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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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Analysis of protrusion dynamics in amoeboid cell motility by means of regularized contour flows
(2021)
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. <br /> 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.
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.
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
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.
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.
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.
Sequential data assimilation of the stochastic SEIR epidemic model for regional COVID-19 dynamics
(2021)
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.
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.
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.
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.
Die Bienaymé-Galton-Watson Prozesse können für die Untersuchung von speziellen und sich entwickelnden Populationen verwendet werden. Die Populationen umfassen Individuen, welche sich identisch, zufällig, selbstständig und unabhängig voneinander fortpflanzen und die jeweils nur eine Generation existieren. Die n-te Generation ergibt sich als zufällige Summe der Individuen der (n-1)-ten Generation. Die Relevanz dieser Prozesse begründet sich innerhalb der Historie und der inner- und außermathematischen Bedeutung. Die Geschichte der Bienaymé-Galton-Watson-Prozesse wird anhand der Entwicklung des Konzeptes bis heute dargestellt. Dabei werden die Wissenschaftler:innen verschiedener Disziplinen angeführt, die Erkenntnisse zu dem Themengebiet beigetragen und das Konzept in ihren Fachbereichen angeführt haben. Somit ergibt sich die außermathematische Signifikanz. Des Weiteren erhält man die innermathematische Bedeutsamkeit mittels des Konzeptes der Verzweigungsprozesse, welches auf die Bienaymé-Galton-Watson Prozesse zurückzuführen ist. Die Verzweigungsprozesse stellen eines der aussagekräftigsten Modelle für die Beschreibung des Populationswachstums dar. Darüber hinaus besteht die derzeitige Wichtigkeit durch die Anwendungsmöglichkeit der Verzweigungsprozesse und der Bienaymé-Galton-Watson Prozesse innerhalb der Epidemiologie. Es werden die Ebola- und die Corona-Pandemie als Anwendungsfelder angeführt. Die Prozesse dienen als Entscheidungsstütze für die Politik und ermöglichen Aussagen über die Auswirkungen von Maßnahmen bezüglich der Pandemien. Neben den Prozessen werden ebenfalls der bedingte Erwartungswert bezüglich diskreter Zufallsvariablen, die wahrscheinlichkeitserzeugende Funktion und die zufällige Summe eingeführt. Die Konzepte vereinfachen die Beschreibung der Prozesse und bilden somit die Grundlage der Betrachtungen. Außerdem werden die benötigten und weiterführenden Eigenschaften der grundlegenden Themengebiete und der Prozesse aufgeführt und bewiesen. Das Kapitel erreicht seinen Höhepunkt bei dem Beweis des Kritikalitätstheorems, wodurch eine Aussage über das Aussterben des Prozesses in verschiedenen Fällen und somit über die Aussterbewahrscheinlichkeit getätigt werden kann. Die Fälle 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 größer 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 Mäusen und die Entstehungsfragestellung der Prozesse, angeführt. Diese werden mithilfe der erlangten Ergebnisse untersucht und einige ausgewählte zufällige 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 Kritikalitätsfällen der Prozesse dar. Zudem werden die Häufigkeiten der einzelnen Populationsgrößen in Form von Histogrammen angebracht. Dabei lässt sich der Unterschied zwischen den einzelnen Fällen bestätigen und es wird die Anwendungsmöglichkeit der Bienaymé-Galton-Watson Prozesse bei komplexeren Problemen deutlich. Histogramme bekräftigen, dass die einzelnen Populationsgrößen nur endlich oft vorkommen. Diese Aussage wurde von Galton aufgeworfen und in der Extinktions-Explosions-Dichotomie verwendet. Die dargestellten Erkenntnisse über das Themengebiet und die Betrachtung des Konzeptes werden mit einer didaktischen Analyse abgeschlossen. Die Untersuchung beinhaltet die Berücksichtigung der Fundamentalen Ideen, der Fundamentalen Ideen der Stochastik und der Leitidee „Daten und Zufall“. Dabei ergibt sich, dass in Abhängigkeit der gewählten Perspektive die Anwendung der Bienaymé-Galton-Watson Prozesse innerhalb der Schule plausibel ist und von Vorteil für die Schüler:innen sein kann. Für die Behandlung wird exemplarisch der Rahmenlehrplan für Berlin und Brandenburg analysiert und mit dem Kernlehrplan Nordrhein-Westfalens verglichen. Die Konzeption des Lehrplans aus Berlin und Brandenburg lässt nicht den Schluss zu, dass die Bienaymé-Galton-Watson Prozesse angewendet werden sollten. Es lässt sich feststellen, dass die zugrunde liegende Leitidee nicht vollumfänglich mit manchen Fundamentalen Ideen der Stochastik vereinbar ist. Somit würde eine Modifikation hinsichtlich einer stärkeren Orientierung des Lehrplans an den Fundamentalen Ideen die Anwendung der Prozesse ermöglichen. Die Aussage wird durch die Betrachtung und Übertragung eines nordrhein-westfälischen Unterrichtsentwurfes für stochastische Prozesse auf die Bienaymé-Galton-Watson Prozesse unterstützt. Darüber hinaus werden eine Concept Map und ein Vernetzungspentagraph nach von der Bank konzipiert um diesen Aspekt hervorzuheben.
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 für den pädagogischen Kontext wenig klassifiziert und für Lehrende kaum zugänglich gemacht worden.
Daher zielt diese Bachelorarbeit darauf ab, eine systematische Strukturierung und Aufarbeitung von Gamification sowie innovative Ansätze für die Verwendung spieltypischer Elemente im Unterricht, konkret dem Mathematikunterricht, zu präsentieren. Dies kann eine Grundlage für andere Fachgebiete, aber auch andere Lehrformen bieten und so die Umsetzbarkeit von Gamification in eigenen Lehrveranstaltungen aufzeigen.
In der Arbeit wird begründet, weshalb und mithilfe welcher Elemente Gamification die Motivation und Leistungsbereitschaft der Lernenden langfristig erhöhen, die Sozial- und Personalkompetenzen fördern sowie die Lernenden zu mehr Aktivität anregen kann. Zudem wird Gamification explizit mit grundlegenden mathematikdidaktischen Prinzipien in Verbindung gesetzt und somit die Relevanz für den Mathematikunterricht hervorgehoben.
Anschließend werden die einzelnen Elemente von Gamification wie Punkte, Level, Abzeichen, Charaktere und Rahmengeschichte entlang einer eigens für den pädagogischen Kontext entwickelten Klassifikation „FUN“ (Feedback – User specific elements – Neutral elements) schematisch beschrieben, ihre Funktionen und Wirkung dargestellt sowie Einsatzmöglichkeiten im Unterricht aufgezeigt. Dies beinhaltet Ideen zu lernförderlichem Feedback, Differenzierungsmöglichkeiten und Unterrichtsrahmengestaltung, die in Lehrveranstaltungen aller Art umsetzbar sein können. Die Bachelorarbeit umfasst zudem ein spezifisches Beispiel, einen Unterrichtsentwurf einer gamifizierten Mathematikstunde inklusive des zugehörigen Arbeitsmaterials, anhand dessen die Verwendung von Gamification deutlich wird.
Gamification offeriert oftmals Vorteile gegenüber dem traditionellen Unterricht, muss jedoch wie jede Methode an den Inhalt und die Zielgruppe angepasst werden. Weiterführende Forschung könnte sich mit konkreten motivationalen Strukturen, personenspezifischen Unterschieden sowie mit mathematischen Inhalten wie dem Problemlösen oder dem Wechsel zwischen verschiedenen Darstellungen hinsichtlich gamifizierter Lehrformen beschäftigen.
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.
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.
Contributions to the theoretical analysis of the algorithms with adversarial and dependent data
(2021)
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).
Im Zuge der Covid-19 Pandemie werden zwei Werte täglich 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. Für die Schätzung dieses Wertes gibt es viele Möglichkeiten - auch das Robert Koch-Institut gibt in seinem täglichen Situationsbericht stets zwei R-Werte an: Einen 4-Tage-R-Wert und einen weniger schwankenden 7-Tage-R-Wert. Diese Arbeit soll eine weitere Möglichkeit vorstellen, einige Aspekte der Pandemie zu modellieren und die Reproduktionsrate zu schätzen.
In der ersten Hälfte der Arbeit werden die mathematischen Grundlagen vorgestellt, die man für die Modellierung benötigt. Hierbei wird davon ausgegangen, dass der Leser bereits ein Basisverständnis von stochastischen Prozessen hat. Im Abschnitt Grundlagen werden Verzweigungsprozesse mit einigen Beispielen eingeführt und die Ergebnisse aus diesem Themengebiet, die für diese Arbeit wichtig sind, präsentiert. Dabei gehen wir zuerst auf einfache Verzweigungsprozesse ein und erweitern diese dann auf Verzweigungsprozesse mit mehreren Typen. Um die Notation zu erleichtern, beschränken wir uns auf zwei Typen. Das Prinzip lässt 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 über einen längeren Zeitraum. In der Anwendung auf die Pandemie hat der Parameter λ die gleiche Rolle wie die Reproduktionsrate R.
In der zweiten Hälfte dieser Arbeit stellen wir eine Anwendung der Theorie über Multitype Verzweigungsprozesse vor. Professor Yanev und seine Mitarbeiter modellieren in ihrer Veröffentlichung Branching stochastic processes as models of Covid-19 epidemic development die Ausbreitung des Corona Virus' über einen Verzweigungsprozess mit zwei Typen. Wir werden dieses Modell diskutieren und Schätzer daraus ableiten: Ziel ist es, die Reproduktionsrate zu ermitteln. Außerdem analysieren wir die Möglichkeiten, die Dunkelziffer (die Zahl nicht gemeldeter Krankheitsfälle) zu schätzen. Wir wenden die Schätzer auf die Zahlen von Deutschland an und werten diese schließlich aus.
Die Vielfältigkeit des Winkelbegriffs ist gleichermaßen spannend wie herausfordernd in Hinblick auf seine Zugänge 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 für den Schulunterricht überführt.
Dabei erfolgt zunächst eine stoffdidaktische Auseinandersetzung mit dem Winkelbegriff, die von einer informationstheoretischen Winkeldefinition begleitet wird. In dieser wird eine Definition für den Winkelbegriff unter der Fragestellung entwickelt, welche Informationen man über einen Winkel benötigt, um ihn beschreiben zu können. So können 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 können, so dass Schlussfolgerungen aus der informationstheoretischen Winkeldefinition beispielsweise in dynamischen Geometriesystemen zur Verfügung 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 Verknüpfung der beiden Theorien wird ein grundsätzlicher 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 ermöglichen soll. Hierfür wird die Lehrstrategie angepasst, um insbesondere den Übergang von Winkelsituationen zu Winkelkontexten zu realisieren. Explizit für den Aspekt des Winkelfeldes werden, anhand der Untersuchung der Sichtfelder von Tieren, Lernhandlungen und Forderungen an ein Lernmodell beschrieben, die Schülerinnen und Schüler bei der Begriffsaneignung unterstützen.
Die Tätigkeitstheorie, 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 münden. Hierzu wird u. a. das Modell der Artifact-Centric Activity Theory genutzt, das das Beziehungsgefüge aus Schülerinnen und Schülern, 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. Gemäß dem Ansatz der Fachdidaktischen Entwicklungsforschung wird die Lernumgebung anschließend in mehreren Zyklen erprobt, evaluiert und überarbeitet. Dabei wird ein qualitatives Setting angewandt, das sich der Semiotischen Vermittlung bedient und untersucht, inwiefern sich die Qualität der von den Schülerinnen und Schülern gezeigten Lernhandlungen durch die Designprinzipien und deren Umsetzung erklären lässt. Am Ende der Arbeit stehen eine finale Version der Designprinzipien und eine sich daraus ergebende Lernumgebung zur Einführung des Winkelfeldbegriffs in der vierten Klassenstufe.
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.
Geomagnetic field modeling using spherical harmonics requires the inversion for hundreds to thousands of parameters. This large-scale problem can always be formulated as an optimization problem, where a global minimum of a certain cost function has to be calculated. A variety of approaches is known in order to solve this inverse problem, e.g. derivative-based methods or least-squares methods and their variants. Each of these methods has its own advantages and disadvantages, which affect for example the applicability to non-differentiable functions or the runtime of the corresponding algorithm.
In this work, we pursue the goal to find an algorithm which is faster than the established methods and which is applicable to non-linear problems. Such non-linear problems occur for example when estimating Euler angles or when the more robust L_1 norm is applied. Therefore, we will investigate the usability of stochastic optimization methods from the CMAES family for modeling the geomagnetic field of Earth's core. On one hand, basics of core field modeling and their parameterization are discussed using some examples from the literature. On the other hand, the theoretical background of the stochastic methods are provided. A specific CMAES algorithm was successfully applied in order to invert data of the Swarm satellite mission and to derive the core field model EvoMag. The EvoMag model agrees well with established models and observatory data from Niemegk. Finally, we present some observed difficulties and discuss the results of our model.