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We consider the initial value problem for the Navier-Stokes equations over R-3 x [0, T] with time T > 0 in the spatially periodic setting.
We prove that it induces open injective mappings A(s): B-1(s) -> B-2(s-1) where B-1(s), B-2(s-1) are elements from scales of specially constructed function spaces of Bochner-Sobolev typeparametrized with the smoothness index s is an element of N.
Finally, we prove that a map Asis surjective if and only if the inverse image A(s)(- 1) (K) of any pre compact set K from the range of the map Asis bounded in the Bochner space L-s([0, T], L-r(T-3))with the Ladyzhenskaya-Prodi-Serrin numbers s, r.
We consider Bayesian inference for large-scale inverse problems, where computational challenges arise from the need for repeated evaluations of an expensive forward model.
This renders most Markov chain Monte Carlo approaches infeasible, since they typically require O(10(4)) model runs, or more.
Moreover, the forward model is often given as a black box or is impractical to differentiate.
Therefore derivative-free algorithms are highly desirable. We propose a framework, which is built on Kalman methodology, to efficiently perform Bayesian inference in such inverse problems.
The basic method is based on an approximation of the filtering distribution of a novel mean-field dynamical system, into which the inverse problem is embedded as an observation operator.
Theoretical properties are established for linear inverse problems, demonstrating that the desired Bayesian posterior is given by the steady state of the law of the filtering distribution of the mean-field dynamical system, and proving exponential convergence to it.
This suggests that, for nonlinear problems which are close to Gaussian, sequentially computing this law provides the basis for efficient iterative methods to approximate the Bayesian posterior.
Ensemble methods are applied to obtain interacting particle system approximations of the filtering distribution of the mean-field model; and practical strategies to further reduce the computational and memory cost of the methodology are presented, including low-rank approximation and a bi-fidelity approach.
The effectiveness of the framework is demonstrated in several numerical experiments, including proof-of-concept linear/nonlinear examples and two large-scale applications: learning of permeability parameters in subsurface flow; and learning subgrid-scale parameters in a global climate model.
Moreover, the stochastic ensemble Kalman filter and various ensemble square-root Kalman filters are all employed and are compared numerically.
The results demonstrate that the proposed method, based on exponential convergence to the filtering distribution of a mean-field dynamical system, is competitive with pre-existing Kalman-based methods for inverse problems.
In this paper, we define a variant of Roe algebras for spaces with cylindrical ends and use this to study questions regarding existence and classification of metrics of positive scalar curvature on such manifolds which are collared on the cylindrical end.
We discuss how our constructions are related to relative higher index theory as developed by Chang, Weinberger, and Yu and use this relationship to define higher rho-invariants for positive scalar curvature metrics on manifolds with boundary.
This paves the way for the classification of these metrics.
Finally, we use the machinery developed here to give a concise proof of a result of Schick and the author, which relates the relative higher index with indices defined in the presence of positive scalar curvature on the boundary.
In this paper we consider surfaces which are critical points of the Willmore functional subject to constrained area.
In the case of small area we calculate the corrections to the intrinsic geometry induced by the ambient curvature.
These estimates together with the choice of an adapted geometric center of mass lead to refined position estimates in relation to the scalar curvature of the ambient manifold.
As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the "top degree component " of a differential form on it.
In this paper, we show that a formula from finite dimensions generalizes to assign a sensible "top degree component " to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical presymplectic 2-form on the loop space.
This construction is a crucial ingredient for the definition of the supersymmetric path integral on the loop space.
State space models enjoy wide popularity in mathematical and statistical modelling across disciplines and research fields. Frequent solutions to problems of estimation and forecasting of a latent signal such as the celebrated Kalman filter hereby rely on a set of strong assumptions such as linearity of system dynamics and Gaussianity of noise terms.
We investigate fallacy in mis-specification of the noise terms, that is signal noise
and observation noise, regarding heavy tailedness in that the true dynamic frequently produces observation outliers or abrupt jumps of the signal state due to realizations of these heavy tails not considered by the model. We propose a formalisation of observation noise mis-specification in terms of Huber’s ε-contamination as well as a computationally cheap solution via generalised Bayesian posteriors with a diffusion Stein divergence loss resulting in the diffusion score matching Kalman filter - a modified algorithm akin in complexity to the regular Kalman filter. For this new filter interpretations of novel terms, stability and an ensemble variant are discussed. Regarding signal noise mis-specification, we propose a formalisation in the frame work of change point detection and join ideas from the popular CUSUM algo-
rithm with ideas from Bayesian online change point detection to combine frequent reliability constraints and online inference resulting in a Gaussian mixture model variant of multiple Kalman filters. We hereby exploit open-end sequential probability ratio tests on the evidence of Kalman filters on observation sub-sequences for aggregated inference under notions of plausibility.
Both proposed methods are combined to investigate the double mis-specification problem and discussed regarding their capabilities in reliable and well-tuned uncertainty quantification. Each section provides an introduction to required terminology and tools as well as simulation experiments on the popular target tracking task and the non-linear, chaotic Lorenz-63 system to showcase practical performance of theoretical considerations.
Hardy inequalities on graphs
(2024)
The dissertation deals with a central inequality of non-linear potential theory, the Hardy inequality. It states that the non-linear energy functional can be estimated from below by a pth power of a weighted p-norm, p>1. The energy functional consists of a divergence part and an arbitrary potential part. Locally summable infinite graphs were chosen as the underlying space. Previous publications on Hardy inequalities on graphs have mainly considered the special case p=2, or locally finite graphs without a potential part.
Two fundamental questions now arise quite naturally: For which graphs is there a Hardy inequality at all? And, if it exists, is there a way to obtain an optimal weight? Answers to these questions are given in Theorem 10.1 and Theorem 12.1. Theorem 10.1 gives a number of characterizations; among others, there is a Hardy inequality on a graph if and only if there is a Green's function. Theorem 12.1 gives an explicit formula to compute optimal Hardy weights for locally finite graphs under some additional technical assumptions. Examples show that Green's functions are good candidates to be used in the formula.
Emphasis is also placed on illustrating the theory with examples. The focus is on natural numbers, Euclidean lattices, trees and star graphs. Finally, a non-linear version of the Heisenberg uncertainty principle and a Rellich inequality are derived from the Hardy inequality.
We present general existence and uniqueness results for marked models with pair interactions, exemplified through Gibbs point processes on path space.
More precisely, we study a class of infinite-dimensional diffusions under Gibbsian interactions, in the context of marked point configurations: the starting points belong to R-d, and the marks are the paths of Langevin diffusions.
We use the entropy method to prove existence of an infinite-volume Gibbs point process and use cluster expansion tools to provide an explicit activity domain in which uniqueness holds.
The Gutenberg-Richter (GR) and the Omori-Utsu (OU) law describe the earthquakes' energy release and temporal clustering and are thus of great importance for seismic hazard assessment. Motivated by experimental results, which indicate stress-dependent parameters, we consider a combined global data set of 127 main shock-aftershock sequences and perform a systematic study of the relationship between main shock-induced stress changes and associated seismicity patterns. For this purpose, we calculate space-dependent Coulomb Stress (& UDelta;CFS) and alternative receiver-independent stress metrics in the surrounding of the main shocks. Our results indicate a clear positive correlation between the GR b-value and the induced stress, contrasting expectations from laboratory experiments and suggesting a crucial role of structural heterogeneity and strength variations. Furthermore, we demonstrate that the aftershock productivity increases nonlinearly with stress, while the OU parameters c and p systematically decrease for increasing stress changes. Our partly unexpected findings can have an important impact on future estimations of the aftershock hazard.
This paper deals with the long-term behavior of positive operator semigroups on spaces of bounded functions and of signed measures, which have applications to parabolic equations with unbounded coefficients and to stochas-tic analysis. The main results are a Tauberian type theorem characterizing the convergence to equilibrium of strongly Feller semigroups and a generalization of a classical convergence theorem of Doob. None of these results requires any kind of time regularity of the semigroup.
Deriving mechanism-based pharmacodynamic models by reducing quantitative systems pharmacology models
(2023)
Quantitative systems pharmacology (QSP) models integrate comprehensive qualitative and quantitative knowledge about pharmacologically relevant processes. We previously proposed a first approach to leverage the knowledge in QSP models to derive simpler, mechanism-based pharmacodynamic (PD) models. Their complexity, however, is typically still too large to be used in the population analysis of clinical data. Here, we extend the approach beyond state reduction to also include the simplification of reaction rates, elimination of reactions, and analytic solutions. We additionally ensure that the reduced model maintains a prespecified approximation quality not only for a reference individual but also for a diverse virtual population. We illustrate the extended approach for the warfarin effect on blood coagulation. Using the model-reduction approach, we derive a novel small-scale warfarin/international normalized ratio model and demonstrate its suitability for biomarker identification. Due to the systematic nature of the approach in comparison with empirical model building, the proposed model-reduction algorithm provides an improved rationale to build PD models also from QSP models in other applications.
Cell-level systems biology model to study inflammatory bowel diseases and their treatment options
(2023)
To help understand the complex and therapeutically challenging inflammatory bowel diseases (IBDs), we developed a systems biology model of the intestinal immune system that is able to describe main aspects of IBD and different treatment modalities thereof. The model, including key cell types and processes of the mucosal immune response, compiles a large amount of isolated experimental findings from literature into a larger context and allows for simulations of different inflammation scenarios based on the underlying data and assumptions. In the context of a large and diverse virtual IBD population, we characterized the patients based on their phenotype (in contrast to healthy individuals, they developed persistent inflammation after a trigger event) rather than on a priori assumptions on parameter differences to a healthy individual. This allowed to reproduce the enormous diversity of predispositions known to lead to IBD. Analyzing different treatment effects, the model provides insight into characteristics of individual drug therapy. We illustrate for anti-TNF-alpha therapy, how the model can be used (i) to decide for alternative treatments with best prospects in the case of nonresponse, and (ii) to identify promising combination therapies with other available treatment options.
We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier-Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by three cases which are the steady flow around a backward step, the flow around a circular cylinder and the unsteady turbulent flow around a surface mounted cubic obstacle.
Introduction:
Hydrocortisone is the standard of care in cortisol replacement therapy for congenital adrenal hyperplasia patients. Challenges in mimicking cortisol circadian rhythm and dosing individualization can be overcome by the support of mathematical modelling. Previously, a non-linear mixed-effects (NLME) model was developed based on clinical hydrocortisone pharmacokinetic (PK) pediatric and adult data. Additionally, a physiologically-based pharmacokinetic (PBPK) model was developed for adults and a pediatric model was obtained using maturation functions for relevant processes. In this work, a middle-out approach was applied. The aim was to investigate whether PBPK-derived maturation functions could provide a better description of hydrocortisone PK inter-individual variability when implemented in the NLME framework, with the goal of providing better individual predictions towards precision dosing at the patient level.
Methods:
Hydrocortisone PK data from 24 adrenal insufficiency pediatric patients and 30 adult healthy volunteers were used for NLME model development, while the PBPK model and maturation functions of clearance and cortisol binding globulin (CBG) were developed based on previous studies published in the literature.
Results:
Clearance (CL) estimates from both approaches were similar for children older than 1 year (CL/F increasing from around 150 L/h to 500 L/h), while CBG concentrations differed across the whole age range (CBG(NLME) stable around 0.5 mu M vs. steady increase from 0.35 to 0.8 mu M for CBG (PBPK)). PBPK-derived maturation functions were subsequently included in the NLME model. After inclusion of the maturation functions, none, a part of, or all parameters were re-estimated. However, the inclusion of CL and/or CBG maturation functions in the NLME model did not result in improved model performance for the CL maturation function (& UDelta;OFV > -15.36) and the re-estimation of parameters using the CBG maturation function most often led to unstable models or individual CL prediction bias.
Discussion:
Three explanations for the observed discrepancies could be postulated, i) non-considered maturation of processes such as absorption or first-pass effect, ii) lack of patients between 1 and 12 months, iii) lack of correction of PBPK CL maturation functions derived from urinary concentration ratio data for the renal function relative to adults. These should be investigated in the future to determine how NLME and PBPK methods can work towards deriving insights into pediatric hydrocortisone PK.
The objectives of this study were the identification in (morbidly) obese and nonobese patients of (i) the most appropriate body size descriptor for fosfomycin dose adjustments and (ii) adequacy of the currently employed dosing regimens. Plasma and target site (interstitial fluid of subcutaneous adipose tissue) concentrations after fosfomycin administration (8 g) to 30 surgery patients (15 obese/15 nonobese) were obtained from a prospective clinical trial. After characterization of plasma and microdialysis-derived target site pharmacokinetics via population analysis, short-term infusions of fosfomycin 3 to 4 times daily were simulated. The adequacy of therapy was assessed by probability of pharmacokinetic/pharmacodynamic target attainment (PTA) analysis based on the unbound drug-related targets of an %fT(>= MIC) (the fraction of time that unbound fosfomycin concentrations exceed the MIC during 24 h) of 70 and an fAUC(0-24h)/MIC (the area under the concentration-time curve from 0 to 24 h for the unbound fraction of fosfomycin relative to the MIC) of 40.8 to 83.3. Lean body weight, fat mass, and creatinine clearance calculated via adjusted body weight (ABW) (CLCRCG_ABW) of all patients (body mass index [BMI] = 20.1 to 52.0 kg/m(2)) explained a considerable proportion of between-patient pharmacokinetic variability (up to 31.0% relative reduction). The steady-state unbound target site/plasma concentration ratio was 26.3% lower in (morbidly) obese than nonobese patients. For infections with fosfomycin-susceptible pathogens (MIC <= 16 mg/L), intermittent "high-dosage" intravenous (i.v.) fosfomycin (8 g, three times daily) was sufficient to treat patients with a CLCRCG_ABW of,130 mL/min, irrespective of the pharmacokinetic/pharmacodynamic indices considered. For infections by Pseudomonas aeruginosa with a MIC of 32 mg/L, when the index fAUC0-24h/MIC is applied, fosfomycin might represent a promising treatment option in obese and nonobese patients, especially in combination therapy to complement beta-lactams, in which carbapenem-resistant P. aeruginosa is critical. In conclusion, fosfomycin showed excellent target site penetration in obese and nonobese patients. Dosing should be guided by renal function rather than obesity status.
The drug concentrations targeted in meropenem and piperacillin/tazobactam therapy also depend on the susceptibility of the pathogen. Yet, the pathogen is often unknown, and antibiotic therapy is guided by empirical targets. To reliably achieve the targeted concentrations, dosing needs to be adjusted for renal function. We aimed to evaluate a meropenem and piperacillin/tazobactam monitoring program in intensive care unit (ICU) patients by assessing (i) the adequacy of locally selected empirical targets, (ii) if dosing is adequately adjusted for renal function and individual target, and (iii) if dosing is adjusted in target attainment (TA) failure. In a prospective, observational clinical trial of drug concentrations, relevant patient characteristics and microbiological data (pathogen, minimum inhibitory concentration (MIC)) for patients receiving meropenem or piperacillin/tazobactam treatment were collected. If the MIC value was available, a target range of 1-5 x MIC was selected for minimum drug concentrations of both drugs. If the MIC value was not available, 8-40 mg/L and 16-80 mg/L were selected as empirical target ranges for meropenem and piperacillin, respectively. A total of 356 meropenem and 216 piperacillin samples were collected from 108 and 96 ICU patients, respectively. The vast majority of observed MIC values was lower than the empirical target (meropenem: 90.0%, piperacillin: 93.9%), suggesting empirical target value reductions. TA was found to be low (meropenem: 35.7%, piperacillin 50.5%) with the lowest TA for severely impaired renal function (meropenem: 13.9%, piperacillin: 29.2%), and observed drug concentrations did not significantly differ between patients with different targets, indicating dosing was not adequately adjusted for renal function or target. Dosing adjustments were rare for both drugs (meropenem: 6.13%, piperacillin: 4.78%) and for meropenem irrespective of TA, revealing that concentration monitoring alone was insufficient to guide dosing adjustment. Empirical targets should regularly be assessed and adjusted based on local susceptibility data. To improve TA, scientific knowledge should be translated into easy-to-use dosing strategies guiding antibiotic dosing.
The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative
(2022)
The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg-Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.
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.
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.
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.
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.
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.
Ground motion with strong-velocity pulses can cause significant damage to buildings and structures at certain periods; hence, knowing the period and velocity amplitude of such pulses is critical for earthquake structural engineering.
However, the physical factors relating the scaling of pulse periods with magnitude are poorly understood.
In this study, we investigate moderate but damaging earthquakes (M-w 6-7) and characterize ground- motion pulses using the method of Shahi and Baker (2014) while considering the potential static-offset effects.
We confirm that the within-event variability of the pulses is large. The identified pulses in this study are mostly from strike-slip-like earthquakes. We further perform simulations using the freq uency-wavenumber algorithm to investigate the causes of the variability of the pulse periods within and between events for moderate strike-slip earthquakes.
We test the effect of fault dips, and the impact of the asperity locations and sizes. The simulations reveal that the asperity properties have a high impact on the pulse periods and amplitudes at nearby stations.
Our results emphasize the importance of asperity characteristics, in addition to earthquake magnitudes for the occurrence and properties of pulses produced by the forward directivity effect.
We finally quantify and discuss within- and between-event variabilities of pulse properties at short distances.
The spatio-temporal epidemic type aftershock sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a fully Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l'Aquila region, Italy, with the devastating event on 6 April 2009, is presented.
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.
Background:
Anti-TNFα monoclonal antibodies (mAbs) are a well-established treatment for patients with Crohn’s disease (CD). However, subtherapeutic concentrations of mAbs have been related to a loss of response during the first year of therapy1. Therefore, an appropriate dosing strategy is crucial to prevent the underexposure of mAbs for those patients. The aim of our study was to assess the impact of different dosing strategies (fixed dose or body size descriptor adapted) on drug exposure and the target concentration attainment for two different anti-TNFα mAbs: infliximab (IFX, body weight (BW)-based dosing) and certolizumab pegol (CZP, fixed dosing). For this purpose, a comprehensive pharmacokinetic (PK) simulation study was performed.
Methods:
A virtual population of 1000 clinically representative CD patients was generated based on the distribution of CD patient characteristics from an in-house clinical database (n = 116). Seven dosing regimens were investigated: fixed dose and per BW, lean BW (LBW), body surface area, height, body mass index and fat-free mass. The individual body size-adjusted doses were calculated from patient generated body size descriptor values. Then, using published PK models for IFX and CZP in CD patients2,3, for each patient, 1000 concentration–time profiles were simulated to consider the typical profile of a specific patient as well as the range of possible individual profiles due to unexplained PK variability across patients. For each dosing strategy, the variability in maximum and minimum mAb concentrations (Cmax and Cmin, respectively), area under the concentration-time curve (AUC) and the per cent of patients reaching target concentration were assessed during maintenance therapy.
Results:
For IFX and CZP, Cmin showed the highest variability between patients (CV ≈110% and CV ≈80%, respectively) with a similar extent across all dosing strategies. For IFX, the per cent of patients reaching the target (Cmin = 5 µg/ml) was similar across all dosing strategies (~15%). For CZP, the per cent of patients reaching the target average concentration of 17 µg/ml ranged substantially (52–71%), being the highest for LBW-adjusted dosing.
Conclusion:
By using a PK simulation approach, different dosing regimen of IFX and CZP revealed the highest variability for Cmin, the most commonly used PK parameter guiding treatment decisions, independent upon dosing regimen. Our results demonstrate similar target attainment with fixed dosing of IFX compared with currently recommended BW-based dosing. For CZP, the current fixed dosing strategy leads to comparable percentage of patients reaching target as the best performing body size-adjusted dosing (66% vs. 71%, respectively).
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.
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.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
Model uncertainty quantification is an essential component of effective data assimilation. Model errors associated with sub-grid scale processes are often represented through stochastic parameterizations of the unresolved process. Many existing Stochastic Parameterization schemes are only applicable when knowledge of the true sub-grid scale process or full observations of the coarse scale process are available, which is typically not the case in real applications. We present a methodology for estimating the statistics of sub-grid scale processes for the more realistic case that only partial observations of the coarse scale process are available. Model error realizations are estimated over a training period by minimizing their conditional sum of squared deviations given some informative covariates (e.g., state of the system), constrained by available observations and assuming that the observation errors are smaller than the model errors. From these realizations a conditional probability distribution of additive model errors given these covariates is obtained, allowing for complex non-Gaussian error structures. Random draws from this density are then used in actual ensemble data assimilation experiments. We demonstrate the efficacy of the approach through numerical experiments with the multi-scale Lorenz 96 system using both small and large time scale separations between slow (coarse scale) and fast (fine scale) variables. The resulting error estimates and forecasts obtained with this new method are superior to those from two existing methods.
We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.
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.
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.
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.
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 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.
Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects
(2022)
Traditionally, Hawkes processes are used to model time-continuous point processes with history dependence. Here, we propose an extended model where the self-effects are of both excitatory and inhibitory types and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and derive an inference algorithm by performing inference on an aggregated sum of Gaussian Processes. Approximate Bayesian inference is achieved via data augmentation, and we describe a mean-field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from different domains and compare it to previously reported results.
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.
We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.
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.
Let X be an infinite linearly ordered set and let Y be a nonempty subset of X. We calculate the relative rank of the semigroup OP(X,Y) of all orientation-preserving transformations on X with restricted range Y modulo the semigroup O(X,Y) of all order-preserving transformations on X with restricted range Y. For Y = X, we characterize the relative generating sets of minimal size.
Forecast verification
(2021)
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.
In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.
The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method, the repeated solution to a shifted linear system of equations is necessary. For very large systems this solve is usually implemented using iterative methods, leading to inexact solves within this inner iteration (and therefore to "inexact methods"). We will show that one can terminate this inner iteration before full precision has been reached and still obtain very good accuracy in the final solution to the Lyapunov equation. In particular, for both the RKSM and the LR-ADI method we derive theory for a relaxation strategy (e.g. increasing the solve tolerance of the inner iteration, as the outer iteration proceeds) within the iterative methods for solving the large linear systems. These theoretical choices involve unknown quantities, therefore practical criteria for relaxing the solution tolerance within the inner linear system are then provided. The theory is supported by several numerical examples, which show that the total amount of work for solving Lyapunov equations can be reduced significantly.
Nonparametric goodness-of-fit testing for parametric covariate models in pharmacometric analyses
(2021)
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.
Alpine ecosystems on the Tibetan Plateau are being threatened by ongoing climate warming and intensified human activities. Ecological time-series obtained from sedimentary ancient DNA (sedaDNA) are essential for understanding past ecosystem and biodiversity dynamics on the Tibetan Plateau and their responses to climate change at a high taxonomic resolution. Hitherto only few but promising studies have been published on this topic. The potential and limitations of using sedaDNA on the Tibetan Plateau are not fully understood. Here, we (i) provide updated knowledge of and a brief introduction to the suitable archives, region-specific taphonomy, state-of-the-art methodologies, and research questions of sedaDNA on the Tibetan Plateau; (ii) review published and ongoing sedaDNA studies from the Tibetan Plateau; and (iii) give some recommendations for future sedaDNA study designs. Based on the current knowledge of taphonomy, we infer that deep glacial lakes with freshwater and high clay sediment input, such as those from the southern and southeastern Tibetan Plateau, may have a high potential for sedaDNA studies. Metabarcoding (for microorganisms and plants), metagenomics (for ecosystems), and hybridization capture (for prehistoric humans) are three primary sedaDNA approaches which have been successfully applied on the Tibetan Plateau, but their power is still limited by several technical issues, such as PCR bias and incompleteness of taxonomic reference databases. Setting up high-quality and open-access regional taxonomic reference databases for the Tibetan Plateau should be given priority in the future. To conclude, the archival, taphonomic, and methodological conditions of the Tibetan Plateau are favorable for performing sedaDNA studies. More research should be encouraged to address questions about long-term ecological dynamics at ecosystem scale and to bring the paleoecology of the Tibetan Plateau into a new era.
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.
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.
The Kramers problem for SDEs driven by small, accelerated Lévy noise with exponentially light jumps
(2021)
We establish Freidlin-Wentzell results for a nonlinear ordinary differential equation starting close to the stable state 0, say, subject to a perturbation by a stochastic integral which is driven by an epsilon-small and (1/epsilon)-accelerated Levy process with exponentially light jumps. For this purpose, we derive a large deviations principle for the stochastically perturbed system using the weak convergence approach developed by Budhiraja, Dupuis, Maroulas and collaborators in recent years. In the sequel, we solve the associated asymptotic first escape problem from the bounded neighborhood of 0 in the limit as epsilon -> 0 which is also known as the Kramers problem in the literature.
Randomised one-step time integration methods for deterministic operator differential equations
(2022)
Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings.
Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.
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.
The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in R-d with the formal Hamiltonian defined as the volume of Ux epsilon omega B1(x), where. is a locally finite configuration of points and B-1(x) denotes the unit closed ball centred at x. The model is also tuned by two other parameters: the activity z > 0 related to the intensity of the process and the inverse temperature beta >= 0 related to the strength of the interaction. In the present paper we investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than 2r > 0, we show that for any beta >= 0, there exists 0 <(similar to a)(zc) (beta, r) < +infinity such that an exponential decay of connectivity at distance n occurs in the subcritical phase (i.e. z <(similar to a)(zc) (beta, r)) and a linear lower bound of the connection at infinity holds in the supercritical case (i.e. z >(similar to a)(zc) (beta, r)). These results are in the spirit of recent works using the theory of randomised tree algorithms (Probab. Theory Related Fields 173 (2019) 479-490, Ann. of Math. 189 (2019) 75-99, Duminil-Copin, Raoufi and Tassion (2018)). Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters z, beta. Old results (Phys. Rev. Lett. 27 (1971) 1040-1041, J. Chem. Phys. 52 (1970) 1670-1684) claim that a non-uniqueness regime occurs for z = beta large enough and it is conjectured that the uniqueness should hold outside such an half line ( z = beta >= beta(c) > 0). We solve partially this conjecture in any dimension by showing that for beta large enough the non-uniqueness holds if and only if z = beta. We show also that this critical value z = beta corresponds to the percolation threshold (similar to a)(zc) (beta, r) = beta for beta large enough, providing a straight connection between these two notions of phase transition.
In June 2018, after 4 years of cruise, the Japanese space probe Hayabusa2 [1-Watanabe S. et al.: Hayabusa2 Mission Overview. (2017)] reached the Near-Earth Asteroid (162173) Ryugu. Hayabusa2 carried a small Lander named MASCOT (Mobile Asteroid Surface Scout) [2-Ho T. M. et al.: MASCOT-The Mobile Asteroid Surface Scout onboard the Hayabusa2 mission. (2017)], jointly developed by the German Aerospace Center (DLR) and the French Space Agency (CNES), to investigate Ryugu's surface structure, composition and physical properties including its thermal behaviour and magnetization in-situ. The Microgravity User Support Centre (DLR-MUSC) in Cologne was in charge of providing all thermal conditions and constraints necessary for the selection of the final landing site and for the final operations of the Lander MASCOT on the surface of the asteroid Ryugu. This article provides a comprehensive assessment of these thermal conditions and constraints, based on predictions performed with the Thermal Mathematical Model (TMM) of MASCOT using different asteroid surface thermal models, ephemeris data for approach as well as descent and hopping trajectories, the related operation sequences and scenarios and the possible environmental conditions driven by the Hayabusa2 spacecraft. A comparison with the real telemetry data confirms the analysis and provides further information about the asteroid characteristics.
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.
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.
Point processes are a common methodology to model sets of events. From earthquakes to social media posts, from the arrival times of neuronal spikes to the timing of crimes, from stock prices to disease spreading -- these phenomena can be reduced to the occurrences of events concentrated in points. Often, these events happen one after the other defining a time--series.
Models of point processes can be used to deepen our understanding of such events and for classification and prediction. Such models include an underlying random process that generates the events. This work uses Bayesian methodology to infer the underlying generative process from observed data. Our contribution is twofold -- we develop new models and new inference methods for these processes.
We propose a model that extends the family of point processes where the occurrence of an event depends on the previous events. This family is known as Hawkes processes. Whereas in most existing models of such processes, past events are assumed to have only an excitatory effect on future events, we focus on the newly developed nonlinear Hawkes process, where past events could have excitatory and inhibitory effects. After defining the model, we present its inference method and apply it to data from different fields, among others, to neuronal activity.
The second model described in the thesis concerns a specific instance of point processes --- the decision process underlying human gaze control. This process results in a series of fixated locations in an image. We developed a new model to describe this process, motivated by the known Exploration--Exploitation dilemma. Alongside the model, we present a Bayesian inference algorithm to infer the model parameters.
Remaining in the realm of human scene viewing, we identify the lack of best practices for Bayesian inference in this field. We survey four popular algorithms and compare their performances for parameter inference in two scan path models.
The novel models and inference algorithms presented in this dissertation enrich the understanding of point process data and allow us to uncover meaningful insights.
Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies.
Amoeboid cell motility takes place in a variety of biomedical processes such as cancer metastasis, embryonic morphogenesis, and wound healing. In contrast to other forms of cell motility, it is mainly driven by substantial cell shape changes. Based on the interplay of explorative membrane protrusions at the front and a slower-acting membrane retraction at the rear, the cell moves in a crawling kind of way. Underlying these protrusions and retractions are multiple physiological processes resulting in changes of the cytoskeleton, a meshwork of different multi-functional proteins. The complexity and versatility of amoeboid cell motility raise the need for novel computational models based on a profound theoretical framework to analyze and simulate the dynamics of the cell shape.
The objective of this thesis is the development of (i) a mathematical framework to describe contour dynamics in time and space, (ii) a computational model to infer expansion and retraction characteristics of individual cell tracks and to produce realistic contour dynamics, (iii) and a complementing Open Science approach to make the above methods fully accessible and easy to use.
In this work, we mainly used single-cell recordings of the model organism Dictyostelium discoideum. Based on stacks of segmented microscopy images, we apply a Bayesian approach to obtain smooth representations of the cell membrane, so-called cell contours. We introduce a one-parameter family of regularized contour flows to track reference points on the contour (virtual markers) in time and space. This way, we define a coordinate system to visualize local geometric and dynamic quantities of individual contour dynamics in so-called kymograph plots. In particular, we introduce the local marker dispersion as a measure to identify membrane protrusions and retractions in a fully automated way.
This mathematical framework is the basis of a novel contour dynamics model, which consists of three biophysiologically motivated components: one stochastic term, accounting for membrane protrusions, and two deterministic terms to control the shape and area of the contour, which account for membrane retractions. Our model provides a fully automated approach to infer protrusion and retraction characteristics from experimental cell tracks while being also capable of simulating realistic and qualitatively different contour dynamics. Furthermore, the model is used to classify two different locomotion types: the amoeboid and a so-called fan-shaped type.
With the complementing Open Science approach, we ensure a high standard regarding the usability of our methods and the reproducibility of our research. In this context, we introduce our software publication named AmoePy, an open-source Python package to segment, analyze, and simulate amoeboid cell motility. Furthermore, we describe measures to improve its usability and extensibility, e.g., by detailed run instructions and an automatically generated source code documentation, and to ensure its functionality and stability, e.g., by automatic software tests, data validation, and a hierarchical package structure.
The mathematical approaches of this work provide substantial improvements regarding the modeling and analysis of amoeboid cell motility. We deem the above methods, due to their generalized nature, to be of greater value for other scientific applications, e.g., varying organisms and experimental setups or the transition from unicellular to multicellular movement. Furthermore, we enable other researchers from different fields, i.e., mathematics, biophysics, and medicine, to apply our mathematical methods. By following Open Science standards, this work is of greater value for the cell migration community and a potential role model for other Open Science contributions.
We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.
A characterization of the essential spectrum of Schrodinger operators on infinite graphs is derived involving the concept of R-limits. This concept, which was introduced previously for operators on N and Z(d) as "right-limits," captures the behaviour of the operator at infinity. For graphs with sub-exponential growth rate, we show that each point in sigma(ss)(H) corresponds to a bounded generalized eigenfunction of a corresponding R-limit of H. If, additionally, the graph is of uniform sub-exponential growth, also the converse inclusion holds.
We describe a new, original approach to the modelling of the Earth's magnetic field. The overall objective of this study is to reliably render fast variations of the core field and its secular variation. This method combines a sequential modelling approach, a Kalman filter, and a correlation-based modelling step. Sources that most significantly contribute to the field measured at the surface of the Earth are modelled. Their separation is based on strong prior information on their spatial and temporal behaviours. We obtain a time series of model distributions which display behaviours similar to those of recent models based on more classic approaches, particularly at large temporal and spatial scales. Interesting new features and periodicities are visible in our models at smaller time and spatial scales. An important aspect of our method is to yield reliable error bars for all model parameters. These errors, however, are only as reliable as the description of the different sources and the prior information used are realistic. Finally, we used a slightly different version of our method to produce candidate models for the thirteenth edition of the International Geomagnetic Reference Field.
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.
Ulcerative colitis (UC) is part of the inflammatory bowels diseases, and moderate to severe UC patients can be treated with anti-tumour necrosis alpha monoclonal antibodies, including infliximab (IFX). Even though treatment of UC patients by IFX has been in place for over a decade, many gaps in modelling of IFX PK in this population remain. This is even more true for acute severe UC (ASUC) patients for which early prediction of IFX pharmacokinetic (PK) could highly improve treatment outcome. Thus, this review aims to compile and analyse published population PK models of IFX in UC and ASUC patients, and to assess the current knowledge on disease activity impact on IFX PK. For this, a semi-systematic literature search was conducted, from which 26 publications including a population PK model analysis of UC patients receiving IFX therapy were selected. Amongst those, only four developed a model specifically for UC patients, and only three populations included severe UC patients. Investigations of disease activity impact on PK were reported in only 4 of the 14 models selected. In addition, the lack of reported model codes and assessment of predictive performance make the use of published models in a clinical setting challenging. Thus, more comprehensive investigation of PK in UC and ASUC is needed as well as more adequate reports on developed models and their evaluation in order to apply them in a clinical setting.
We present a new model of the geomagnetic field spanning the last 20 years and called Kalmag. Deriving from the assimilation of CHAMP and Swarm vector field measurements, it separates the different contributions to the observable field through parameterized prior covariance matrices. To make the inverse problem numerically feasible, it has been sequentialized in time through the combination of a Kalman filter and a smoothing algorithm. The model provides reliable estimates of past, present and future mean fields and associated uncertainties. The version presented here is an update of our IGRF candidates; the amount of assimilated data has been doubled and the considered time window has been extended from [2000.5, 2019.74] to [2000.5, 2020.33].
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.
Background
Cytochrome P450 (CYP) 3A contributes to the metabolism of many approved drugs. CYP3A perpetrator drugs can profoundly alter the exposure of CYP3A substrates. However, effects of such drug-drug interactions are usually reported as maximum effects rather than studied as time-dependent processes. Identification of the time course of CYP3A modulation can provide insight into when significant changes to CYP3A activity occurs, help better design drug-drug interaction studies, and manage drug-drug interactions in clinical practice.
Objective
We aimed to quantify the time course and extent of the in vivo modulation of different CYP3A perpetrator drugs on hepatic CYP3A activity and distinguish different modulatory mechanisms by their time of onset, using pharmacologically inactive intravenous microgram doses of the CYP3A-specific substrate midazolam, as a marker of CYP3A activity.
Methods
Twenty-four healthy individuals received an intravenous midazolam bolus followed by a continuous infusion for 10 or 36 h. Individuals were randomized into four arms: within each arm, two individuals served as a placebo control and, 2 h after start of the midazolam infusion, four individuals received the CYP3A perpetrator drug: voriconazole (inhibitor, orally or intravenously), rifampicin (inducer, orally), or efavirenz (activator, orally). After midazolam bolus administration, blood samples were taken every hour (rifampicin arm) or every 15 min (remaining study arms) until the end of midazolam infusion. A total of 1858 concentrations were equally divided between midazolam and its metabolite, 1'-hydroxymidazolam. A nonlinear mixed-effects population pharmacokinetic model of both compounds was developed using NONMEM (R). CYP3A activity modulation was quantified over time, as the relative change of midazolam clearance encountered by the perpetrator drug, compared to the corresponding clearance value in the placebo arm.
Results
Time course of CYP3A modulation and magnitude of maximum effect were identified for each perpetrator drug. While efavirenz CYP3A activation was relatively fast and short, reaching a maximum after approximately 2-3 h, the induction effect of rifampicin could only be observed after 22 h, with a maximum after approximately 28-30 h followed by a steep drop to almost baseline within 1-2 h. In contrast, the inhibitory impact of both oral and intravenous voriconazole was prolonged with a steady inhibition of CYP3A activity followed by a gradual increase in the inhibitory effect until the end of sampling at 8 h. Relative maximum clearance changes were +59.1%, +46.7%, -70.6%, and -61.1% for efavirenz, rifampicin, oral voriconazole, and intravenous voriconazole, respectively.
Conclusions
We could distinguish between different mechanisms of CYP3A modulation by the time of onset. Identification of the time at which clearance significantly changes, per perpetrator drug, can guide the design of an optimal sampling schedule for future drug-drug interaction studies. The impact of a short-term combination of different perpetrator drugs on the paradigm CYP3A substrate midazolam was characterized and can define combination intervals in which no relevant interaction is to be expected.
The purpose of this paper is to build an algebraic framework suited to regularize branched structures emanating from rooted forests and which encodes the locality principle. This is achieved by means of the universal properties in the locality framework of properly decorated rooted forests. These universal properties are then applied to derive the multivariate regularization of integrals indexed by rooted forests. We study their renormalization, along the lines of Kreimer's toy model for Feynman integrals.
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.
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.
For the time stationary global geomagnetic field, a new modelling concept is presented. A Bayesian non-parametric approach provides realistic location dependent uncertainty estimates. Modelling related variabilities are dealt with systematically by making little subjective apriori assumptions. Rather than parametrizing the model by Gauss coefficients, a functional analytic approach is applied. The geomagnetic potential is assumed a Gaussian process to describe a distribution over functions. Apriori correlations are given by an explicit kernel function with non-informative dipole contribution. A refined modelling strategy is proposed that accommodates non-linearities of archeomagnetic observables: First, a rough field estimate is obtained considering only sites that provide full field vector records. Subsequently, this estimate supports the linearization that incorporates the remaining incomplete records. The comparison of results for the archeomagnetic field over the past 1000 yr is in general agreement with previous models while improved model uncertainty estimates are provided.
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.
In this paper 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 for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, 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. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.
In this paper 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 for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, 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. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.
In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.
In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.
Global numerical weather prediction (NWP) models have begun to resolve the mesoscale k(-5/3) range of the energy spectrum, which is known to impose an inherently finite range of deterministic predictability per se as errors develop more rapidly on these scales than on the larger scales. However, the dynamics of these errors under the influence of the synoptic-scale k(-3) range is little studied. Within a perfect-model context, the present work examines the error growth behavior under such a hybrid spectrum in Lorenz's original model of 1969, and in a series of identical-twin perturbation experiments using an idealized two-dimensional barotropic turbulence model at a range of resolutions. With the typical resolution of today's global NWP ensembles, error growth remains largely uniform across scales. The theoretically expected fast error growth characteristic of a k(-5/3) spectrum is seen to be largely suppressed in the first decade of the mesoscale range by the synoptic-scale k(-3) range. However, it emerges once models become fully able to resolve features on something like a 20-km scale, which corresponds to a grid resolution on the order of a few kilometers.
This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson-Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay-Rudin-Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.
This thesis bridges two areas of mathematics, algebra on the one hand with the Milnor-Moore theorem (also called Cartier-Quillen-Milnor-Moore theorem) as well as the Poincaré-Birkhoff-Witt theorem, and analysis on the other hand with Shintani zeta functions which generalise multiple zeta functions.
The first part is devoted to an algebraic formulation of the locality principle in physics and generalisations of classification theorems such as Milnor-Moore and Poincaré-Birkhoff-Witt theorems to the locality framework. The locality principle roughly says that events that take place far apart in spacetime do not infuence each other. The algebraic formulation of this principle discussed here is useful when analysing singularities which arise from events located far apart in space, in order to renormalise them while keeping a memory of the fact that they do not influence each other. We start by endowing a vector space with a symmetric relation, named the locality relation, which keeps track of elements that are "locally independent". The pair of a vector space together with such relation is called a pre-locality vector space. This concept is extended to tensor products allowing only tensors made of locally independent elements. We extend this concept to the locality tensor algebra, and locality symmetric algebra of a pre-locality vector space and prove the universal properties of each of such structures. We also introduce the pre-locality Lie algebras, together with their associated locality universal enveloping algebras and prove their universal property. We later upgrade all such structures and results from the pre-locality to the locality context, requiring the locality relation to be compatible with the linear structure of the vector space. This allows us to define locality coalgebras, locality bialgebras, and locality Hopf algebras. Finally, all the previous results are used to prove the locality version of the Milnor-Moore and the Poincaré-Birkhoff-Witt theorems. It is worth noticing that the proofs presented, not only generalise the results in the usual (non-locality) setup, but also often use less tools than their counterparts in their non-locality counterparts.
The second part is devoted to study the polar structure of the Shintani zeta functions. Such functions, which generalise the Riemman zeta function, multiple zeta functions, Mordell-Tornheim zeta functions, among others, are parametrised by matrices with real non-negative arguments. It is known that Shintani zeta functions extend to meromorphic functions with poles on afine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coeficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.
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 show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
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 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
Purpose The anatomy of the circle of Willis (CoW), the brain's main arterial blood supply system, strongly differs between individuals, resulting in highly variable flow fields and intracranial vascularization patterns. To predict subject-specific hemodynamics with high certainty, we propose a data assimilation (DA) approach that merges fully 4D phase-contrast magnetic resonance imaging (PC-MRI) data with a numerical model in the form of computational fluid dynamics (CFD) simulations. Methods To the best of our knowledge, this study is the first to provide a transient state estimate for the three-dimensional velocity field in a subject-specific CoW geometry using DA. High-resolution velocity state estimates are obtained using the local ensemble transform Kalman filter (LETKF). Results Quantitative evaluation shows a considerable reduction (up to 90%) in the uncertainty of the velocity field state estimate after the data assimilation step. Velocity values in vessel areas that are below the resolution of the PC-MRI data (e.g., in posterior communicating arteries) are provided. Furthermore, the uncertainty of the analysis-based wall shear stress distribution is reduced by a factor of 2 for the data assimilation approach when compared to the CFD model alone. Conclusion This study demonstrates the potential of data assimilation to provide detailed information on vascular flow, and to reduce the uncertainty in such estimates by combining various sources of data in a statistically appropriate fashion.
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.
The estimation of a log-concave density on R is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.
Conventional embeddings of the edge-graphs of Platonic polyhedra, {f,z}, where f,z denote the number of edges in each face and the edge-valence at each vertex, respectively, are untangled in that they can be placed on a sphere (S-2) such that distinct edges do not intersect, analogous to unknotted loops, which allow crossing-free drawings of S-1 on the sphere. The most symmetric (flag-transitive) realizations of those polyhedral graphs are those of the classical Platonic polyhedra, whose symmetries are *2fz, according to Conway's two-dimensional (2D) orbifold notation (equivalent to Schonflies symbols I-h, O-h, and T-d). Tangled Platonic {f,z} polyhedra-which cannot lie on the sphere without edge-crossings-are constructed as windings of helices with three, five, seven,... strands on multigenus surfaces formed by tubifying the edges of conventional Platonic polyhedra, have (chiral) symmetries 2fz (I, O, and T), whose vertices, edges, and faces are symmetrically identical, realized with two flags. The analysis extends to the "theta(z)" polyhedra, {2,z}. The vertices of these symmetric tangled polyhedra overlap with those of the Platonic polyhedra; however, their helicity requires curvilinear (or kinked) edges in all but one case. We show that these 2fz polyhedral tangles are maximally symmetric; more symmetric embeddings are necessarily untangled. On one hand, their topologies are very constrained: They are either self-entangled graphs (analogous to knots) or mutually catenated entangled compound polyhedra (analogous to links). On the other hand, an endless variety of entanglements can be realized for each topology. Simpler examples resemble patterns observed in synthetic organometallic materials and clathrin coats in vivo.