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Model-informed precision dosing (MIPD) is a quantitative dosing framework that combines prior knowledge on the drug-disease-patient system with patient data from therapeutic drug/ biomarker monitoring (TDM) to support individualized dosing in ongoing treatment. Structural models and prior parameter distributions used in MIPD approaches typically build on prior clinical trials that involve only a limited number of patients selected according to some exclusion/inclusion criteria. Compared to the prior clinical trial population, the patient population in clinical practice can be expected to also include altered behavior and/or increased interindividual variability, the extent of which, however, is typically unknown. Here, we address the question of how to adapt and refine models on the level of the model parameters to better reflect this real-world diversity. We propose an approach for continued learning across patients during MIPD using a sequential hierarchical Bayesian framework. The approach builds on two stages to separate the update of the individual patient parameters from updating the population parameters. Consequently, it enables continued learning across hospitals or study centers, because only summary patient data (on the level of model parameters) need to be shared, but no individual TDM data. We illustrate this continued learning approach with neutrophil-guided dosing of paclitaxel. The present study constitutes an important step toward building confidence in MIPD and eventually establishing MIPD increasingly in everyday therapeutic use.
We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.
Extreme value statistics is a popular and frequently used tool to model the occurrence of large earthquakes. The problem of poor statistics arising from rare events is addressed by taking advantage of the validity of general statistical properties in asymptotic regimes. In this note, I argue that the use of extreme value statistics for the purpose of practically modeling the tail of the frequency-magnitude distribution of earthquakes can produce biased and thus misleading results because it is unknown to what degree the tail of the true distribution is sampled by data. Using synthetic data allows to quantify this bias in detail. The implicit assumption that the true M-max is close to the maximum observed magnitude M-max,M-observed restricts the class of the potential models a priori to those with M-max = M-max,M-observed + Delta M with an increment Delta M approximate to 0.5... 1.2. This corresponds to the simple heuristic method suggested by Wheeler (2009) and labeled :M-max equals M-obs plus an increment." The incomplete consideration of the entire model family for the frequency-magnitude distribution neglects, however, the scenario of a large so far unobserved earthquake.
A rigorous construction of the supersymmetric path integral associated to a compact spin manifold
(2022)
We give a rigorous construction of the path integral in N = 1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Guneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaume, Atiyah, Bismut and Witten.
An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions
(2022)
We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature beta. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.
We propose a global geomagnetic field model for the last 14 thousand years, based on thermoremanent records. We call the model ArchKalmag14k. ArchKalmag14k is constructed by modifying recently proposed algorithms, based on space-time correlations. Due to the amount of data and complexity of the model, the full Bayesian posterior is numerically intractable. To tackle this, we sequentialize the inversion by implementing a Kalman-filter with a fixed time step. Every step consists of a prediction, based on a degree dependent temporal covariance, and a correction via Gaussian process regression. Dating errors are treated via a noisy input formulation. Cross correlations are reintroduced by a smoothing algorithm and model parameters are inferred from the data. Due to the specific statistical nature of the proposed algorithms, the model comes with space and time-dependent uncertainty estimates. The new model ArchKalmag14k shows less variation in the large-scale degrees than comparable models. Local predictions represent the underlying data and agree with comparable models, if the location is sampled well. Uncertainties are bigger for earlier times and in regions of sparse data coverage. We also use ArchKalmag14k to analyze the appearance and evolution of the South Atlantic anomaly together with reverse flux patches at the core-mantle boundary, considering the model uncertainties. While we find good agreement with earlier models for recent times, our model suggests a different evolution of intensity minima prior to 1650 CE. In general, our results suggest that prior to 6000 BCE the data is not sufficient to support global models.
We study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary. The adapted boundary operator need not be selfadjoint and the boundary condition need not be pseudo-local.We show the equivalence of various characterisations of elliptic boundary conditions and demonstrate how the boundary conditions traditionally considered in the literature fit in our framework. The regularity of the solutions up to the boundary is proven. We show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact. We provide examples which are conveniently treated by our methods.
The index theorem for elliptic operators on a closed Riemannian manifold by Atiyah and Singer has many applications in analysis, geometry and topology, but it is not suitable for a generalization to a Lorentzian setting.
In the case where a boundary is present Atiyah, Patodi and Singer provide an index theorem for compact Riemannian manifolds by introducing non-local boundary conditions obtained via the spectral decomposition of an induced boundary operator, so called APS boundary conditions. Bär and Strohmaier prove a Lorentzian version of this index theorem for the Dirac operator on a manifold with boundary by utilizing results from APS and the characterization of the spectral flow by Phillips. In their case the Lorentzian manifold is assumed to be globally hyperbolic and spatially compact, and the induced boundary operator is given by the Riemannian Dirac operator on a spacelike Cauchy hypersurface. Their results show that imposing APS boundary conditions for these boundary operator will yield a Fredholm operator with a smooth kernel and its index can be calculated by a formula similar to the Riemannian case.
Back in the Riemannian setting, Bär and Ballmann provide an analysis of the most general kind of boundary conditions that can be imposed on a first order elliptic differential operator that will still yield regularity for solutions as well as Fredholm property for the resulting operator. These boundary conditions can be thought of as deformations to the graph of a suitable operator mapping APS boundary conditions to their orthogonal complement.
This thesis aims at applying the boundary conditions found by Bär and Ballmann to a Lorentzian setting to understand more general types of boundary conditions for the Dirac operator, conserving Fredholm property as well as providing regularity results and relative index formulas for the resulting operators. As it turns out, there are some differences in applying these graph-type boundary conditions to the Lorentzian Dirac operator when compared to the Riemannian setting. It will be shown that in contrast to the Riemannian case, going from a Fredholm boundary condition to its orthogonal complement works out fine in the Lorentzian setting. On the other hand, in order to deduce Fredholm property and regularity of solutions for graph-type boundary conditions, additional assumptions for the deformation maps need to be made.
The thesis is organized as follows. In chapter 1 basic facts about Lorentzian and Riemannian spin manifolds, their spinor bundles and the Dirac operator are listed. These will serve as a foundation to define the setting and prove the results of later chapters.
Chapter 2 defines the general notion of boundary conditions for the Dirac operator used in this thesis and introduces the APS boundary conditions as well as their graph type deformations. Also the role of the wave evolution operator in finding Fredholm boundary conditions is analyzed and these boundary conditions are connected to notion of Fredholm pairs in a given Hilbert space.
Chapter 3 focuses on the principal symbol calculation of the wave evolution operator and the results are used to proof Fredholm property as well as regularity of solutions for suitable graph-type boundary conditions. Also sufficient conditions are derived for (pseudo-)local boundary conditions imposed on the Dirac operator to yield a Fredholm operator with a smooth solution space.
In the last chapter 4, a few examples of boundary conditions are calculated applying the results of previous chapters. Restricting to special geometries and/or boundary conditions, results can be obtained that are not covered by the more general statements, and it is shown that so-called transmission conditions behave very differently than in the Riemannian setting.
In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.
The motivation for this work was the question of reliability and robustness of seismic tomography. The problem is that many earth models exist which can describe the underlying ground motion records equally well. Most algorithms for reconstructing earth models provide a solution, but rarely quantify their variability. If there is no way to verify the imaged structures, an interpretation is hardly reliable. The initial idea was to explore the space of equivalent earth models using Bayesian inference. However, it quickly became apparent that the rigorous quantification of tomographic uncertainties could not be accomplished within the scope of a dissertation.
In order to maintain the fundamental concept of statistical inference, less complex problems from the geosciences are treated instead. This dissertation aims to anchor Bayesian inference more deeply in the geosciences and to transfer knowledge from applied mathematics. The underlying idea is to use well-known methods and techniques from statistics to quantify the uncertainties of inverse problems in the geosciences. This work is divided into three parts:
Part I introduces the necessary mathematics and should be understood as a kind of toolbox. With a physical application in mind, this section provides a compact summary of all methods and techniques used. The introduction of Bayesian inference makes the beginning. Then, as a special case, the focus is on regression with Gaussian processes under linear transformations. The chapters on the derivation of covariance functions and the approximation of non-linearities are discussed in more detail.
Part II presents two proof of concept studies in the field of seismology. The aim is to present the conceptual application of the introduced methods and techniques with moderate complexity. The example about traveltime tomography applies the approximation of non-linear relationships. The derivation of a covariance function using the wave equation is shown in the example of a damped vibrating string. With these two synthetic applications, a consistent concept for the quantification of modeling uncertainties has been developed.
Part III presents the reconstruction of the Earth's archeomagnetic field. This application uses the whole toolbox presented in Part I and is correspondingly complex. The modeling of the past 1000 years is based on real data and reliably quantifies the spatial modeling uncertainties. The statistical model presented is widely used and is under active development.
The three applications mentioned are intentionally kept flexible to allow transferability to similar problems. The entire work focuses on the non-uniqueness of inverse problems in the geosciences. It is intended to be of relevance to those interested in the concepts of Bayesian inference.
The geomagnetic main field is vital for live on Earth, as it shields our habitat against the solar wind and cosmic rays. It is generated by the geodynamo in the Earth’s outer core and has a rich dynamic on various timescales. Global models of the field are used to study the interaction of the field and incoming charged particles, but also to infer core dynamics and to feed numerical simulations of the geodynamo. Modern satellite missions, such as the SWARM or the CHAMP mission, support high resolution reconstructions of the global field. From the 19 th century on, a global network of magnetic observatories has been established. It is growing ever since and global models can be constructed from the data it provides. Geomagnetic field models that extend further back in time rely on indirect observations of the field, i.e. thermoremanent records such as burnt clay or volcanic rocks and sediment records from lakes and seas. These indirect records come with (partially very large) uncertainties, introduced by the complex measurement methods and the dating procedure.
Focusing on thermoremanent records only, the aim of this thesis is the development of a new modeling strategy for the global geomagnetic field during the Holocene, which takes the uncertainties into account and produces realistic estimates of the reliability of the model. This aim is approached by first considering snapshot models, in order to address the irregular spatial distribution of the records and the non-linear relation of the indirect observations to the field itself. In a Bayesian setting, a modeling algorithm based on Gaussian process regression is developed and applied to binned data. The modeling algorithm is then extended to the temporal domain and expanded to incorporate dating uncertainties. Finally, the algorithm is sequentialized to deal with numerical challenges arising from the size of the Holocene dataset.
The central result of this thesis, including all of the aspects mentioned, is a new global geomagnetic field model. It covers the whole Holocene, back until 12000 BCE, and we call it ArchKalmag14k. When considering the uncertainties that are produced together with the model, it is evident that before 6000 BCE the thermoremanent database is not sufficient to support global models. For times more recent, ArchKalmag14k can be used to analyze features of the field under consideration of posterior uncertainties. The algorithm for generating ArchKalmag14k can be applied to different datasets and is provided to the community as an open source python package.
Dynamical models make specific assumptions about cognitive processes that generate human behavior. In data assimilation, these models are tested against timeordered data. Recent progress on Bayesian data assimilation demonstrates that this approach combines the strengths of statistical modeling of individual differences with the those of dynamical cognitive models.
We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems.
Congenital adrenal hyperplasia (CAH) is the most common form of adrenal insufficiency in childhood; it requires cortisol replacement therapy with hydrocortisone (HC, synthetic cortisol) from birth and therapy monitoring for successful treatment. In children, the less invasive dried blood spot (DBS) sampling with whole blood including red blood cells (RBCs) provides an advantageous alternative to plasma sampling.
Potential differences in binding/association processes between plasma and DBS however need to be considered to correctly interpret DBS measurements for therapy monitoring. While capillary DBS samples would be used in clinical practice, venous cortisol DBS samples from children with adrenal insufficiency were analyzed due to data availability and to directly compare and thus understand potential differences between venous DBS and plasma. A previously published HC plasma pharmacokinetic (PK) model was extended by leveraging these DBS concentrations.
In addition to previously characterized binding of cortisol to albumin (linear process) and corticosteroid-binding globulin (CBG; saturable process), DBS data enabled the characterization of a linear cortisol association with RBCs, and thereby providing a quantitative link between DBS and plasma cortisol concentrations. The ratio between the observed cortisol plasma and DBS concentrations varies highly from 2 to 8. Deterministic simulations of the different cortisol binding/association fractions demonstrated that with higher blood cortisol concentrations, saturation of cortisol binding to CBG was observed, leading to an increase in all other cortisol binding fractions.
In conclusion, a mathematical PK model was developed which links DBS measurements to plasma exposure and thus allows for quantitative interpretation of measurements of DBS samples.
Hidden semi-Markov models generalise hidden Markov models by explicitly modelling the time spent in a given state, the so-called dwell time, using some distribution defined on the natural numbers. While the (shifted) Poisson and negative binomial distribution provide natural choices for such distributions, in practice, parametric distributions can lack the flexibility to adequately model the dwell times. To overcome this problem, a penalised maximum likelihood approach is proposed that allows for a flexible and data-driven estimation of the dwell-time distributions without the need to make any distributional assumption. This approach is suitable for direct modelling purposes or as an exploratory tool to investigate the latent state dynamics. The feasibility and potential of the suggested approach is illustrated in a simulation study and by modelling muskox movements in northeast Greenland using GPS tracking data. The proposed method is implemented in the R-package PHSMM which is available on CRAN.
We construct and examine the prototype of a deep learning-based ground-motion model (GMM) that is both fully data driven and nonergodic. We formulate ground-motion modeling as an image processing task, in which a specific type of neural network, the U-Net, relates continuous, horizontal maps of earthquake predictive parameters to sparse observations of a ground-motion intensity measure (IM). The processing of map-shaped data allows the natural incorporation of absolute earthquake source and observation site coordinates, and is, therefore, well suited to include site-, source-, and path-specific amplification effects in a nonergodic GMM. Data-driven interpolation of the IM between observation points is an inherent feature of the U-Net and requires no a priori assumptions. We evaluate our model using both a synthetic dataset and a subset of observations from the KiK-net strong motion network in the Kanto basin in Japan. We find that the U-Net model is capable of learning the magnitude???distance scaling, as well as site-, source-, and path-specific amplification effects from a strong motion dataset. The interpolation scheme is evaluated using a fivefold cross validation and is found to provide on average unbiased predictions. The magnitude???distance scaling as well as the site amplification of response spectral acceleration at a period of 1 s obtained for the Kanto basin are comparable to previous regional studies.
According to Radzikowski’s celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF−G˜F+G˜A−G˜R) in the sense of Duistermaat and Hörmander [Acta Math. 128, 183–269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by Bär, Ginoux, and Pfäffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Zürich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF−G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied.
In this work, we present Raman lidar data (from a Nd:YAG operating at 355 nm, 532 nm and 1064 nm) from the international research village Ny-Alesund for the time period of January to April 2020 during the Arctic haze season of the MOSAiC winter. We present values of the aerosol backscatter, the lidar ratio and the backscatter Angstrom exponent, though the latter depends on wavelength. The aerosol polarization was generally below 2%, indicating mostly spherical particles. We observed that events with high backscatter and high lidar ratio did not coincide. In fact, the highest lidar ratios (LR > 75 sr at 532 nm) were already found by January and may have been caused by hygroscopic growth, rather than by advection of more continental aerosol. Further, we performed an inversion of the lidar data to retrieve a refractive index and a size distribution of the aerosol. Our results suggest that in the free troposphere (above approximate to 2500 m) the aerosol size distribution is quite constant in time, with dominance of small particles with a modal radius well below 100 nm. On the contrary, below approximate to 2000 m in altitude, we frequently found gradients in aerosol backscatter and even size distribution, sometimes in accordance with gradients of wind speed, humidity or elevated temperature inversions, as if the aerosol was strongly modified by vertical displacement in what we call the "mechanical boundary layer". Finally, we present an indication that additional meteorological soundings during MOSAiC campaign did not necessarily improve the fidelity of air backtrajectories.
We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.
The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K is an element of Double-struck capital R every manifold of dimension at least 2 carries a complete C-1,C- 1-metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C-2-metrics in general.