@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} }