen. Zum Einen der expliziten Berechnung des ersten Zeitpunkts einer Katastrophe, wenn die Population zu Beginn der Beobachtung von instabiler Größe ist. KW - Markov chains KW - graph theory KW - complex systems KW - interacting particle systems KW - Markovketten KW - komplexe Systeme KW - Graphentheorie KW - Systeme interagierender Partikel Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-553725 ER - TY - THES A1 - Schanner, Maximilian Arthus T1 - Correlation based modeling of the archeomagnetic field T1 - Korrelationsbasierte Modellierung des archäomagnetischen Feldes N2 - 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. N2 - Das geomagnetische Hauptfeld ist essenziell für das Leben auf der Erde, da es unseren Lebensraum gegen den Sonnenwind und kosmische Strahlung abschirmt. Es wird vom Geodynamo im Erdkern erzeugt und zeigt eine komplexe Dynamik auf unterschiedlichen Zeitskalen. Globale Modelle des Magnetfelds werden zur Studie der Wechselwirkung von einströmenden geladenen Teilchen genutzt, aber auch um Kerndynamiken zu untersuchen und um sie in numerische Simulationen des Geodynamos einzuspeisen. Moderne Satellitenmissionen, wie SWARM und CHAMP, stützen hochauflösende Rekonstruktionen des globalen Felds. Seit dem 19. Jahrhundert wird ein globales Netzwerk von magnetischen Observatorien aufgebaut. Es wächst stetig und globale Modelle können aus den Daten, die es liefert, konstruiert werden. Geomagnetische Feldmodelle, die weiter in der Zeit zurückreichen, basieren auf indirekten Beobachtungen des Felds, d.h. auf thermoremanenten Daten, wie gebrannten Tonen oder vulkanischen Gesteinen, und auf Sedimentdaten aus Seen und Meeren. Diese indirekten Beobachtungen werden mit (teilweise sehr hohen) Unsicherheiten geliefert, die aus den komplexen Datierungs- und Messmethoden resultieren. Ziel dieser Arbeit ist die Entwicklung einer neuen Modellierungsmethode für das globale geomagnetische Feld während des Holozäns, welche die Unsicherheiten berücksichtigt und realistische Schätzungen für die Verlässlichkeit des Modells liefert. Dabei werden lediglich thermoremanente Daten betrachtet. Diesem Ziel wird sich zunächst genähert, indem ein Schnappschuss-Modell konstruiert wird, um die unregelmäßige räumliche Verteilung der Daten und die nichtlineare Beziehung zwischen Daten und Magnetfeld zu untersuchen. In einem Bayesianischen Rahmen wird ein auf Gaussprozessen basierender Algorithmus entwickelt und zunächst auf diskretisierte Daten angewendet. Dieser Algorithmus wird dann um eine zeitabhängige Komponente ergänzt und erweitert, um Datierungsfehler zu berücksichtigen. Zuletzt wird der Algorithmus sequenzialisiert, um mit numerischen Herausforderungen umzugehen, die aufgrund der Größe des Holozän-Datensatzes bestehen. Das zentrale Ergebnis dieser Arbeit, welches alle genannten Aspekte beinhaltet, ist ein neues globales geomagnetisches Feldmodell. Es deckt das gesamte Holozän ab, bis ins Jahr 12000 BCE, und wir nennen es ArchKalmag14k. Bei Betrachtung der Unsicherheiten, die gemeinsam mit dem Modell ermittelt werden, wird deutlich, dass die thermoremanente Datenbasis nicht ausreicht, um globale Modelle vor dem Jahr 6000 BCE zu stützen. Für jüngere Zeiträume kann ArchKalmag14k genutzt werden, um Merkmale des Erdmagnetfelds unter Berücksichtigung der a posteriori Unsicherheiten zu analysieren. Der Algorithmus, mit dem ArchKalmag14k erzeugt wurde, kann auf weitere Datensätze angewendet werden und wird als quelloffenes python-Paket zur Verfügung gestellt. KW - geomagnetism KW - applied mathematics KW - Gaussian processes KW - Kalman filter KW - Gauß-Prozesse KW - Kalman Filter KW - angewandte Mathematik KW - Geomagnetismus Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-555875 ER - TY - GEN A1 - Evans, Myfanwy E. A1 - Hyde, Stephen T. T1 - Symmetric Tangling of Honeycomb Networks T2 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - Symmetric, elegantly entangled structures are a curious mathematical construction that has found their way into the heart of the chemistry lab and the toolbox of constructive geometry. Of particular interest are those structures—knots, links and weavings—which are composed locally of simple twisted strands and are globally symmetric. This paper considers the symmetric tangling of multiple 2-periodic honeycomb networks. We do this using a constructive methodology borrowing elements of graph theory, low-dimensional topology and geometry. The result is a wide-ranging enumeration of symmetric tangled honeycomb networks, providing a foundation for their exploration in both the chemistry lab and the geometers toolbox. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1282 KW - tangles KW - knots KW - networks KW - periodic entanglement KW - molecular weaving KW - graphs Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-570842 SN - 1866-8372 IS - 1282 ER - TY - THES A1 - Hannes, Sebastian T1 - Boundary Value Problems for the Lorentzian Dirac Operator N2 - 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. Bär 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, Bär 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 Bär 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. N2 - Der Indexsatz für elliptische Operatoren auf geschlossenen Riemannschen Mannigfaltigkeiten von Atiyah und Singer hat zahlreiche Anwendungen in Analysis, Geometrie und Topologie, ist aber ungeeignet für eine Verallgemeinerung auf Lorentz-Mannigfaltigkeiten. Durch die Einführung nicht-lokaler Randbedingungen, gewonnen aus der Spektralzerlegung eines induzierten Randoperators, beweisen Atiyah, Patodi und Singer (APS) einen Indexsatz für den Fall kompakter Riemannscher Mannigfaltigkeiten mit Rand. Aufbauend auf diesem Resultat und mit Hilfe der Charakterisierung des Spektralflusses durch Philipps gelangen Bär und Strohmaier zu einem Indexsatz für den Dirac-Operator auf global hyperbolischen Lorentz-Mannigfaltigkeiten mit kompakten und raumartigen Cauchy-Hyperflächen. Ihr Ergebnis zeigt unter anderem, dass der Dirac Operator auf solchen Mannigfaltigkeiten und unter APS Randbedingungen ein Fredholm-Operator mit glattem Kern ist und das sein Index sich aus einer zum Riemannschen Fall analogen Formel berechnen lässt. Zurück im Riemannschen Setup zeigen Bär und Ballmann eine allgemeine Charakterisierung von Randbedingungen für elliptische Differentialoperatoren erster Ordnung die sowohl die Regularität von Lösungen, als auch Fredholm-Eigenschaft des resultierenden Operators garantieren. Die dort entwickelten Randbedingungen können als Deformation auf den Graphen einer geeigneten Abbildung der APS-Randbedingung auf ihr orthogonales Komplement verstanden werden. Die vorliegende Arbeit hat das Ziel die von Bär und Ballmann beschriebenen Randbedingungen auf den Dirac-Operator von global hyperbolischen Lorentz-Mannigfaltigkeiten zu übertragen um eine allgemeinere Klasse von Randbedingungen zu finden unter denen der resultierende Dirac-Operator Fredholm ist und einen glatten Lösungsraum hat. Weiterhin wird analysiert wie sich derartige Deformation von APS-Randbedingungen auf den Index solcher Operatoren auswirken und wie dieser aus den bekannten Resultaten für den APS-Index berechnet werden kann. Es wird unter anderem gezeigt, dass im Gegensatz zum Riemannschen Fall beim Übergang von Randbedingungen zu ihrem orthogonalen Komplement die Fredholm-Eigenschaft des Operators erhalten bleibt. Andererseits sind zusätzliche Annahme nötig um die Regularität von Lösungen, sowie die Fredholm-Eigenschaft für Graph-Deformationen im Fall von Lorentz-Mannigfaltigkeiten zu erhalten. Die Arbeit ist dabei wie folgt aufgebaut. In Kapitel 1 werden grundlegende Fakten zu Lorentzschen und Riemannschen Spin-Mannigfaltigkeiten, ihren Spinor-Bündeln und Dirac-Operatoren zusammengetragen. Diese Informationen dienen als Ausgangspunkt zur Definition und Analyse von Randbedingungen in späteren Kapiteln der Arbeit. Kapitel 2 definiert allgemein den Begriff der Randbedingung wie er in dieser Arbeit verwendet wird und führt zudem den sogenannten ''wave-evolution-Operator'' ein, der eine wichtige Rolle im Finden und Analysieren von Fredholm-Randbedingungen für den Dirac-Operator spielen wird. Zuletzt wird der Zusammenhang zwischen Fredholm-Paaren eines Hilbert-Raumes und Fredholm-Randbedingungen für den Dirac-Operator erklärt. Kapitel 3 beschäftigt sich mit der Berechnung des Hauptsymbols des wave-evolution-Operators und die dort erzielten Resultate werden verwendet um Fredholm-Eigenschaft, sowie Regularität von Lösungen für geeignete Deformationen von APS-Randbedingungen zu beweisen. Weiterhin werden hinreichende Bedingungen für (pseudo-)lokale Randbedingungen abgeleitet, die Fredholm-Eigenschaft und Regularität für den resultierenden Dirac-Operator garantieren. Kapitel 4 zeigt, aufbauend auf den Ergebnissen der Kapitel 1-3, einige Beispiele von lokalen und nicht-lokalen Randbedingungen für den Dirac-Operator. Unter gewissen Einschränkungen an die Geometrie der zugrunde liegenden Mannigfaltigkeit bzw. den gestellten Randbedingungen können Ergebnisse erzielt werden die in den allgemeineren Resultaten der vorangehenden Kapitel nicht enthalten sind. Zuletzt werden sogenannte Transmission-Bedingungen analysiert und die Unterschiede dieser Randbedingungen zum Riemannschen Fall aufgezeigt. T2 - Randwertprobleme für den Lorentschen Diracoperator KW - Dirac Operator KW - Boundary Value Problems KW - Lorentzian Geometry KW - Randwertprobleme KW - Diracoperator KW - Lorentzgeometrie Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-548391 ER - TY - JOUR A1 - Bär, Christian A1 - Bandara, Lashi T1 - Boundary value problems for general first-order elliptic differential operators JF - Journal of functional analysis N2 - We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods. KW - elliptic differential operators of firstorder KW - elliptic boundary KW - conditions KW - boundary regularity KW - Fredholm property KW - H-infinity-functional calculus KW - maximal regularity KW - Rarita-Schwinger KW - operator Y1 - 2022 U6 - https://doi.org/10.1016/j.jfa.2022.109445 SN - 0022-1236 SN - 1096-0783 VL - 282 IS - 12 PB - Elsevier CY - Amsterdam [u.a.] ER - TY - JOUR A1 - Mera, Azal Jaafar Musa A1 - Tarkhanov, Nikolai T1 - An elliptic equation of finite index in a domain JF - Boletin de la Sociedad Matemática Mexicana N2 - We give an example of first order elliptic equation for a complex-valued function in a plane domain which has a finite number of linearly independent solutions for any right-hand side. No boundary value conditions are thus required. KW - elliptic equation KW - Fredholm operator KW - index Y1 - 2022 U6 - https://doi.org/10.1007/s40590-022-00442-7 SN - 1405-213X SN - 2296-4495 VL - 28 IS - 2 PB - Springer International CY - New York [u.a.] ER - TY - JOUR A1 - Engbert, Ralf A1 - Rabe, Maximilian Michael A1 - Schwetlick, Lisa A1 - Seelig, Stefan A. A1 - Reich, Sebastian A1 - Vasishth, Shravan T1 - Data assimilation in dynamical cognitive science JF - Trends in cognitive sciences N2 - Dynamical models make specific assumptions about cognitive processes that generate human behavior. In data assimilation, these models are tested against timeordered data. Recent progress on Bayesian data assimilation demonstrates that this approach combines the strengths of statistical modeling of individual differences with the those of dynamical cognitive models. Y1 - 2022 U6 - https://doi.org/10.1016/j.tics.2021.11.006 SN - 1364-6613 SN - 1879-307X VL - 26 IS - 2 SP - 99 EP - 102 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Hinz, Michael A1 - Schwarz, Michael T1 - A note on Neumann problems on graphs JF - Positivity N2 - We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations. KW - Graphs KW - Discrete Dirichlet forms KW - Neumann problem KW - Royden boundary Y1 - 2022 U6 - https://doi.org/10.1007/s11117-022-00930-0 SN - 1385-1292 SN - 1572-9281 VL - 26 IS - 4 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Bär, Christian A1 - Hanke, Bernhard T1 - Local flexibility for open partial differential relations JF - Communications on pure and applied mathematics / issued by the Courant Institute of Mathematical Sciences, New York Univ. N2 - We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry. The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K is an element of Double-struck capital R every manifold of dimension at least 2 carries a complete C-1,C- 1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C-2-metrics in general. Y1 - 2021 U6 - https://doi.org/10.1002/cpa.21982 SN - 0010-3640 SN - 1097-0312 VL - 75 IS - 6 SP - 1377 EP - 1415 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Kolbe, Benedikt Maximilian A1 - Evans, Myfanwy E. T1 - Enumerating isotopy classes of tilings guided by the symmetry of triply JF - Siam journal on applied algebra and geometry N2 - We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems. KW - isotopic tiling theory KW - mapping class group KW - orbifolds KW - group KW - presentations KW - representations of groups as automorphism groups of KW - algebraic systems KW - triply periodic minimal surface KW - Delaney--Dress KW - tiling theory KW - hyperbolic tilings KW - two-dimensional topology Y1 - 2022 U6 - https://doi.org/10.1137/20M1358943 SN - 2470-6566 VL - 6 IS - 1 SP - 1 EP - 40 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Hanisch, Florian A1 - Ludewig, Matthias T1 - A rigorous construction of the supersymmetric path integral associated to a compact spin manifold JF - Communications in mathematical physics N2 - We give a rigorous construction of the path integral in N = 1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Guneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaume, Atiyah, Bismut and Witten. Y1 - 2022 U6 - https://doi.org/10.1007/s00220-022-04336-7 SN - 0010-3616 SN - 1432-0916 VL - 391 IS - 3 SP - 1209 EP - 1239 PB - Springer CY - Berlin ; Heidelberg ER - TY - JOUR A1 - Nassar, Yomna M. A1 - Hohmann, Nicolas A1 - Michelet, Robin A1 - Gottwalt, Katharina A1 - Meid, Andreas D. A1 - Burhenne, Jürgen A1 - Huisinga, Wilhelm A1 - Haefeli, Walter E. A1 - Mikus, Gerd A1 - Kloft, Charlotte T1 - Quantification of the Time Course of CYP3A Inhibition, Activation, and Induction Using a Population Pharmacokinetic Model of Microdosed Midazolam Continuous Infusion JF - Clinical Pharmacokinetics N2 - Background Cytochrome P450 (CYP) 3A contributes to the metabolism of many approved drugs. CYP3A perpetrator drugs can profoundly alter the exposure of CYP3A substrates. However, effects of such drug-drug interactions are usually reported as maximum effects rather than studied as time-dependent processes. Identification of the time course of CYP3A modulation can provide insight into when significant changes to CYP3A activity occurs, help better design drug-drug interaction studies, and manage drug-drug interactions in clinical practice. Objective We aimed to quantify the time course and extent of the in vivo modulation of different CYP3A perpetrator drugs on hepatic CYP3A activity and distinguish different modulatory mechanisms by their time of onset, using pharmacologically inactive intravenous microgram doses of the CYP3A-specific substrate midazolam, as a marker of CYP3A activity. Methods Twenty-four healthy individuals received an intravenous midazolam bolus followed by a continuous infusion for 10 or 36 h. Individuals were randomized into four arms: within each arm, two individuals served as a placebo control and, 2 h after start of the midazolam infusion, four individuals received the CYP3A perpetrator drug: voriconazole (inhibitor, orally or intravenously), rifampicin (inducer, orally), or efavirenz (activator, orally). After midazolam bolus administration, blood samples were taken every hour (rifampicin arm) or every 15 min (remaining study arms) until the end of midazolam infusion. A total of 1858 concentrations were equally divided between midazolam and its metabolite, 1'-hydroxymidazolam. A nonlinear mixed-effects population pharmacokinetic model of both compounds was developed using NONMEM (R). CYP3A activity modulation was quantified over time, as the relative change of midazolam clearance encountered by the perpetrator drug, compared to the corresponding clearance value in the placebo arm. Results Time course of CYP3A modulation and magnitude of maximum effect were identified for each perpetrator drug. While efavirenz CYP3A activation was relatively fast and short, reaching a maximum after approximately 2-3 h, the induction effect of rifampicin could only be observed after 22 h, with a maximum after approximately 28-30 h followed by a steep drop to almost baseline within 1-2 h. In contrast, the inhibitory impact of both oral and intravenous voriconazole was prolonged with a steady inhibition of CYP3A activity followed by a gradual increase in the inhibitory effect until the end of sampling at 8 h. Relative maximum clearance changes were +59.1%, +46.7%, -70.6%, and -61.1% for efavirenz, rifampicin, oral voriconazole, and intravenous voriconazole, respectively. Conclusions We could distinguish between different mechanisms of CYP3A modulation by the time of onset. Identification of the time at which clearance significantly changes, per perpetrator drug, can guide the design of an optimal sampling schedule for future drug-drug interaction studies. The impact of a short-term combination of different perpetrator drugs on the paradigm CYP3A substrate midazolam was characterized and can define combination intervals in which no relevant interaction is to be expected. Y1 - 2022 U6 - https://doi.org/10.1007/s40262-022-01175-6 SN - 0312-5963 SN - 1179-1926 VL - 61 IS - 11 SP - 1595 EP - 1607 PB - Springer CY - Northcote ER - TY - JOUR A1 - Démaris, Alix A1 - Widigson, Ella S. K. A1 - Ilvemark, Johan F. K. F. A1 - Steenholdt, Casper A1 - Seidelin, Jakob B. A1 - Huisinga, Wilhelm A1 - Michelet, Robin A1 - Aulin, Linda B. S. A1 - Kloft, Charlotte T1 - Ulcerative colitis and acute severe ulcerative colitis patients are overlooked in infliximab population pharmacokinetic models BT - results from a comprehensive review JF - Pharmaceutics / Molecular Diversity Preservation International N2 - Ulcerative colitis (UC) is part of the inflammatory bowels diseases, and moderate to severe UC patients can be treated with anti-tumour necrosis alpha monoclonal antibodies, including infliximab (IFX). Even though treatment of UC patients by IFX has been in place for over a decade, many gaps in modelling of IFX PK in this population remain. This is even more true for acute severe UC (ASUC) patients for which early prediction of IFX pharmacokinetic (PK) could highly improve treatment outcome. Thus, this review aims to compile and analyse published population PK models of IFX in UC and ASUC patients, and to assess the current knowledge on disease activity impact on IFX PK. For this, a semi-systematic literature search was conducted, from which 26 publications including a population PK model analysis of UC patients receiving IFX therapy were selected. Amongst those, only four developed a model specifically for UC patients, and only three populations included severe UC patients. Investigations of disease activity impact on PK were reported in only 4 of the 14 models selected. In addition, the lack of reported model codes and assessment of predictive performance make the use of published models in a clinical setting challenging. Thus, more comprehensive investigation of PK in UC and ASUC is needed as well as more adequate reports on developed models and their evaluation in order to apply them in a clinical setting. KW - infliximab KW - inflammatory bowel disease KW - ulcerative colitis KW - acute severe KW - disease activity KW - pharmacokinetic KW - pharmacometrics Y1 - 2022 U6 - https://doi.org/10.3390/pharmaceutics14102095 SN - 1999-4923 VL - 14 IS - 10 PB - MDPI CY - Basel ER - TY - JOUR A1 - Schanner, Maximilian A1 - Korte, Monika A1 - Holschneider, Matthias T1 - ArchKalmag14k: A kalman-filter based global geomagnetic model for the holocene JF - Journal of geophysical research : Solid earth N2 - We propose a global geomagnetic field model for the last 14 thousand years, based on thermoremanent records. We call the model ArchKalmag14k. ArchKalmag14k is constructed by modifying recently proposed algorithms, based on space-time correlations. Due to the amount of data and complexity of the model, the full Bayesian posterior is numerically intractable. To tackle this, we sequentialize the inversion by implementing a Kalman-filter with a fixed time step. Every step consists of a prediction, based on a degree dependent temporal covariance, and a correction via Gaussian process regression. Dating errors are treated via a noisy input formulation. Cross correlations are reintroduced by a smoothing algorithm and model parameters are inferred from the data. Due to the specific statistical nature of the proposed algorithms, the model comes with space and time-dependent uncertainty estimates. The new model ArchKalmag14k shows less variation in the large-scale degrees than comparable models. Local predictions represent the underlying data and agree with comparable models, if the location is sampled well. Uncertainties are bigger for earlier times and in regions of sparse data coverage. We also use ArchKalmag14k to analyze the appearance and evolution of the South Atlantic anomaly together with reverse flux patches at the core-mantle boundary, considering the model uncertainties. While we find good agreement with earlier models for recent times, our model suggests a different evolution of intensity minima prior to 1650 CE. In general, our results suggest that prior to 6000 BCE the data is not sufficient to support global models. Y1 - 2022 U6 - https://doi.org/10.1029/2021JB023166 SN - 2169-9313 SN - 2169-9356 VL - 127 IS - 2 PB - American Geophysical Union CY - Washington ER - TY - JOUR A1 - Dimitrova, Ilinka A1 - Koppitz, Jörg T1 - On relative ranks of the semigroup of orientation-preserving transformations on infinite chain with restricted range JF - Communications in algebra N2 - Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X,Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X,Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size. KW - Order-preserving transformations KW - orientation-preserving KW - transformations KW - relative rank KW - restricted range KW - transformation KW - semigroups on infinite chain Y1 - 2022 U6 - https://doi.org/10.1080/00927872.2021.2000998 SN - 0092-7872 SN - 1532-4125 VL - 50 IS - 5 SP - 2157 EP - 2168 PB - Taylor & Francis Group CY - Philadelphia ER - TY - JOUR A1 - Jia, Weihan A1 - Anslan, Sten A1 - Chen, Fahu A1 - Cao, Xianyong A1 - Dong, Hailiang A1 - Dulias, Katharina A1 - Gu, Zhengquan A1 - Heinecke, Liv A1 - Jiang, Hongchen A1 - Kruse, Stefan A1 - Kang, Wengang A1 - Li, Kai A1 - Liu, Sisi A1 - Liu, Xingqi A1 - Liu, Ying A1 - Ni, Jian A1 - Schwalb, Antje A1 - Stoof-Leichsenring, Kathleen R. A1 - Shen, Wei A1 - Tian, Fang A1 - Wang, Jing A1 - Wang, Yongbo A1 - Wang, Yucheng A1 - Xu, Hai A1 - Yang, Xiaoyan A1 - Zhang, Dongju A1 - Herzschuh, Ulrike T1 - Sedimentary ancient DNA reveals past ecosystem and biodiversity changes on the Tibetan Plateau: overview and prospects JF - Quaternary science reviews : the international multidisciplinary research and review journal N2 - Alpine ecosystems on the Tibetan Plateau are being threatened by ongoing climate warming and intensified human activities. Ecological time-series obtained from sedimentary ancient DNA (sedaDNA) are essential for understanding past ecosystem and biodiversity dynamics on the Tibetan Plateau and their responses to climate change at a high taxonomic resolution. Hitherto only few but promising studies have been published on this topic. The potential and limitations of using sedaDNA on the Tibetan Plateau are not fully understood. Here, we (i) provide updated knowledge of and a brief introduction to the suitable archives, region-specific taphonomy, state-of-the-art methodologies, and research questions of sedaDNA on the Tibetan Plateau; (ii) review published and ongoing sedaDNA studies from the Tibetan Plateau; and (iii) give some recommendations for future sedaDNA study designs. Based on the current knowledge of taphonomy, we infer that deep glacial lakes with freshwater and high clay sediment input, such as those from the southern and southeastern Tibetan Plateau, may have a high potential for sedaDNA studies. Metabarcoding (for microorganisms and plants), metagenomics (for ecosystems), and hybridization capture (for prehistoric humans) are three primary sedaDNA approaches which have been successfully applied on the Tibetan Plateau, but their power is still limited by several technical issues, such as PCR bias and incompleteness of taxonomic reference databases. Setting up high-quality and open-access regional taxonomic reference databases for the Tibetan Plateau should be given priority in the future. To conclude, the archival, taphonomic, and methodological conditions of the Tibetan Plateau are favorable for performing sedaDNA studies. More research should be encouraged to address questions about long-term ecological dynamics at ecosystem scale and to bring the paleoecology of the Tibetan Plateau into a new era. KW - Sedimentary ancient DNA (sedaDNA) KW - Tibetan Plateau KW - Environmental DNA KW - Taphonomy KW - Ecosystem KW - Biodiversity KW - Paleoecology KW - Paleogeography Y1 - 2022 U6 - https://doi.org/10.1016/j.quascirev.2022.107703 SN - 0277-3791 SN - 1873-457X VL - 293 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Bellingeri, Carlo A1 - Friz, Peter A1 - Paycha, Sylvie A1 - Preiß, Rosa Lili Dora T1 - Smooth rough paths, their geometry and algebraic renormalization JF - Vietnam journal of mathematics N2 - We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting. KW - Signatures KW - Rough paths KW - Cartan's development KW - Renormalization Y1 - 2022 U6 - https://doi.org/10.1007/s10013-022-00570-7 SN - 2305-221X SN - 2305-2228 VL - 50 IS - 3 SP - 719 EP - 761 PB - Springer CY - Singapore ER - TY - JOUR A1 - Lie, Han Cheng A1 - Stahn, Martin A1 - Sullivan, Tim J. T1 - Randomised one-step time integration methods for deterministic operator differential equations JF - Calcolo N2 - Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings. KW - Time integration KW - Operator differential equations KW - Randomisation KW - Uncertainty quantification Y1 - 2022 U6 - https://doi.org/10.1007/s10092-022-00457-6 SN - 0008-0624 SN - 1126-5434 VL - 59 IS - 1 PB - Springer CY - Milano ER - TY - JOUR A1 - Malem-Shinitski, Noa A1 - Ojeda, Cesar A1 - Opper, Manfred T1 - Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects JF - Entropy N2 - Traditionally, Hawkes processes are used to model time-continuous point processes with history dependence. Here, we propose an extended model where the self-effects are of both excitatory and inhibitory types and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and derive an inference algorithm by performing inference on an aggregated sum of Gaussian Processes. Approximate Bayesian inference is achieved via data augmentation, and we describe a mean-field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from different domains and compare it to previously reported results. KW - Bayesian inference KW - point process KW - Gaussian process Y1 - 2022 U6 - https://doi.org/10.3390/e24030356 SN - 1099-4300 VL - 24 IS - 3 PB - MDPI CY - Basel ER - TY - JOUR A1 - Pathiraja, Sahani Darschika A1 - Leeuwen, Peter Jan van T1 - Multiplicative Non-Gaussian model error estimation in data assimilation JF - Journal of advances in modeling earth systems : JAMES N2 - Model uncertainty quantification is an essential component of effective data assimilation. Model errors associated with sub-grid scale processes are often represented through stochastic parameterizations of the unresolved process. Many existing Stochastic Parameterization schemes are only applicable when knowledge of the true sub-grid scale process or full observations of the coarse scale process are available, which is typically not the case in real applications. We present a methodology for estimating the statistics of sub-grid scale processes for the more realistic case that only partial observations of the coarse scale process are available. Model error realizations are estimated over a training period by minimizing their conditional sum of squared deviations given some informative covariates (e.g., state of the system), constrained by available observations and assuming that the observation errors are smaller than the model errors. From these realizations a conditional probability distribution of additive model errors given these covariates is obtained, allowing for complex non-Gaussian error structures. Random draws from this density are then used in actual ensemble data assimilation experiments. We demonstrate the efficacy of the approach through numerical experiments with the multi-scale Lorenz 96 system using both small and large time scale separations between slow (coarse scale) and fast (fine scale) variables. The resulting error estimates and forecasts obtained with this new method are superior to those from two existing methods. KW - model uncertainty KW - non-Gaussian KW - data-driven KW - uncertainty KW - quantification KW - Lorenz 96 KW - sub-grid scale Y1 - 2022 U6 - https://doi.org/10.1029/2021MS002564 SN - 1942-2466 VL - 14 IS - 4 PB - American Geophysical Union CY - Washington ER - TY - JOUR A1 - Pohle, Jennifer A1 - Adam, Timo A1 - Beumer, Larissa T1 - Flexible estimation of the state dwell-time distribution in hidden semi-Markov models JF - Computational statistics & data analysis N2 - Hidden semi-Markov models generalise hidden Markov models by explicitly modelling the time spent in a given state, the so-called dwell time, using some distribution defined on the natural numbers. While the (shifted) Poisson and negative binomial distribution provide natural choices for such distributions, in practice, parametric distributions can lack the flexibility to adequately model the dwell times. To overcome this problem, a penalised maximum likelihood approach is proposed that allows for a flexible and data-driven estimation of the dwell-time distributions without the need to make any distributional assumption. This approach is suitable for direct modelling purposes or as an exploratory tool to investigate the latent state dynamics. The feasibility and potential of the suggested approach is illustrated in a simulation study and by modelling muskox movements in northeast Greenland using GPS tracking data. The proposed method is implemented in the R-package PHSMM which is available on CRAN. KW - Penalized likelihood KW - Smoothing KW - Time series KW - Animal movement modeling Y1 - 2022 U6 - https://doi.org/10.1016/j.csda.2022.107479 SN - 0167-9473 SN - 1872-7352 VL - 172 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Yen, Ming-Hsuan A1 - von Specht, Sebastian A1 - Lin, Yen-Yu A1 - Cotton, Fabrice A1 - Ma, Kuo-Fong T1 - Within- and between-event variabilities of strong-velocity pulses of moderate earthquakes within dense seismic arrays JF - Bulletin of the Seismological Society of America N2 - Ground motion with strong-velocity pulses can cause significant damage to buildings and structures at certain periods; hence, knowing the period and velocity amplitude of such pulses is critical for earthquake structural engineering. However, the physical factors relating the scaling of pulse periods with magnitude are poorly understood. In this study, we investigate moderate but damaging earthquakes (M-w 6-7) and characterize ground- motion pulses using the method of Shahi and Baker (2014) while considering the potential static-offset effects. We confirm that the within-event variability of the pulses is large. The identified pulses in this study are mostly from strike-slip-like earthquakes. We further perform simulations using the freq uency-wavenumber algorithm to investigate the causes of the variability of the pulse periods within and between events for moderate strike-slip earthquakes. We test the effect of fault dips, and the impact of the asperity locations and sizes. The simulations reveal that the asperity properties have a high impact on the pulse periods and amplitudes at nearby stations. Our results emphasize the importance of asperity characteristics, in addition to earthquake magnitudes for the occurrence and properties of pulses produced by the forward directivity effect. We finally quantify and discuss within- and between-event variabilities of pulse properties at short distances. Y1 - 2021 U6 - https://doi.org/10.1785/0120200376 SN - 0037-1106 SN - 1943-3573 VL - 112 IS - 1 SP - 361 EP - 380 PB - Seismological Society of America CY - El Cerito, Calif. ER - TY - JOUR A1 - Dube, Jonas A1 - Böckmann, Christine A1 - Ritter, Christoph T1 - Lidar-Derived Aerosol Properties from Ny-Ålesund, Svalbard during the MOSAiC Spring 2020 JF - Remote sensing / Molecular Diversity Preservation International (MDPI) N2 - In this work, we present Raman lidar data (from a Nd:YAG operating at 355 nm, 532 nm and 1064 nm) from the international research village Ny-Alesund for the time period of January to April 2020 during the Arctic haze season of the MOSAiC winter. We present values of the aerosol backscatter, the lidar ratio and the backscatter Angstrom exponent, though the latter depends on wavelength. The aerosol polarization was generally below 2%, indicating mostly spherical particles. We observed that events with high backscatter and high lidar ratio did not coincide. In fact, the highest lidar ratios (LR > 75 sr at 532 nm) were already found by January and may have been caused by hygroscopic growth, rather than by advection of more continental aerosol. Further, we performed an inversion of the lidar data to retrieve a refractive index and a size distribution of the aerosol. Our results suggest that in the free troposphere (above approximate to 2500 m) the aerosol size distribution is quite constant in time, with dominance of small particles with a modal radius well below 100 nm. On the contrary, below approximate to 2000 m in altitude, we frequently found gradients in aerosol backscatter and even size distribution, sometimes in accordance with gradients of wind speed, humidity or elevated temperature inversions, as if the aerosol was strongly modified by vertical displacement in what we call the "mechanical boundary layer". Finally, we present an indication that additional meteorological soundings during MOSAiC campaign did not necessarily improve the fidelity of air backtrajectories. KW - aerosol KW - Arctic haze KW - lidar KW - microphysical properties KW - backtrajectories; KW - Ny-Alesund KW - Svalbard KW - MOSAiC KW - aerosol-boundary layer interactions Y1 - 2022 U6 - https://doi.org/10.3390/rs14112578 SN - 2072-4292 VL - 14 IS - 11 PB - MDPI CY - Basel ER - TY - JOUR A1 - Stachanow, Viktoria A1 - Neumann, Uta A1 - Blankenstein, Oliver A1 - Bindellini, Davide A1 - Melin, Johanna A1 - Ross, Richard A1 - Whitaker, Martin J. J. A1 - Huisinga, Wilhelm A1 - Michelet, Robin A1 - Kloft, Charlotte T1 - Exploring dried blood spot cortisol concentrations as an alternative for monitoring pediatric adrenal insufficiency patients BT - a model-based analysis JF - Frontiers in pharmacology N2 - Congenital adrenal hyperplasia (CAH) is the most common form of adrenal insufficiency in childhood; it requires cortisol replacement therapy with hydrocortisone (HC, synthetic cortisol) from birth and therapy monitoring for successful treatment. In children, the less invasive dried blood spot (DBS) sampling with whole blood including red blood cells (RBCs) provides an advantageous alternative to plasma sampling. Potential differences in binding/association processes between plasma and DBS however need to be considered to correctly interpret DBS measurements for therapy monitoring. While capillary DBS samples would be used in clinical practice, venous cortisol DBS samples from children with adrenal insufficiency were analyzed due to data availability and to directly compare and thus understand potential differences between venous DBS and plasma. A previously published HC plasma pharmacokinetic (PK) model was extended by leveraging these DBS concentrations. In addition to previously characterized binding of cortisol to albumin (linear process) and corticosteroid-binding globulin (CBG; saturable process), DBS data enabled the characterization of a linear cortisol association with RBCs, and thereby providing a quantitative link between DBS and plasma cortisol concentrations. The ratio between the observed cortisol plasma and DBS concentrations varies highly from 2 to 8. Deterministic simulations of the different cortisol binding/association fractions demonstrated that with higher blood cortisol concentrations, saturation of cortisol binding to CBG was observed, leading to an increase in all other cortisol binding fractions. In conclusion, a mathematical PK model was developed which links DBS measurements to plasma exposure and thus allows for quantitative interpretation of measurements of DBS samples. KW - adrenal insufficiency KW - cortisol KW - dried blood spots KW - pediatrics KW - pharmacokinetics KW - binding KW - association KW - red blood cells Y1 - 2022 U6 - https://doi.org/10.3389/fphar.2022.819590 SN - 1663-9812 VL - 13 PB - Frontiers Media CY - Lausanne ER - TY - JOUR A1 - Pornsawad, Pornsarp A1 - Böckmann, Christine A1 - Panitsupakamon, Wannapa T1 - The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative JF - Electronic transactions on numerical analysis - ETNA N2 - The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg-Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided. KW - ill-posed problems KW - time-fractional derivative KW - backward heat problem KW - Levenberg-Marquardt method KW - a posteriori stopping rule KW - optimal order Y1 - 2022 SN - 978-3-7001-8258-0 U6 - https://doi.org/10.1553/etna_vol57s67 SN - 1068-9613 VL - 57 SP - 67 EP - 79 PB - Kent State University CY - Kent ER - TY - THES A1 - Mauerberger, Stefan T1 - Correlation based Bayesian modeling T1 - Korrelationsbasierte Bayesianische Modellierung BT - with applications in travel time tomography, seismic source inversion and magnetic field modeling BT - mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung N2 - 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. N2 - Die Motivation für diese Arbeit war die Frage nach Verlässlichkeit und Belastbarkeit der seismischen Tomographie. Das Problem besteht darin, dass sehr viele Erdmodelle existieren welche die zugrundeliegenden seismischen Aufzeichnungen gleich gut beschreiben können. Die meisten Algorithmen zur Rekonstruktion von Erdmodellen liefern zwar eine Lösung, quantifizierten jedoch kaum deren Variabilität. Wenn es keine Möglichkeit gibt die abgebildeten Strukturen zu verifizieren, so ist eine Interpretation kaum verlässlich. Der ursprüngliche Gedanke war den Raum äquivalenter Erdmodelle mithilfe Bayesianische Inferenz zu erkunden. Es stellte sich jedoch schnell heraus, dass die vollständige Quantifizierung tomographischer Unsicherheiten im Rahmen einer Promotion nicht zu bewältigen ist. Um das wesentliche Konzept der statistischen Inferenz beizubehalten werden stattdessen weniger komplexe Problemstellungen aus den Geowissenschaften behandelt. Diese Dissertation hat das Ziel die Bayesianische Inferenz tiefer in den Geowissenschaften zu verankern und Wissen aus der angewandten Mathematik zu transferieren. Die zugrundeliegende Idee besteht darin auf bekannte Methoden und Techniken der Statistik zurückzugreifen um die Unsicherheiten inverser Probleme in den Geowissenschaften zu quantifizieren. Diese Arbeit gliedert sich in drei Teile: Teil I führt die notwendige Mathematik ein und soll als eine Art Werkzeugkasten verstanden werden. In Hinblick auf eine physikalische Anwendung bietet dieser Abschnitt eine kompakte Zusammenfassung aller eingesetzter Methoden und Techniken. Den Anfang macht die Einführung der Bayesianische Inferenz. Danach steht als Spezialfall die Regression mit Gauß-Prozessen unter linearen Transformationen im Vordergrund. Die Kapitel zur Herleitung von Kovarianzfunktionen und die Approximation von Nichtlinearitäten gehen etwas weiter in die Tiefe. Teil II präsentiert zwei Konzeptstudien aus dem Bereich der Seismologie. Ziel ist es bei moderater Komplexität die prinzipielle Anwendung der eingeführten Methoden und Techniken zu präsentieren. Das Beispiel zur Laufzeittomographie wendet die Näherungs\-methoden für nichtlineare Zusammenhänge an. Die Herleitung einer Kovarianzfunktion mithilfe der Wellengleichung ist am Beispiel der gedämpften Saitenschwingung gezeigt. Mit diesen beiden synthetischen Anwendungen wurde ein konsistentes Konzept zur Quantifizierung von Modellierungsunsicherheiten erarbeitet. Teil III präsentiert die Rekonstruktion des archeomagnetischen Feldes unserer Erde. Diese Anwendung nutzt den gesamten Werkzeugkasten aus Teil I und ist entsprechend umfangreich. Die Modellierung der vergangenen 1000 Jahre basiert auf echten Daten und quantifiziert zuverlässig die räumlichen Modellierungsunsicherheiten. Das präsentierte statistische Modell findet breite Anwendung und wird aktiv weiter entwickelt. Die drei genannten Anwendungen sind bewusst flexibel gehalten um die Übertragbarkeit auf ähnliche Problemstellungen zu ermöglichen. Die gesamte Arbeit legt den Fokus auf die nicht-Eindeutigkeit inverser Probleme in den Geowissenschaften. Sie will für all Jene von Relevanz sein, die sich für die Konzepte der Bayesianischen Inferenz interessieren. KW - statistical inference KW - Bayesian inversion KW - travel time tomography KW - seismic source inversion KW - magnetic field modeling KW - mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung KW - Magnetfeldmodellierung KW - seismische Quellinversion KW - statistische Inferenz KW - Laufzeittomographie Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-537827 ER - TY - JOUR A1 - Lilienkamp, Henning A1 - von Specht, Sebastian A1 - Weatherill, Graeme A1 - Caire, Giuseppe A1 - Cotton, Fabrice T1 - Ground-Motion modeling as an image processing task BT - introducing a neural network based, fully data-driven, and nonergodic JF - Bulletin of the Seismological Society of America N2 - We construct and examine the prototype of a deep learning-based ground-motion model (GMM) that is both fully data driven and nonergodic. We formulate ground-motion modeling as an image processing task, in which a specific type of neural network, the U-Net, relates continuous, horizontal maps of earthquake predictive parameters to sparse observations of a ground-motion intensity measure (IM). The processing of map-shaped data allows the natural incorporation of absolute earthquake source and observation site coordinates, and is, therefore, well suited to include site-, source-, and path-specific amplification effects in a nonergodic GMM. Data-driven interpolation of the IM between observation points is an inherent feature of the U-Net and requires no a priori assumptions. We evaluate our model using both a synthetic dataset and a subset of observations from the KiK-net strong motion network in the Kanto basin in Japan. We find that the U-Net model is capable of learning the magnitude???distance scaling, as well as site-, source-, and path-specific amplification effects from a strong motion dataset. The interpolation scheme is evaluated using a fivefold cross validation and is found to provide on average unbiased predictions. The magnitude???distance scaling as well as the site amplification of response spectral acceleration at a period of 1 s obtained for the Kanto basin are comparable to previous regional studies. Y1 - 2022 U6 - https://doi.org/10.1785/0120220008 SN - 0037-1106 SN - 1943-3573 VL - 112 IS - 3 SP - 1565 EP - 1582 PB - Seismological Society of America CY - Albany ER - TY - JOUR A1 - Molkenthin, Christian A1 - Donner, Christian A1 - Reich, Sebastian A1 - Zöller, Gert A1 - Hainzl, Sebastian A1 - Holschneider, Matthias A1 - Opper, Manfred T1 - GP-ETAS: semiparametric Bayesian inference for the spatio-temporal epidemic type aftershock sequence model JF - Statistics and Computing N2 - The spatio-temporal epidemic type aftershock sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a fully Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l'Aquila region, Italy, with the devastating event on 6 April 2009, is presented. KW - Self-exciting point process KW - Hawkes process KW - Spatio-temporal ETAS model KW - Bayesian inference KW - Sampling KW - Earthquake modeling KW - Gaussian process KW - Data augmentation Y1 - 2022 U6 - https://doi.org/10.1007/s11222-022-10085-3 SN - 0960-3174 SN - 1573-1375 VL - 32 IS - 2 PB - Springer CY - Dordrecht ER -