@article{FischerKeller2021, author = {Fischer, Florian and Keller, Matthias}, title = {Riesz decompositions for Schr{\"o}dinger operators on graphs}, series = {Journal of mathematical analysis and applications}, volume = {495}, journal = {Journal of mathematical analysis and applications}, number = {1}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0022-247X}, doi = {10.1016/j.jmaa.2020.124674}, pages = {22}, year = {2021}, abstract = {We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.}, language = {en} } @article{Baer2021, author = {B{\"a}r, Christian}, title = {The Faddeev-LeVerrier algorithm and the Pfaffian}, series = {Linear algebra and its applications}, volume = {630}, journal = {Linear algebra and its applications}, publisher = {Elsevier}, address = {New York}, issn = {0024-3795}, doi = {10.1016/j.laa.2021.07.023}, pages = {39 -- 55}, year = {2021}, abstract = {We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.}, language = {en} } @article{Bandara2021, author = {Bandara, Lashi}, title = {Functional calculus and harmonic analysis in geometry}, series = {S{\~a}o Paulo journal of mathematical sciences / Instituto de Matem{\´a}tica e Estat{\´i}stica da Universidade de S{\~a}o Paulo}, volume = {15}, journal = {S{\~a}o Paulo journal of mathematical sciences / Instituto de Matem{\´a}tica e Estat{\´i}stica da Universidade de S{\~a}o Paulo}, number = {1}, publisher = {Springer}, address = {Cham}, issn = {1982-6907}, doi = {10.1007/s40863-019-00149-0}, pages = {20 -- 53}, year = {2021}, abstract = {In this short survey article, we showcase a number of non-trivial geometric problems that have recently been resolved by marrying methods from functional calculus and real-variable harmonic analysis. We give a brief description of these methods as well as their interplay. This is a succinct survey that hopes to inspire geometers and analysts alike to study these methods so that they can be further developed to be potentially applied to a broader range of questions.}, language = {en} } @article{KleinRosenberger2021, author = {Klein, Markus and Rosenberger, Elke}, title = {The tunneling effect for Schr{\"o}dinger operators on a vector bundle}, series = {Analysis and mathematical physics}, volume = {11}, journal = {Analysis and mathematical physics}, number = {2}, publisher = {Springer International Publishing AG}, address = {Cham (ZG)}, issn = {1664-2368}, doi = {10.1007/s13324-021-00485-5}, pages = {35}, year = {2021}, abstract = {In the semiclassical limit (h) over bar -> 0, we analyze a class of self-adjoint Schrodinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V center dot id(E) acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m(1),... m(r) is an element of M, called potential wells. Using quasimodes of WKB-type near m(j) for eigenfunctions associated with the low lying eigenvalues of H-(h) over bar, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension l + 1. This dimension l determines the polynomial prefactor for exponentially small eigenvalue splitting.}, language = {en} } @phdthesis{Etzold2021, author = {Etzold, Heiko}, title = {Neue Zug{\"a}nge zum Winkelbegriff}, doi = {10.25932/publishup-50418}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-504187}, school = {Universit{\"a}t Potsdam}, pages = {300}, year = {2021}, abstract = {Die Vielf{\"a}ltigkeit des Winkelbegriffs ist gleichermaßen spannend wie herausfordernd in Hinblick auf seine Zug{\"a}nge im Mathematikunterricht der Schule. Ausgehend von verschiedenen Vorstellungen zum Winkelbegriff wird in dieser Arbeit ein Lehrgang zur Vermittlung des Winkelbegriffs entwickelt und letztlich in konkrete Umsetzungen f{\"u}r den Schulunterricht {\"u}berf{\"u}hrt. Dabei erfolgt zun{\"a}chst eine stoffdidaktische Auseinandersetzung mit dem Winkelbegriff, die von einer informationstheoretischen Winkeldefinition begleitet wird. In dieser wird eine Definition f{\"u}r den Winkelbegriff unter der Fragestellung entwickelt, welche Informationen man {\"u}ber einen Winkel ben{\"o}tigt, um ihn beschreiben zu k{\"o}nnen. So k{\"o}nnen die in der fachdidaktischen Literatur auftretenden Winkelvorstellungen aus fachmathematischer Perspektive erneut abgeleitet und validiert werden. Parallel dazu wird ein Verfahren beschrieben, wie Winkel - auch unter dynamischen Aspekten - informationstechnisch verarbeitet werden k{\"o}nnen, so dass Schlussfolgerungen aus der informationstheoretischen Winkeldefinition beispielsweise in dynamischen Geometriesystemen zur Verf{\"u}gung stehen. Unter dem Gesichtspunkt, wie eine Abstraktion des Winkelbegriffs im Mathematikunterricht vonstatten gehen kann, werden die Grundvorstellungsidee sowie die Lehrstrategie des Aufsteigens vom Abstrakten zum Konkreten miteinander in Beziehung gesetzt. Aus der Verkn{\"u}pfung der beiden Theorien wird ein grunds{\"a}tzlicher Weg abgeleitet, wie im Rahmen der Lehrstrategie eine Ausgangsabstraktion zu einzelnen Winkelaspekten aufgebaut werden kann, was die Generierung von Grundvorstellungen zu den Bestandteilen des jeweiligen Winkelaspekts und zum Operieren mit diesen Begriffsbestandteilen erm{\"o}glichen soll. Hierf{\"u}r wird die Lehrstrategie angepasst, um insbesondere den {\"U}bergang von Winkelsituationen zu Winkelkontexten zu realisieren. Explizit f{\"u}r den Aspekt des Winkelfeldes werden, anhand der Untersuchung der Sichtfelder von Tieren, Lernhandlungen und Forderungen an ein Lernmodell beschrieben, die Sch{\"u}lerinnen und Sch{\"u}ler bei der Begriffsaneignung unterst{\"u}tzen. Die T{\"a}tigkeitstheorie, der die genannte Lehrstrategie zuzuordnen ist, zieht sich als roter Faden durch die weitere Arbeit, wenn nun theoriebasiert Designprinzipien generiert werden, die in die Entwicklung einer interaktiven Lernumgebung m{\"u}nden. Hierzu wird u. a. das Modell der Artifact-Centric Activity Theory genutzt, das das Beziehungsgef{\"u}ge aus Sch{\"u}lerinnen und Sch{\"u}lern, dem mathematischen Gegenstand und einer zu entwickelnden App als vermittelndes Medium beschreibt, wobei der Einsatz der App im Unterrichtskontext sowie deren regelgeleitete Entwicklung Bestandteil des Modells sind. Gem{\"a}ß dem Ansatz der Fachdidaktischen Entwicklungsforschung wird die Lernumgebung anschließend in mehreren Zyklen erprobt, evaluiert und {\"u}berarbeitet. Dabei wird ein qualitatives Setting angewandt, das sich der Semiotischen Vermittlung bedient und untersucht, inwiefern sich die Qualit{\"a}t der von den Sch{\"u}lerinnen und Sch{\"u}lern gezeigten Lernhandlungen durch die Designprinzipien und deren Umsetzung erkl{\"a}ren l{\"a}sst. Am Ende der Arbeit stehen eine finale Version der Designprinzipien und eine sich daraus ergebende Lernumgebung zur Einf{\"u}hrung des Winkelfeldbegriffs in der vierten Klassenstufe.}, language = {de} } @phdthesis{Zass2021, author = {Zass, Alexander}, title = {A multifaceted study of marked Gibbs point processes}, doi = {10.25932/publishup-51277}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-512775}, school = {Universit{\"a}t Potsdam}, pages = {vii, 104}, year = {2021}, abstract = {This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation. We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight. We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution. Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation. Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry.}, language = {en} } @phdthesis{Oancea2021, author = {Oancea, Marius-Adrian}, title = {Spin Hall effects in general relativity}, doi = {10.25932/publishup-50229}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-502293}, school = {Universit{\"a}t Potsdam}, pages = {vii, 123}, year = {2021}, abstract = {The propagation of test fields, such as electromagnetic, Dirac or linearized gravity, on a fixed spacetime manifold is often studied by using the geometrical optics approximation. In the limit of infinitely high frequencies, the geometrical optics approximation provides a conceptual transition between the test field and an effective point-particle description. The corresponding point-particles, or wave rays, coincide with the geodesics of the underlying spacetime. For most astrophysical applications of interest, such as the observation of celestial bodies, gravitational lensing, or the observation of cosmic rays, the geometrical optics approximation and the effective point-particle description represent a satisfactory theoretical model. However, the geometrical optics approximation gradually breaks down as test fields of finite frequency are considered. In this thesis, we consider the propagation of test fields on spacetime, beyond the leading-order geometrical optics approximation. By performing a covariant Wentzel-Kramers-Brillouin analysis for test fields, we show how higher-order corrections to the geometrical optics approximation can be considered. The higher-order corrections are related to the dynamics of the spin internal degree of freedom of the considered test field. We obtain an effective point-particle description, which contains spin-dependent corrections to the geodesic motion obtained using geometrical optics. This represents a covariant generalization of the well-known spin Hall effect, usually encountered in condensed matter physics and in optics. Our analysis is applied to electromagnetic and massive Dirac test fields, but it can easily be extended to other fields, such as linearized gravity. In the electromagnetic case, we present several examples where the gravitational spin Hall effect of light plays an important role. These include the propagation of polarized light rays on black hole spacetimes and cosmological spacetimes, as well as polarization-dependent effects on the shape of black hole shadows. Furthermore, we show that our effective point-particle equations for polarized light rays reproduce well-known results, such as the spin Hall effect of light in an inhomogeneous medium, and the relativistic Hall effect of polarized electromagnetic wave packets encountered in Minkowski spacetime.}, language = {en} } @misc{Moehring2021, type = {Master Thesis}, author = {M{\"o}hring, Jan}, title = {Stochastic inversion for core field modeling using satellite data}, doi = {10.25932/publishup-49807}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-498072}, school = {Universit{\"a}t Potsdam}, pages = {vii, 55}, year = {2021}, abstract = {Magnetfeldmodellierung mit Kugelfl{\"a}chenfunktionen basiert auf der Inversion nach hunderten bis tausenden von Parametern. Dieses hochdimensionale Problem kann grunds{\"a}tzlich als ein Optimierungsproblem formuliert werden, bei dem ein globales Minimum einer gewissen Zielfunktion berechnet werden soll. Um dieses Problem zu l{\"o}sen, gibt es eine Reihe bekannter Ans{\"a}tze, dazu z{\"a}hlen etwa gradientenbasierte Verfahren oder die Methode der kleinsten Quadrate und deren Varianten. Jede dieser Methoden hat verschiedene Vor- und Nachteile, beispielsweise bez{\"u}glich der Anwendbarkeit auf nicht-differenzierbare Funktionen oder der Laufzeit zugeh{\"o}riger Algorithmen. In dieser Arbeit verfolgen wir das Ziel, einen Algorithmus zu finden, der schneller als die etablierten Verfahren ist und sich auch f{\"u}r nichtlineare Probleme anwenden l{\"a}sst. Solche nichtlinearen Probleme treten beispielsweise bei der Absch{\"a}tzung von Euler-Winkeln oder bei der Verwendung der robusteren L_1-Norm auf. Dazu untersuchen wir die Anwendbarkeit stochastischer Optimierungsverfahren aus der CMAES-Familie auf die Modellierung des geomagnetischen Feldes des Erdkerns. Es werden sowohl die Grundlagen der Kernfeldmodellierung und deren Parametrisierung anhand einiger Beispiele aus der Literatur besprochen, als auch die theoretischen Hintergr{\"u}nde der stochastischen Verfahren gegeben. Ein CMAES-Algorithmus wurde erfolgreich angewendet, um Daten der Swarm-Satellitenmission zu invertieren und daraus das Magnetfeldmodell EvoMag abzuleiten. EvoMag zeigt gute {\"U}bereinstimmung mit etablierten Modellen, sowie mit Observatoriumsdaten aus Niemegk. Wir thematisieren einige beobachtete Schwierigkeiten und pr{\"a}sentieren und diskutieren die Ergebnisse unserer Modellierung.}, language = {en} } @article{BaerMazzeo2021, author = {B{\"a}r, Christian and Mazzeo, Rafe}, title = {Manifolds with many Rarita-Schwinger fields}, series = {Communications in mathematical physics}, volume = {384}, journal = {Communications in mathematical physics}, number = {1}, publisher = {Springer}, address = {Berlin}, issn = {0010-3616}, doi = {10.1007/s00220-021-04030-0}, pages = {533 -- 548}, year = {2021}, abstract = {The Rarita-Schwinger operator is the twisted Dirac operator restricted to 3/2-spinors. Rarita-Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita-Schwinger fields tends to infinity. These manifolds are either simply connected Kahler-Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi-Yau manifolds of even complex dimension with more linearly independent Rarita-Schwinger fields than flat tori of the same dimension.}, language = {en} } @phdthesis{Zadorozhnyi2021, author = {Zadorozhnyi, Oleksandr}, title = {Contributions to the theoretical analysis of the algorithms with adversarial and dependent data}, school = {Universit{\"a}t Potsdam}, pages = {144}, year = {2021}, abstract = {In this work I present the concentration inequalities of Bernstein's type for the norms of Banach-valued random sums under a general functional weak-dependency assumption (the so-called \$\cC-\$mixing). The latter is then used to prove, in the asymptotic framework, excess risk upper bounds of the regularised Hilbert valued statistical learning rules under the τ-mixing assumption on the underlying training sample. These results (of the batch statistical setting) are then supplemented with the regret analysis over the classes of Sobolev balls of the type of kernel ridge regression algorithm in the setting of online nonparametric regression with arbitrary data sequences. Here, in particular, a question of robustness of the kernel-based forecaster is investigated. Afterwards, in the framework of sequential learning, the multi-armed bandit problem under \$\cC-\$mixing assumption on the arm's outputs is considered and the complete regret analysis of a version of Improved UCB algorithm is given. Lastly, probabilistic inequalities of the first part are extended to the case of deviations (both of Azuma-Hoeffding's and of Burkholder's type) to the partial sums of real-valued weakly dependent random fields (under the type of projective dependence condition).}, language = {en} } @article{GottwaldReich2021, author = {Gottwald, Georg A. and Reich, Sebastian}, title = {Combining machine learning and data assimilation to forecast dynamical systems from noisy partial observations}, series = {Chaos : an interdisciplinary journal of nonlinear science}, volume = {31}, journal = {Chaos : an interdisciplinary journal of nonlinear science}, number = {10}, publisher = {AIP}, address = {Melville}, issn = {1054-1500}, doi = {10.1063/5.0066080}, pages = {8}, year = {2021}, abstract = {We present a supervised learning method to learn the propagator map of a dynamical system from partial and noisy observations. In our computationally cheap and easy-to-implement framework, a neural network consisting of random feature maps is trained sequentially by incoming observations within a data assimilation procedure. By employing Takens's embedding theorem, the network is trained on delay coordinates. We show that the combination of random feature maps and data assimilation, called RAFDA, outperforms standard random feature maps for which the dynamics is learned using batch data.}, language = {en} } @article{Clavier2021, author = {Clavier, Pierre J.}, title = {Borel-{\´E}calle resummation of a two-point function}, series = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincar{\´e} and by the Swiss Physical Society}, volume = {22}, journal = {Annales Henri Poincar{\´e} : a journal of theoretical and mathematical physics / ed. jointly by the Institut Henri Poincar{\´e} and by the Swiss Physical Society}, number = {6}, publisher = {Springer}, address = {Cham}, issn = {1424-0637}, doi = {10.1007/s00023-021-01057-w}, pages = {2103 -- 2136}, year = {2021}, abstract = {We provide an overview of the tools and techniques of resurgence theory used in the Borel-ecalle resummation method, which we then apply to the massless Wess-Zumino model. Starting from already known results on the anomalous dimension of the Wess-Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and that its Borel transform is resurgent. The Schwinger-Dyson equation of the model is then used to prove an asymptotic exponential bound for the Borel transformed two-point function on a star-shaped domain of a suitable ramified complex plane. This proves that the two-point function of the Wess-Zumino model is Borel-ecalle summable.}, language = {en} } @masterthesis{Dahl2021, type = {Bachelor Thesis}, author = {Dahl, Dorothee Sophie}, title = {Let's have FUN! Gamification im Mathematikunterricht}, doi = {10.25932/publishup-51593}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-515937}, school = {Universit{\"a}t Potsdam}, pages = {78}, year = {2021}, abstract = {Spiele und spieltypische Elemente wie das Sammeln von Treuepunkten sind aus dem Alltag kaum wegzudenken. Zudem werden sie zunehmend in Unternehmen oder in Lernumgebungen eingesetzt. Allerdings ist die Methode Gamification bisher f{\"u}r den p{\"a}dagogischen Kontext wenig klassifiziert und f{\"u}r Lehrende kaum zug{\"a}nglich gemacht worden. Daher zielt diese Bachelorarbeit darauf ab, eine systematische Strukturierung und Aufarbeitung von Gamification sowie innovative Ans{\"a}tze f{\"u}r die Verwendung spieltypischer Elemente im Unterricht, konkret dem Mathematikunterricht, zu pr{\"a}sentieren. Dies kann eine Grundlage f{\"u}r andere Fachgebiete, aber auch andere Lehrformen bieten und so die Umsetzbarkeit von Gamification in eigenen Lehrveranstaltungen aufzeigen. In der Arbeit wird begr{\"u}ndet, weshalb und mithilfe welcher Elemente Gamification die Motivation und Leistungsbereitschaft der Lernenden langfristig erh{\"o}hen, die Sozial- und Personalkompetenzen f{\"o}rdern sowie die Lernenden zu mehr Aktivit{\"a}t anregen kann. Zudem wird Gamification explizit mit grundlegenden mathematikdidaktischen Prinzipien in Verbindung gesetzt und somit die Relevanz f{\"u}r den Mathematikunterricht hervorgehoben. Anschließend werden die einzelnen Elemente von Gamification wie Punkte, Level, Abzeichen, Charaktere und Rahmengeschichte entlang einer eigens f{\"u}r den p{\"a}dagogischen Kontext entwickelten Klassifikation „FUN" (Feedback - User specific elements - Neutral elements) schematisch beschrieben, ihre Funktionen und Wirkung dargestellt sowie Einsatzm{\"o}glichkeiten im Unterricht aufgezeigt. Dies beinhaltet Ideen zu lernf{\"o}rderlichem Feedback, Differenzierungsm{\"o}glichkeiten und Unterrichtsrahmengestaltung, die in Lehrveranstaltungen aller Art umsetzbar sein k{\"o}nnen. Die Bachelorarbeit umfasst zudem ein spezifisches Beispiel, einen Unterrichtsentwurf einer gamifizierten Mathematikstunde inklusive des zugeh{\"o}rigen Arbeitsmaterials, anhand dessen die Verwendung von Gamification deutlich wird. Gamification offeriert oftmals Vorteile gegen{\"u}ber dem traditionellen Unterricht, muss jedoch wie jede Methode an den Inhalt und die Zielgruppe angepasst werden. Weiterf{\"u}hrende Forschung k{\"o}nnte sich mit konkreten motivationalen Strukturen, personenspezifischen Unterschieden sowie mit mathematischen Inhalten wie dem Probleml{\"o}sen oder dem Wechsel zwischen verschiedenen Darstellungen hinsichtlich gamifizierter Lehrformen besch{\"a}ftigen.}, language = {de} } @masterthesis{Engelhardt2021, type = {Bachelor Thesis}, author = {Engelhardt, Max Angel Ronan}, title = {Zwischen Simulation und Beweis - eine mathematische Analyse des Bienaym{\´e}-Galton-Watson-Prozesses und sein Einsatz innerhalb des Mathematikunterrichts}, doi = {10.25932/publishup-52447}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-524474}, school = {Universit{\"a}t Potsdam}, pages = {117}, year = {2021}, abstract = {Die Bienaym{\´e}-Galton-Watson Prozesse k{\"o}nnen f{\"u}r die Untersuchung von speziellen und sich entwickelnden Populationen verwendet werden. Die Populationen umfassen Individuen, welche sich identisch, zuf{\"a}llig, selbstst{\"a}ndig und unabh{\"a}ngig voneinander fortpflanzen und die jeweils nur eine Generation existieren. Die n-te Generation ergibt sich als zuf{\"a}llige Summe der Individuen der (n-1)-ten Generation. Die Relevanz dieser Prozesse begr{\"u}ndet sich innerhalb der Historie und der inner- und außermathematischen Bedeutung. Die Geschichte der Bienaym{\´e}-Galton-Watson-Prozesse wird anhand der Entwicklung des Konzeptes bis heute dargestellt. Dabei werden die Wissenschaftler:innen verschiedener Disziplinen angef{\"u}hrt, die Erkenntnisse zu dem Themengebiet beigetragen und das Konzept in ihren Fachbereichen angef{\"u}hrt haben. Somit ergibt sich die außermathematische Signifikanz. Des Weiteren erh{\"a}lt man die innermathematische Bedeutsamkeit mittels des Konzeptes der Verzweigungsprozesse, welches auf die Bienaym{\´e}-Galton-Watson Prozesse zur{\"u}ckzuf{\"u}hren ist. Die Verzweigungsprozesse stellen eines der aussagekr{\"a}ftigsten Modelle f{\"u}r die Beschreibung des Populationswachstums dar. Dar{\"u}ber hinaus besteht die derzeitige Wichtigkeit durch die Anwendungsm{\"o}glichkeit der Verzweigungsprozesse und der Bienaym{\´e}-Galton-Watson Prozesse innerhalb der Epidemiologie. Es werden die Ebola- und die Corona-Pandemie als Anwendungsfelder angef{\"u}hrt. Die Prozesse dienen als Entscheidungsst{\"u}tze f{\"u}r die Politik und erm{\"o}glichen Aussagen {\"u}ber die Auswirkungen von Maßnahmen bez{\"u}glich der Pandemien. Neben den Prozessen werden ebenfalls der bedingte Erwartungswert bez{\"u}glich diskreter Zufallsvariablen, die wahrscheinlichkeitserzeugende Funktion und die zuf{\"a}llige Summe eingef{\"u}hrt. Die Konzepte vereinfachen die Beschreibung der Prozesse und bilden somit die Grundlage der Betrachtungen. Außerdem werden die ben{\"o}tigten und weiterf{\"u}hrenden Eigenschaften der grundlegenden Themengebiete und der Prozesse aufgef{\"u}hrt und bewiesen. Das Kapitel erreicht seinen H{\"o}hepunkt bei dem Beweis des Kritikalit{\"a}tstheorems, wodurch eine Aussage {\"u}ber das Aussterben des Prozesses in verschiedenen F{\"a}llen und somit {\"u}ber die Aussterbewahrscheinlichkeit get{\"a}tigt werden kann. Die F{\"a}lle werden anhand der zu erwartenden Anzahl an Nachkommen eines Individuums unterschieden. Es zeigt sich, dass ein Prozess bei einer zu erwartenden Anzahl kleiner gleich Eins mit Sicherheit ausstirbt und bei einer Anzahl gr{\"o}ßer als Eins, die Population nicht in jedem Fall aussterben muss. Danach werden einzelne Beispiele, wie der linear fractional case, die Population von Fibroblasten (Bindegewebszellen) von M{\"a}usen und die Entstehungsfragestellung der Prozesse, angef{\"u}hrt. Diese werden mithilfe der erlangten Ergebnisse untersucht und einige ausgew{\"a}hlte zuf{\"a}llige Dynamiken werden im nachfolgenden Kapitel simuliert. Die Simulationen erfolgen durch ein in Python erstelltes Programm und werden mithilfe der Inversionsmethode realisiert. Die Simulationen stellen beispielhaft die Entwicklungen in den verschiedenen Kritikalit{\"a}tsf{\"a}llen der Prozesse dar. Zudem werden die H{\"a}ufigkeiten der einzelnen Populationsgr{\"o}ßen in Form von Histogrammen angebracht. Dabei l{\"a}sst sich der Unterschied zwischen den einzelnen F{\"a}llen best{\"a}tigen und es wird die Anwendungsm{\"o}glichkeit der Bienaym{\´e}-Galton-Watson Prozesse bei komplexeren Problemen deutlich. Histogramme bekr{\"a}ftigen, dass die einzelnen Populationsgr{\"o}ßen nur endlich oft vorkommen. Diese Aussage wurde von Galton aufgeworfen und in der Extinktions-Explosions-Dichotomie verwendet. Die dargestellten Erkenntnisse {\"u}ber das Themengebiet und die Betrachtung des Konzeptes werden mit einer didaktischen Analyse abgeschlossen. Die Untersuchung beinhaltet die Ber{\"u}cksichtigung der Fundamentalen Ideen, der Fundamentalen Ideen der Stochastik und der Leitidee „Daten und Zufall". Dabei ergibt sich, dass in Abh{\"a}ngigkeit der gew{\"a}hlten Perspektive die Anwendung der Bienaym{\´e}-Galton-Watson Prozesse innerhalb der Schule plausibel ist und von Vorteil f{\"u}r die Sch{\"u}ler:innen sein kann. F{\"u}r die Behandlung wird exemplarisch der Rahmenlehrplan f{\"u}r Berlin und Brandenburg analysiert und mit dem Kernlehrplan Nordrhein-Westfalens verglichen. Die Konzeption des Lehrplans aus Berlin und Brandenburg l{\"a}sst nicht den Schluss zu, dass die Bienaym{\´e}-Galton-Watson Prozesse angewendet werden sollten. Es l{\"a}sst sich feststellen, dass die zugrunde liegende Leitidee nicht vollumf{\"a}nglich mit manchen Fundamentalen Ideen der Stochastik vereinbar ist. Somit w{\"u}rde eine Modifikation hinsichtlich einer st{\"a}rkeren Orientierung des Lehrplans an den Fundamentalen Ideen die Anwendung der Prozesse erm{\"o}glichen. Die Aussage wird durch die Betrachtung und {\"U}bertragung eines nordrhein-westf{\"a}lischen Unterrichtsentwurfes f{\"u}r stochastische Prozesse auf die Bienaym{\´e}-Galton-Watson Prozesse unterst{\"u}tzt. Dar{\"u}ber hinaus werden eine Concept Map und ein Vernetzungspentagraph nach von der Bank konzipiert um diesen Aspekt hervorzuheben.}, language = {de} } @article{RoosOtoba2021, author = {Roos, Saskia and Otoba, Nobuhiko}, title = {Scalar curvature and the multiconformal class of a direct product Riemannian manifold}, series = {Geometriae dedicata}, volume = {214}, journal = {Geometriae dedicata}, number = {1}, publisher = {Springer}, address = {Dordrecht}, issn = {0046-5755}, doi = {10.1007/s10711-021-00636-9}, pages = {801 -- 829}, year = {2021}, abstract = {For a closed, connected direct product Riemannian manifold (M, g) = (M-1, g(1)) x ... x (M-l, g(l)), we define its multiconformal class [[g]] as the totality {integral(2)(1)g(1) circle plus center dot center dot center dot integral(2)(l)g(l)} of all Riemannian metrics obtained from multiplying the metric gi of each factor Mi by a positive function fi on the total space M. A multiconformal class [[ g]] contains not only all warped product type deformations of g but also the whole conformal class [(g) over tilde] of every (g) over tilde is an element of[[ g]]. In this article, we prove that [[g]] contains a metric of positive scalar curvature if and only if the conformal class of some factor (Mi, gi) does, under the technical assumption dim M-i = 2. We also show that, even in the case where every factor (M-i, g(i)) has positive scalar curvature, [[g]] contains a metric of scalar curvature constantly equal to -1 and with arbitrarily large volume, provided l = 2 and dim M = 3.}, language = {en} } @article{GottwaldReich2021, author = {Gottwald, Georg A. and Reich, Sebastian}, title = {Supervised learning from noisy observations}, series = {Physica : D, Nonlinear phenomena}, volume = {423}, journal = {Physica : D, Nonlinear phenomena}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2021.132911}, pages = {15}, year = {2021}, abstract = {Data-driven prediction and physics-agnostic machine-learning methods have attracted increased interest in recent years achieving forecast horizons going well beyond those to be expected for chaotic dynamical systems. In a separate strand of research data-assimilation has been successfully used to optimally combine forecast models and their inherent uncertainty with incoming noisy observations. The key idea in our work here is to achieve increased forecast capabilities by judiciously combining machine-learning algorithms and data assimilation. We combine the physics-agnostic data -driven approach of random feature maps as a forecast model within an ensemble Kalman filter data assimilation procedure. The machine-learning model is learned sequentially by incorporating incoming noisy observations. We show that the obtained forecast model has remarkably good forecast skill while being computationally cheap once trained. Going beyond the task of forecasting, we show that our method can be used to generate reliable ensembles for probabilistic forecasting as well as to learn effective model closure in multi-scale systems. (C) 2021 Elsevier B.V. All rights reserved.}, language = {en} } @article{RedmannFreitag2021, author = {Redmann, Martin and Freitag, Melina A.}, title = {Optimization based model order reduction for stochastic systems}, series = {Applied mathematics and computation}, volume = {398}, journal = {Applied mathematics and computation}, publisher = {Elsevier}, address = {New York}, issn = {0096-3003}, doi = {10.1016/j.amc.2020.125783}, pages = {18}, year = {2021}, abstract = {In this paper, we bring together the worlds of model order reduction for stochastic linear systems and H-2-optimal model order reduction for deterministic systems. In particular, we supplement and complete the theory of error bounds for model order reduction of stochastic differential equations. With these error bounds, we establish a link between the output error for stochastic systems (with additive and multiplicative noise) and modified versions of the H-2-norm for both linear and bilinear deterministic systems. When deriving the respective optimality conditions for minimizing the error bounds, we see that model order reduction techniques related to iterative rational Krylov algorithms (IRKA) are very natural and effective methods for reducing the dimension of large-scale stochastic systems with additive and/or multiplicative noise. We apply modified versions of (linear and bilinear) IRKA to stochastic linear systems and show their efficiency in numerical experiments.}, language = {en} } @article{RuchiDubinkinaWiljes2021, author = {Ruchi, Sangeetika and Dubinkina, Svetlana and Wiljes, Jana de}, title = {Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation}, series = {Nonlinear processes in geophysics / European Geosciences Union ; American Geophysical Union}, volume = {28}, journal = {Nonlinear processes in geophysics / European Geosciences Union ; American Geophysical Union}, number = {1}, publisher = {Copernicus}, address = {G{\"o}ttingen}, issn = {1023-5809}, doi = {10.5194/npg-28-23-2021}, pages = {23 -- 41}, year = {2021}, abstract = {Identification of unknown parameters on the basis of partial and noisy data is a challenging task, in particular in high dimensional and non-linear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust and computationally cheap and often produce astonishingly accurate estimations despite the simplifying underlying assumptions. Yet there is a lot of room for improvement, specifically regarding a correct approximation of a non-Gaussian posterior distribution. The tempered ensemble transform particle filter is an adaptive Sequential Monte Carlo (SMC) method, whereby resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion, it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity, and the method is not as robust as ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy-inspired regularisation factor to the underlying optimal transport problem that allows the high computational cost to be considerably reduced via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as a hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov chain Monte Carlo methods results are computed as a benchmark.}, language = {en} } @article{ShlapunovTarchanov2021, author = {Shlapunov, Alexander and Tarchanov, Nikolaj Nikolaevič}, title = {An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n}, series = {Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija}, volume = {18}, journal = {Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija}, number = {2}, publisher = {Institut Matematiki Imeni S. L. Soboleva}, address = {Novosibirsk}, issn = {1813-3304}, doi = {10.33048/semi.2021.18.108}, pages = {1433 -- 1466}, year = {2021}, abstract = {We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.}, language = {en} } @article{BeckusEliaz2021, author = {Beckus, Siegfried and Eliaz, Latif}, title = {Eigenfunctions growth of R-limits on graphs}, series = {Journal of spectral theory / European Mathematical Society}, volume = {11}, journal = {Journal of spectral theory / European Mathematical Society}, number = {4}, publisher = {EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut f{\"u}r Mathematik, Technische Universit{\"a}t}, address = {Berlin}, issn = {1664-039X}, doi = {10.4171/JST/389}, pages = {1895 -- 1933}, year = {2021}, abstract = {A characterization of the essential spectrum of Schrodinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on N and Z(d) as "right-limits," captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in sigma(ss)(H) corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.}, language = {en} } @article{SchannerMauerbergerKorteetal.2021, author = {Schanner, Maximilian Arthus and Mauerberger, Stefan and Korte, Monika and Holschneider, Matthias}, title = {Correlation based time evolution of the archeomagnetic field}, series = {Journal of geophysical research : JGR ; an international quarterly. B, Solid earth}, volume = {126}, journal = {Journal of geophysical research : JGR ; an international quarterly. B, Solid earth}, number = {7}, publisher = {American Geophysical Union}, address = {Washington}, issn = {2169-9313}, doi = {10.1029/2020JB021548}, pages = {22}, year = {2021}, abstract = {In a previous study, a new snapshot modeling concept for the archeomagnetic field was introduced (Mauerberger et al., 2020, ). By assuming a Gaussian process for the geomagnetic potential, a correlation-based algorithm was presented, which incorporates a closed-form spatial correlation function. This work extends the suggested modeling strategy to the temporal domain. A space-time correlation kernel is constructed from the tensor product of the closed-form spatial correlation kernel with a squared exponential kernel in time. Dating uncertainties are incorporated into the modeling concept using a noisy input Gaussian process. All but one modeling hyperparameters are marginalized, to reduce their influence on the outcome and to translate their variability to the posterior variance. The resulting distribution incorporates uncertainties related to dating, measurement and modeling process. Results from application to archeomagnetic data show less variation in the dipole than comparable models, but are in general agreement with previous findings.}, language = {en} } @article{ReichWeissmann2021, author = {Reich, Sebastian and Weissmann, Simon}, title = {Fokker-Planck particle systems for Bayesian inference: computational approaches}, series = {SIAM ASA journal on uncertainty quantification}, volume = {9}, journal = {SIAM ASA journal on uncertainty quantification}, number = {2}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {2166-2525}, doi = {10.1137/19M1303162}, pages = {446 -- 482}, year = {2021}, abstract = {Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies.}, language = {en} } @article{CozzoniMaibaumHamm2021, author = {Cozzoni, Barbara and Maibaum, Michael and Hamm, Maximilian}, title = {Thermal analysis and constraints for the MASCOT landing site selection on the asteroid Ryugu}, series = {Planetary and space science}, volume = {205}, journal = {Planetary and space science}, publisher = {Elsevier}, address = {Oxford}, issn = {0032-0633}, doi = {10.1016/j.pss.2021.105286}, pages = {11}, year = {2021}, abstract = {In June 2018, after 4 years of cruise, the Japanese space probe Hayabusa2 [1-Watanabe S. et al.: Hayabusa2 Mission Overview. (2017)] reached the Near-Earth Asteroid (162173) Ryugu. Hayabusa2 carried a small Lander named MASCOT (Mobile Asteroid Surface Scout) [2-Ho T. M. et al.: MASCOT-The Mobile Asteroid Surface Scout onboard the Hayabusa2 mission. (2017)], jointly developed by the German Aerospace Center (DLR) and the French Space Agency (CNES), to investigate Ryugu's surface structure, composition and physical properties including its thermal behaviour and magnetization in-situ. The Microgravity User Support Centre (DLR-MUSC) in Cologne was in charge of providing all thermal conditions and constraints necessary for the selection of the final landing site and for the final operations of the Lander MASCOT on the surface of the asteroid Ryugu. This article provides a comprehensive assessment of these thermal conditions and constraints, based on predictions performed with the Thermal Mathematical Model (TMM) of MASCOT using different asteroid surface thermal models, ephemeris data for approach as well as descent and hopping trajectories, the related operation sequences and scenarios and the possible environmental conditions driven by the Hayabusa2 spacecraft. A comparison with the real telemetry data confirms the analysis and provides further information about the asteroid characteristics.}, language = {en} } @article{HastermannReinhardtKleinetal.2021, author = {Hastermann, Gottfried and Reinhardt, Maria and Klein, Rupert and Reich, Sebastian}, title = {Balanced data assimilation for highly oscillatory mechanical systems}, series = {Communications in applied mathematics and computational science : CAMCoS}, volume = {16}, journal = {Communications in applied mathematics and computational science : CAMCoS}, number = {1}, publisher = {Mathematical Sciences Publishers}, address = {Berkeley}, issn = {1559-3940}, doi = {10.2140/camcos.2021.16.119}, pages = {119 -- 154}, year = {2021}, abstract = {Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, this filter also has limitations due to its inherent assumptions of Gaussianity and linearity, which can manifest themselves in the form of dynamically inconsistent state estimates. This issue is investigated here for balanced, slowly evolving solutions to highly oscillatory Hamiltonian systems which are prototypical for applications in numerical weather prediction. It is demonstrated that the standard ensemble Kalman filter can lead to state estimates that do not satisfy the pertinent balance relations and ultimately lead to filter divergence. Two remedies are proposed, one in terms of blended asymptotically consistent time-stepping schemes, and one in terms of minimization-based postprocessing methods. The effects of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for balanced motions of two prototypical Hamiltonian reference systems.}, language = {en} } @article{DeneckeHounnon2021, author = {Denecke, Klaus-Dieter and Hounnon, Hippolyte}, title = {Partial Menger algebras of terms}, series = {Asian-European journal of mathematics}, volume = {14}, journal = {Asian-European journal of mathematics}, number = {06}, publisher = {World Scientific}, address = {Singapore}, issn = {1793-5571}, doi = {10.1142/S1793557121500923}, pages = {14}, year = {2021}, abstract = {The superposition operation S-n,S-A, n >= 1, n is an element of N, maps to each (n + 1)-tuple of n-ary operations on a set A an n-ary operation on A and satisfies the so-called superassociative law, a generalization of the associative law. The corresponding algebraic structures are Menger algebras of rank n. A partial algebra of type (n + 1) which satisfies the superassociative law as weak identity is said to be a partial Menger algebra of rank n. As a generalization of linear terms we define r-terms as terms where each variable occurs at most r-times. It will be proved that n-ary r-terms form partial Menger algebras of rank n. In this paper, some algebraic properties of partial Menger algebras such as generating systems, homomorphic images and freeness are investigated. As generalization of hypersubstitutions and linear hypersubstitutions we consider r-hypersubstitutions.U}, language = {en} } @article{DeOliveiraGomesHoegele2021, author = {De Oliveira Gomes, Andr{\´e} and H{\"o}gele, Michael Anton}, title = {The Kramers problem for SDEs driven by small, accelerated L{\´e}vy noise with exponentially light jumps}, series = {Stochastics and dynamics}, volume = {21}, journal = {Stochastics and dynamics}, number = {04}, publisher = {World Scientific}, address = {Singapore}, issn = {0219-4937}, doi = {10.1142/S0219493721500192}, pages = {44}, year = {2021}, abstract = {We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state 0, say, subject to a perturbation by a stochastic integral which is driven by an epsilon-small and (1/epsilon)-accelerated Levy process with exponentially light jumps. For this purpose, we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel, we solve the associated asymptotic first escape problem from the bounded neighborhood of 0 in the limit as epsilon -> 0 which is also known as the Kramers problem in the literature.}, language = {en} } @article{GarmendiaZambon2021, author = {Garmendia, Alfonso and Zambon, Marco}, title = {Quotients of singular foliations and Lie 2-group actions}, series = {Journal of noncommutative geometry}, volume = {15}, journal = {Journal of noncommutative geometry}, number = {4}, publisher = {EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut f{\"u}r Mathematik, Technische Universit{\"a}t Berlin}, address = {Berlin}, issn = {1661-6952}, doi = {10.4171/JNCG/434}, pages = {1251 -- 1283}, year = {2021}, abstract = {Androulidakis-Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold (M, FM ) is the quotient of a foliated manifold (P, FP ) along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid.}, language = {en} } @article{SchickSeyedhosseini2021, author = {Schick, Thomas and Seyedhosseini, Mehran}, title = {On an index theorem of Chang, Weinberger and Yu}, series = {M{\"u}nster journal of mathematics}, volume = {14}, journal = {M{\"u}nster journal of mathematics}, number = {1}, publisher = {WWU, Fachbereich Mathematik und Informatik}, address = {M{\"u}nster}, issn = {1867-5778}, doi = {10.17879/59019522628}, pages = {123 -- 154}, year = {2021}, abstract = {In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that (a slight variation of) the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger and Yu (rather: a slight variation). To take the fundamental groups of the manifold and its boundary into account requires working with maximal C*-completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions. On the other hand, we introduce and propose a more conceptual and more geometric completion, which still has all the required functoriality.}, language = {en} } @article{MatzkaStolleYamazakietal.2021, author = {Matzka, J{\"u}rgen and Stolle, Claudia and Yamazaki, Yosuke and Bronkalla, Oliver and Morschhauser, Achim}, title = {The geomagnetic Kp index and derived indices of geomagnetic activity}, series = {Space weather : the international journal of research and applications}, volume = {19}, journal = {Space weather : the international journal of research and applications}, number = {5}, publisher = {Wiley}, address = {New York}, issn = {1542-7390}, doi = {10.1029/2020SW002641}, pages = {21}, year = {2021}, abstract = {The geomagnetic Kp index is one of the most extensively used indices of geomagnetic activity, both for scientific and operational purposes. This article reviews the properties of the Kp index and provides a reference for users of the Kp index and associated data products as derived and distributed by the GFZ German Research Centre for Geosciences. The near real-time production of the nowcast Kp index is of particular interest for space weather services and here we describe and evaluate its current setup.}, language = {en} } @article{KemptonMuenchYau2021, author = {Kempton, Mark and M{\"u}nch, Florentin and Yau, Shing-Tung}, title = {A homology vanishing theorem for graphs with positive curvature}, series = {Communications in analysis and geometry}, volume = {29}, journal = {Communications in analysis and geometry}, number = {6}, publisher = {International Press of Boston}, address = {Somerville}, issn = {1019-8385}, doi = {10.4310/CAG.2021.v29.n6.a5}, pages = {1449 -- 1473}, year = {2021}, abstract = {We prove a homology vanishing theorem for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Bochner on manifolds [3]. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau [11]. We moreover prove that the fundamental group is finite for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Myers on manifolds [22]. The proofs draw on several separate areas of graph theory, including graph coverings, gain graphs, and cycle spaces, in addition to the Bakry-Emery curvature, path homology, and graph homotopy. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature cannot have a non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau [12], and obtain an alternative proof of their result that the abelianization of the fundamental group of a graph is isomorphic to the first path homology over the integers.}, language = {en} } @article{PathirajaReichStannat2021, author = {Pathiraja, Sahani Darschika and Reich, Sebastian and Stannat, Wilhelm}, title = {McKean-Vlasov SDEs in nonlinear filtering}, series = {SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics}, volume = {59}, journal = {SIAM journal on control and optimization : a publication of the Society for Industrial and Applied Mathematics}, number = {6}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {0363-0129}, doi = {10.1137/20M1355197}, pages = {4188 -- 4215}, year = {2021}, abstract = {Various particle filters have been proposed over the last couple of decades with the common feature that the update step is governed by a type of control law. This feature makes them an attractive alternative to traditional sequential Monte Carlo which scales poorly with the state dimension due to weight degeneracy. This article proposes a unifying framework that allows us to systematically derive the McKean-Vlasov representations of these filters for the discrete time and continuous time observation case, taking inspiration from the smooth approximation of the data considered in [D. Crisan and J. Xiong, Stochastics, 82 (2010), pp. 53-68; J. M. Clark and D. Crisan, Probab. Theory Related Fields, 133 (2005), pp. 43-56]. We consider three filters that have been proposed in the literature and use this framework to derive Ito representations of their limiting forms as the approximation parameter delta -> 0. All filters require the solution of a Poisson equation defined on R-d, for which existence and uniqueness of solutions can be a nontrivial issue. We additionally establish conditions on the signal-observation system that ensures well-posedness of the weighted Poisson equation arising in one of the filters.}, language = {en} } @article{LeungLeutbecherReichetal.2021, author = {Leung, Tsz Yan and Leutbecher, Martin and Reich, Sebastian and Shepherd, Theodore G.}, title = {Forecast verification}, series = {Quarterly journal of the Royal Meteorological Society}, volume = {147}, journal = {Quarterly journal of the Royal Meteorological Society}, number = {739}, publisher = {Wiley}, address = {Hoboken}, issn = {0035-9009}, doi = {10.1002/qj.4120}, pages = {3124 -- 3134}, year = {2021}, abstract = {The philosophy of forecast verification is rather different between deterministic and probabilistic verification metrics: generally speaking, deterministic metrics measure differences, whereas probabilistic metrics assess reliability and sharpness of predictive distributions. This article considers the root-mean-square error (RMSE), which can be seen as a deterministic metric, and the probabilistic metric Continuous Ranked Probability Score (CRPS), and demonstrates that under certain conditions, the CRPS can be mathematically expressed in terms of the RMSE when these metrics are aggregated. One of the required conditions is the normality of distributions. The other condition is that, while the forecast ensemble need not be calibrated, any bias or over/underdispersion cannot depend on the forecast distribution itself. Under these conditions, the CRPS is a fraction of the RMSE, and this fraction depends only on the heteroscedasticity of the ensemble spread and the measures of calibration. The derived CRPS-RMSE relationship for the case of perfect ensemble reliability is tested on simulations of idealised two-dimensional barotropic turbulence. Results suggest that the relationship holds approximately despite the normality condition not being met.}, language = {en} } @article{AyanbayevKlebanovLietal.2021, author = {Ayanbayev, Birzhan and Klebanov, Ilja and Li, Han Cheng and Sullivan, Tim J.}, title = {Gamma-convergence of Onsager-Machlup functionals}, series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data}, volume = {38}, journal = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data}, number = {2}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {0266-5611}, doi = {10.1088/1361-6420/ac3f81}, pages = {32}, year = {2021}, abstract = {The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Gamma-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.}, language = {en} } @article{AyanbayevKlebanovLieetal.2021, author = {Ayanbayev, Birzhan and Klebanov, Ilja and Lie, Han Cheng and Sullivan, Tim J.}, title = {Gamma-convergence of Onsager-Machlup functionals}, series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data}, volume = {38}, journal = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data}, number = {2}, publisher = {IOP Publ. Ltd.}, address = {Bristol}, issn = {0266-5611}, doi = {10.1088/1361-6420/ac3f82}, pages = {35}, year = {2021}, abstract = {We derive Onsager-Machlup functionals for countable product measures on weighted l(p) subspaces of the sequence space R-N. Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Gamma-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 <= p <= 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.}, language = {en} } @article{EshghiMachReichel2021, author = {Eshghi, Nasim and Mach, Thomas and Reichel, Lothar}, title = {New matrix function approximations and quadrature rules based on the Arnoldi process}, series = {Journal of computational and applied mathematics}, volume = {391}, journal = {Journal of computational and applied mathematics}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0377-0427}, doi = {10.1016/j.cam.2021.113442}, pages = {12}, year = {2021}, abstract = {The Arnoldi process can be applied to inexpensively approximate matrix functions of the form f (A)v and matrix functionals of the form v*(f (A))*g(A)v, where A is a large square non-Hermitian matrix, v is a vector, and the superscript * denotes transposition and complex conjugation. Here f and g are analytic functions that are defined in suitable regions in the complex plane. This paper reviews available approximation methods and describes new ones that provide higher accuracy for essentially the same computational effort by exploiting available, but generally not used, moment information. Numerical experiments show that in some cases the modifications of the Arnoldi decompositions proposed can improve the accuracy of v*(f (A))*g(A)v about as much as performing an additional step of the Arnoldi process.}, language = {en} } @misc{Huebner2021, type = {Master Thesis}, author = {H{\"u}bner, Andrea}, title = {Ein multityper Verzweigungsprozess als Modell zur Untersuchung der Ausbreitung von Covid-19}, doi = {10.25932/publishup-50922}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-509225}, school = {Universit{\"a}t Potsdam}, year = {2021}, abstract = {Im Zuge der Covid-19 Pandemie werden zwei Werte t{\"a}glich diskutiert: Die zuletzt gemeldete Zahl der neu Infizierten und die sogenannte Reproduktionsrate. Sie gibt wieder, wie viele weitere Menschen ein an Corona erkranktes Individuum im Durchschnitt ansteckt. F{\"u}r die Sch{\"a}tzung dieses Wertes gibt es viele M{\"o}glichkeiten - auch das Robert Koch-Institut gibt in seinem t{\"a}glichen Situationsbericht stets zwei R-Werte an: Einen 4-Tage-R-Wert und einen weniger schwankenden 7-Tage-R-Wert. Diese Arbeit soll eine weitere M{\"o}glichkeit vorstellen, einige Aspekte der Pandemie zu modellieren und die Reproduktionsrate zu sch{\"a}tzen. In der ersten H{\"a}lfte der Arbeit werden die mathematischen Grundlagen vorgestellt, die man f{\"u}r die Modellierung ben{\"o}tigt. Hierbei wird davon ausgegangen, dass der Leser bereits ein Basisverst{\"a}ndnis von stochastischen Prozessen hat. Im Abschnitt Grundlagen werden Verzweigungsprozesse mit einigen Beispielen eingef{\"u}hrt und die Ergebnisse aus diesem Themengebiet, die f{\"u}r diese Arbeit wichtig sind, pr{\"a}sentiert. Dabei gehen wir zuerst auf einfache Verzweigungsprozesse ein und erweitern diese dann auf Verzweigungsprozesse mit mehreren Typen. Um die Notation zu erleichtern, beschr{\"a}nken wir uns auf zwei Typen. Das Prinzip l{\"a}sst sich aber auf eine beliebige Anzahl von Typen erweitern. Vor allem soll die Wichtigkeit des Parameters λ herausgestellt werden. Dieser Wert kann als durchschnittliche Zahl von Nachfahren eines Individuums interpretiert werden und bestimmt die Dynamik des Prozesses {\"u}ber einen l{\"a}ngeren Zeitraum. In der Anwendung auf die Pandemie hat der Parameter λ die gleiche Rolle wie die Reproduktionsrate R. In der zweiten H{\"a}lfte dieser Arbeit stellen wir eine Anwendung der Theorie {\"u}ber Multitype Verzweigungsprozesse vor. Professor Yanev und seine Mitarbeiter modellieren in ihrer Ver{\"o}ffentlichung Branching stochastic processes as models of Covid-19 epidemic development die Ausbreitung des Corona Virus' {\"u}ber einen Verzweigungsprozess mit zwei Typen. Wir werden dieses Modell diskutieren und Sch{\"a}tzer daraus ableiten: Ziel ist es, die Reproduktionsrate zu ermitteln. Außerdem analysieren wir die M{\"o}glichkeiten, die Dunkelziffer (die Zahl nicht gemeldeter Krankheitsf{\"a}lle) zu sch{\"a}tzen. Wir wenden die Sch{\"a}tzer auf die Zahlen von Deutschland an und werten diese schließlich aus.}, language = {de} } @article{CvetkovićConradLie2021, author = {Cvetković, Nada and Conrad, Tim and Lie, Han Cheng}, title = {A convergent discretization method for transition path theory for diffusion processes}, series = {Multiscale modeling \& simulation : a SIAM interdisciplinary journal}, volume = {19}, journal = {Multiscale modeling \& simulation : a SIAM interdisciplinary journal}, number = {1}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1540-3459}, doi = {10.1137/20M1329354}, pages = {242 -- 266}, year = {2021}, abstract = {Transition path theory (TPT) for diffusion processes is a framework for analyzing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretization of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretization. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.}, language = {en} } @article{ChangKhalilSchulze2021, author = {Chang, Der-Chen and Khalil, Sara and Schulze, Bert-Wolfgang}, title = {Analysis on regular corner spaces}, series = {The journal of geometric analysis}, volume = {31}, journal = {The journal of geometric analysis}, number = {9}, publisher = {Springer}, address = {New York}, issn = {1050-6926}, doi = {10.1007/s12220-021-00614-3}, pages = {9199 -- 9240}, year = {2021}, abstract = {We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.}, language = {en} } @article{WormellReich2021, author = {Wormell, Caroline L. and Reich, Sebastian}, title = {Spectral convergence of diffusion maps}, series = {SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics}, volume = {59}, journal = {SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics}, number = {3}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {0036-1429}, doi = {10.1137/20M1344093}, pages = {1687 -- 1734}, year = {2021}, abstract = {Diffusion maps is a manifold learning algorithm widely used for dimensionality reduction. Using a sample from a distribution, it approximates the eigenvalues and eigenfunctions of associated Laplace-Beltrami operators. Theoretical bounds on the approximation error are, however, generally much weaker than the rates that are seen in practice. This paper uses new approaches to improve the error bounds in the model case where the distribution is supported on a hypertorus. For the data sampling (variance) component of the error we make spatially localized compact embedding estimates on certain Hardy spaces; we study the deterministic (bias) component as a perturbation of the Laplace-Beltrami operator's associated PDE and apply relevant spectral stability results. Using these approaches, we match long-standing pointwise error bounds for both the spectral data and the norm convergence of the operator discretization. We also introduce an alternative normalization for diffusion maps based on Sinkhorn weights. This normalization approximates a Langevin diffusion on the sample and yields a symmetric operator approximation. We prove that it has better convergence compared with the standard normalization on flat domains, and we present a highly efficient rigorous algorithm to compute the Sinkhorn weights.}, language = {en} } @article{KellerLiuPeyerimhoff2021, author = {Keller, Matthias and Liu, Shiping and Peyerimhoff, Norbert}, title = {A note on eigenvalue bounds for non-compact manifolds}, series = {Mathematische Nachrichten}, volume = {294}, journal = {Mathematische Nachrichten}, number = {6}, publisher = {Wiley-VCH}, address = {Weinheim}, issn = {0025-584X}, doi = {10.1002/mana.201900209}, pages = {1134 -- 1139}, year = {2021}, abstract = {In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.}, language = {en} } @article{SaynischWagnerBaerenzungHornschildetal.2021, author = {Saynisch-Wagner, Jan and B{\"a}renzung, Julien and Hornschild, Aaron and Irrgang, Christopher and Thomas, Maik}, title = {Tide-induced magnetic signals and their errors derived from CHAMP and Swarm satellite magnetometer observations}, series = {Earth, planets and space : EPS}, volume = {73}, journal = {Earth, planets and space : EPS}, number = {1}, publisher = {Springer}, address = {Heidelberg}, issn = {1880-5981}, doi = {10.1186/s40623-021-01557-3}, pages = {11}, year = {2021}, abstract = {Satellite-measured tidal magnetic signals are of growing importance. These fields are mainly used to infer Earth's mantle conductivity, but also to derive changes in the oceanic heat content. We present a new Kalman filter-based method to derive tidal magnetic fields from satellite magnetometers: KALMAG. The method's advantage is that it allows to study a precisely estimated posterior error covariance matrix. We present the results of a simultaneous estimation of the magnetic signals of 8 major tides from 17 years of Swarm and CHAMP data. For the first time, robustly derived posterior error distributions are reported along with the reported tidal magnetic fields. The results are compared to other estimates that are either based on numerical forward models or on satellite inversions of the same data. For all comparisons, maximal differences and the corresponding globally averaged RMSE are reported. We found that the inter-product differences are comparable with the KALMAG-based errors only in a global mean sense. Here, all approaches give values of the same order, e.g., 0.09 nT-0.14 nT for M2. Locally, the KALMAG posterior errors are up to one order smaller than the inter-product differences, e.g., 0.12 nT vs. 0.96 nT for M2.}, language = {en} } @article{RodriguezZuluagaStolleYamazakietal.2021, author = {Rodr{\´i}guez Zuluaga, Juan and Stolle, Claudia and Yamazaki, Yosuke and Xiong, Chao and England, Scott L.}, title = {A synoptic-scale wavelike structure in the nighttime equatorial ionization anomaly}, series = {Earth and Space Science : ESS}, volume = {8}, journal = {Earth and Space Science : ESS}, number = {2}, publisher = {American Geophysical Union}, address = {Malden, Mass.}, issn = {2333-5084}, doi = {10.1029/2020EA001529}, pages = {10}, year = {2021}, abstract = {Both ground- and satellite-based airglow imaging have significantly contributed to understanding the low-latitude ionosphere, especially the morphology and dynamics of the equatorial ionization anomaly (EIA). The NASA Global-scale Observations of the Limb and Disk (GOLD) mission focuses on far-ultraviolet airglow images from a geostationary orbit at 47.5 degrees W. This region is of particular interest at low magnetic latitudes because of the high magnetic declination (i.e., about -20 degrees) and proximity of the South Atlantic magnetic anomaly. In this study, we characterize an exciting feature of the nighttime EIA using GOLD observations from October 5, 2018 to June 30, 2020. It consists of a wavelike structure of a few thousand kilometers seen as poleward and equatorward displacements of the EIA-crests. Initial analyses show that the synoptic-scale structure is symmetric about the dip equator and appears nearly stationary with time over the night. In quasi-dipole coordinates, maxima poleward displacements of the EIA-crests are seen at about +/- 12 degrees latitude and around 20 and 60 degrees longitude (i.e., in geographic longitude at the dip equator, about 53 degrees W and 14 degrees W). The wavelike structure presents typical zonal wavelengths of about 6.7 x 10(3) km and 3.3 x 10(3) km. The structure's occurrence and wavelength are highly variable on a day-to-day basis with no apparent dependence on geomagnetic activity. In addition, a cluster or quasi-periodic wave train of equatorial plasma depletions (EPDs) is often detected within the synoptic-scale structure. We further outline the difference in observing these EPDs from FUV images and in situ measurements during a GOLD and Swarm mission conjunction.}, language = {en} } @article{SchindlerMoldenhawerStangeetal.2021, author = {Schindler, Daniel and Moldenhawer, Ted and Stange, Maike and Lepro, Valentino and Beta, Carsten and Holschneider, Matthias and Huisinga, Wilhelm}, title = {Analysis of protrusion dynamics in amoeboid cell motility by means of regularized contour flows}, series = {PLoS Computational Biology : a new community journal}, volume = {17}, journal = {PLoS Computational Biology : a new community journal}, number = {8}, publisher = {PLoS}, address = {San Fransisco}, issn = {1553-734X}, doi = {10.1371/journal.pcbi.1009268}, pages = {33}, year = {2021}, abstract = {Amoeboid cell motility is essential for a wide range of biological processes including wound healing, embryonic morphogenesis, and cancer metastasis. It relies on complex dynamical patterns of cell shape changes that pose long-standing challenges to mathematical modeling and raise a need for automated and reproducible approaches to extract quantitative morphological features from image sequences. Here, we introduce a theoretical framework and a computational method for obtaining smooth representations of the spatiotemporal contour dynamics from stacks of segmented microscopy images. Based on a Gaussian process regression we propose a one-parameter family of regularized contour flows that allows us to continuously track reference points (virtual markers) between successive cell contours. We use this approach to define a coordinate system on the moving cell boundary and to represent different local geometric quantities in this frame of reference. In particular, we introduce the local marker dispersion as a measure to identify localized membrane expansions and provide a fully automated way to extract the properties of such expansions, including their area and growth time. The methods are available as an open-source software package called AmoePy, a Python-based toolbox for analyzing amoeboid cell motility (based on time-lapse microscopy data), including a graphical user interface and detailed documentation. Due to the mathematical rigor of our framework, we envision it to be of use for the development of novel cell motility models. We mainly use experimental data of the social amoeba Dictyostelium discoideum to illustrate and validate our approach.
Author summary Amoeboid motion is a crawling-like cell migration that plays an important key role in multiple biological processes such as wound healing and cancer metastasis. This type of cell motility results from expanding and simultaneously contracting parts of the cell membrane. From fluorescence images, we obtain a sequence of points, representing the cell membrane, for each time step. By using regression analysis on these sequences, we derive smooth representations, so-called contours, of the membrane. Since the number of measurements is discrete and often limited, the question is raised of how to link consecutive contours with each other. In this work, we present a novel mathematical framework in which these links are described by regularized flows allowing a certain degree of concentration or stretching of neighboring reference points on the same contour. This stretching rate, the so-called local dispersion, is used to identify expansions and contractions of the cell membrane providing a fully automated way of extracting properties of these cell shape changes. We applied our methods to time-lapse microscopy data of the social amoeba Dictyostelium discoideum.}, language = {en} } @article{HartungWahlRastogietal.2021, author = {Hartung, Niklas and Wahl, Martin and Rastogi, Abhishake and Huisinga, Wilhelm}, title = {Nonparametric goodness-of-fit testing for parametric covariate models in pharmacometric analyses}, series = {CPT: pharmacometrics \& systems pharmacology}, volume = {10}, journal = {CPT: pharmacometrics \& systems pharmacology}, number = {6}, publisher = {Nature Publ. Group}, address = {London}, issn = {2163-8306}, doi = {10.1002/psp4.12614}, pages = {564 -- 576}, year = {2021}, abstract = {The characterization of covariate effects on model parameters is a crucial step during pharmacokinetic/pharmacodynamic analyses. Although covariate selection criteria have been studied extensively, the choice of the functional relationship between covariates and parameters, however, has received much less attention. Often, a simple particular class of covariate-to-parameter relationships (linear, exponential, etc.) is chosen ad hoc or based on domain knowledge, and a statistical evaluation is limited to the comparison of a small number of such classes. Goodness-of-fit testing against a nonparametric alternative provides a more rigorous approach to covariate model evaluation, but no such test has been proposed so far. In this manuscript, we derive and evaluate nonparametric goodness-of-fit tests for parametric covariate models, the null hypothesis, against a kernelized Tikhonov regularized alternative, transferring concepts from statistical learning to the pharmacological setting. The approach is evaluated in a simulation study on the estimation of the age-dependent maturation effect on the clearance of a monoclonal antibody. Scenarios of varying data sparsity and residual error are considered. The goodness-of-fit test correctly identified misspecified parametric models with high power for relevant scenarios. The case study provides proof-of-concept of the feasibility of the proposed approach, which is envisioned to be beneficial for applications that lack well-founded covariate models.}, language = {en} } @article{HetheyHartungWangorschetal.2021, author = {Hethey, Christoph Philipp and Hartung, Niklas and Wangorsch, Gaby and Weisser, Karin and Huisinga, Wilhelm}, title = {Physiology-based toxicokinetic modelling of aluminium in rat and man}, series = {Archives of toxicology : official journal of EUROTOX}, volume = {95}, journal = {Archives of toxicology : official journal of EUROTOX}, number = {9}, publisher = {Springer}, address = {Berlin ; Heidelberg}, issn = {0340-5761}, doi = {10.1007/s00204-021-03107-y}, pages = {2977 -- 3000}, year = {2021}, abstract = {A sufficient quantitative understanding of aluminium (Al) toxicokinetics (TK) in man is still lacking, although highly desirable for risk assessment of Al exposure. Baseline exposure and the risk of contamination severely limit the feasibility of TK studies administering the naturally occurring isotope Al-27, both in animals and man. These limitations are absent in studies with Al-26 as a tracer, but tissue data are limited to animal studies. A TK model capable of inter-species translation to make valid predictions of Al levels in humans-especially in toxicological relevant tissues like bone and brain-is urgently needed. Here, we present: (i) a curated dataset which comprises all eligible studies with single doses of Al-26 tracer administered as citrate or chloride salts orally and/or intravenously to rats and humans, including ultra-long-term kinetic profiles for plasma, blood, liver, spleen, muscle, bone, brain, kidney, and urine up to 150 weeks; and (ii) the development of a physiology-based (PB) model for Al TK after intravenous and oral administration of aqueous Al citrate and Al chloride solutions in rats and humans. Based on the comprehensive curated Al-26 dataset, we estimated substance-dependent parameters within a non-linear mixed-effect modelling context. The model fitted the heterogeneous Al-26 data very well and was successfully validated against datasets in rats and humans. The presented PBTK model for Al, based on the most extensive and diverse dataset of Al exposure to date, constitutes a major advancement in the field, thereby paving the way towards a more quantitative risk assessment in humans.}, language = {en} } @phdthesis{Perera2021, author = {Perera, Upeksha}, title = {Solutions of direct and inverse Sturm-Liouville problems}, doi = {10.25932/publishup-53006}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-530064}, school = {Universit{\"a}t Potsdam}, pages = {x, 109}, year = {2021}, abstract = {Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm-Liouville Problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular and some singular SLPs of even orders (tested up to order eight), with a mix of boundary conditions (including non-separable and finite singular endpoints), accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. Next, a concrete implementation to the inverse Sturm-Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm-Liouville problems of order n=2,4 is verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides methods that can be adapted successfully for solving a direct (regular/singular) or inverse SLP of an arbitrary order with arbitrary boundary conditions.}, language = {en} } @phdthesis{Maier2021, author = {Maier, Corinna}, title = {Bayesian data assimilation and reinforcement learning for model-informed precision dosing in oncology}, doi = {10.25932/publishup-51587}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-515870}, school = {Universit{\"a}t Potsdam}, pages = {x, 138}, year = {2021}, abstract = {While patients are known to respond differently to drug therapies, current clinical practice often still follows a standardized dosage regimen for all patients. For drugs with a narrow range of both effective and safe concentrations, this approach may lead to a high incidence of adverse events or subtherapeutic dosing in the presence of high patient variability. Model-informedprecision dosing (MIPD) is a quantitative approach towards dose individualization based on mathematical modeling of dose-response relationships integrating therapeutic drug/biomarker monitoring (TDM) data. MIPD may considerably improve the efficacy and safety of many drug therapies. Current MIPD approaches, however, rely either on pre-calculated dosing tables or on simple point predictions of the therapy outcome. These approaches lack a quantification of uncertainties and the ability to account for effects that are delayed. In addition, the underlying models are not improved while applied to patient data. Therefore, current approaches are not well suited for informed clinical decision-making based on a differentiated understanding of the individually predicted therapy outcome. The objective of this thesis is to develop mathematical approaches for MIPD, which (i) provide efficient fully Bayesian forecasting of the individual therapy outcome including associated uncertainties, (ii) integrate Markov decision processes via reinforcement learning (RL) for a comprehensive decision framework for dose individualization, (iii) allow for continuous learning across patients and hospitals. Cytotoxic anticancer chemotherapy with its major dose-limiting toxicity, neutropenia, serves as a therapeutically relevant application example. For more comprehensive therapy forecasting, we apply Bayesian data assimilation (DA) approaches, integrating patient-specific TDM data into mathematical models of chemotherapy-induced neutropenia that build on prior population analyses. The value of uncertainty quantification is demonstrated as it allows reliable computation of the patient-specific probabilities of relevant clinical quantities, e.g., the neutropenia grade. In view of novel home monitoring devices that increase the amount of TDM data available, the data processing of sequential DA methods proves to be more efficient and facilitates handling of the variability between dosing events. By transferring concepts from DA and RL we develop novel approaches for MIPD. While DA-guided dosing integrates individualized uncertainties into dose selection, RL-guided dosing provides a framework to consider delayed effects of dose selections. The combined DA-RL approach takes into account both aspects simultaneously and thus represents a holistic approach towards MIPD. Additionally, we show that RL can be used to gain insights into important patient characteristics for dose selection. The novel dosing strategies substantially reduce the occurrence of both subtherapeutic and life-threatening neutropenia grades in a simulation study based on a recent clinical study (CEPAC-TDM trial) compared to currently used MIPD approaches. If MIPD is to be implemented in routine clinical practice, a certain model bias with respect to the underlying model is inevitable, as the models are typically based on data from comparably small clinical trials that reflect only to a limited extent the diversity in real-world patient populations. We propose a sequential hierarchical Bayesian inference framework that enables continuous cross-patient learning to learn the underlying model parameters of the target patient population. It is important to note that the approach only requires summary information of the individual patient data to update the model. This separation of the individual inference from population inference enables implementation across different centers of care. The proposed approaches substantially improve current MIPD approaches, taking into account new trends in health care and aspects of practical applicability. They enable progress towards more informed clinical decision-making, ultimately increasing patient benefits beyond the current practice.}, language = {en} } @article{EngbertRabeKliegletal.2021, author = {Engbert, Ralf and Rabe, Maximilian Michael and Kliegl, Reinhold and Reich, Sebastian}, title = {Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics}, series = {Bulletin of mathematical biology : official journal of the Society for Mathematical Biology}, volume = {83}, journal = {Bulletin of mathematical biology : official journal of the Society for Mathematical Biology}, number = {1}, publisher = {Springer}, address = {New York}, issn = {0092-8240}, doi = {10.1007/s11538-020-00834-8}, pages = {16}, year = {2021}, abstract = {Newly emerging pandemics like COVID-19 call for predictive models to implement precisely tuned responses to limit their deep impact on society. Standard epidemic models provide a theoretically well-founded dynamical description of disease incidence. For COVID-19 with infectiousness peaking before and at symptom onset, the SEIR model explains the hidden build-up of exposed individuals which creates challenges for containment strategies. However, spatial heterogeneity raises questions about the adequacy of modeling epidemic outbreaks on the level of a whole country. Here, we show that by applying sequential data assimilation to the stochastic SEIR epidemic model, we can capture the dynamic behavior of outbreaks on a regional level. Regional modeling, with relatively low numbers of infected and demographic noise, accounts for both spatial heterogeneity and stochasticity. Based on adapted models, short-term predictions can be achieved. Thus, with the help of these sequential data assimilation methods, more realistic epidemic models are within reach.}, language = {en} }