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Quasi-Monte Carlo for Bayesian design of experiment problems governed by parametric PDEs
No full textThis paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration problems over high-dimensional parameter and data domains arising in Bayesian optimal design problems. We provide a detailed analysis for these double integration problems using two approaches: a full tensor product and a sparse tensor product combination of quasi-Monte Carlo (QMC) cubature rules over the parameter and data domains. Specifically, we show that the latter approach significantly improves the convergence rate, exhibiting performance comparable to that of QMC integration of a single high-dimensional integral. Furthermore, we numerically verify the predicted convergence rates for an elliptic PDE problem with an unknown diffusion coefficient in two spatial dimensions, offering empirical evidence supporting the theoretical results and highlighting practical applicability
A Ridge-based Approach for Extraction and Visualization of 3D Atmospheric Fronts
No full textAn atmospheric front is an imaginary surface that separates two distinct air masses and is commonly defined as the warm-air side of a frontal zone with high gradients of atmospheric temperature and humidity (Fig. 1, left). These fronts are a widely used conceptual model in meteorology, which are often encountered in the literature as two-dimensional (2D) front lines on surface analysis charts. This paper presents a method for computing three-dimensional (3D) atmospheric fronts as surfaces that is capable of extracting continuous and well-confined features suitable for 3D visual analysis, spatiotemporal tracking, and statistical analyses (Fig. 1, middle, right). Recently developed contour-based methods for 3D front extraction rely on computing the third derivative of a moist potential temperature field. Additionally, they require the field to be smoothed to obtain continuous large-scale structures. This paper demonstrates the feasibility of an alternative method to front extraction using ridge surface computation. The proposed method requires only the second derivative of the input field and produces accurate structures even from unsmoothed data. An application of the ridge-based method to a data set corresponding to Cyclone Friederike demonstrates its benefits and utility towards visual analysis of the full 3D structure of fronts
Disease Burden and Inpatient Management of Children with Acute Respiratory Viral Infections during the Pre-COVID Era in Germany: A Cost-of-Illness Study
No full textRespiratory viral infections (RVIs) are common reasons for healthcare consultations. The inpatient management of RVIs consumes significant resources. From 2009 to 2014, we assessed the costs of RVI management in 4776 hospitalized children aged 0–18 years participating in a quality improvement program, where all ILI patients underwent virologic testing at the National Reference Centre followed by detailed recording of their clinical course. The direct (medical or non-medical) and indirect costs of inpatient management outside the ICU (‘non-ICU’) versus management requiring ICU care (‘ICU’) added up to EUR 2767.14 (non-ICU) vs. EUR 29,941.71 (ICU) for influenza, EUR 2713.14 (non-ICU) vs. EUR 16,951.06 (ICU) for RSV infections, and EUR 2767.33 (non-ICU) vs. EUR 14,394.02 (ICU) for human rhinovirus (hRV) infections, respectively. Non-ICU inpatient costs were similar for all eight RVIs studied: influenza, RSV, hRV, adenovirus (hAdV), metapneumovirus (hMPV), parainfluenza virus (hPIV), bocavirus (hBoV), and seasonal coronavirus (hCoV) infections. ICU costs for influenza, however, exceeded all other RVIs. At the time of the study, influenza was the only RVI with antiviral treatment options available for children, but only 9.8 of influenza patients (non-ICU) and 1.5 of ICU patients with influenza received antivirals; only 2.9 were vaccinated. Future studies should investigate the economic impact of treatment and prevention of influenza, COVID-19, and RSV post vaccine introduction
Hybrid PDE-ODE Models for Efficient Simulation of Infection Spread in Epidemiology
No full textThis paper introduces a novel hybrid mathematical modeling approach that effectively couples Partial Differential Equations (PDEs) with Ordinary Differential Equations (ODEs), exemplified through the simulation of epidemiological processes. The hybrid model aims to integrate the spatially detailed representation of disease dynamics provided by PDEs with the computational efficiency of ODEs. In the presented epidemiological use-case, this integration allows for the rapid assessment of public health interventions and the potential impact of infectious diseases across large populations. We discuss the theoretical formulation of the hybrid PDE-ODE model, including the governing equations and boundary conditions. The model's capabilities are demonstrated through detailed simulations of disease spread in synthetic environments and real-world scenarios, specifically focusing on the regions of Lombardy, Italy, and Berlin, Germany. Results indicate that the hybrid model achieves a balance between computational speed and accuracy, making it a valuable tool for policymakers in real-time decision-making and scenario analysis in epidemiology and potentially in other fields requiring similar modeling approaches
Multilevel Parareal Algorithm with Averaging for Oscillatory Problems
No full texthe present study is an extension of the work done by Peddle, Haut, and Wingate [SIAM J. Sci. Comput., 41 (2019), pp. A3476–A3497] and Haut and Wingate [SIAM J. Sci. Comput., 36 (2014), pp. A693–A713], where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples
Quasi-Monte Carlo and discontinuous Galerkin
No full textIn this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field, and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Notably, the parametric regularity bounds for DG, which are developed in this work, are also useful for other methods such as sparse grids. Numerical results underline our analytical findings
Molecular Stokes-Einstein and Stokes-Einstein-Debye relations for water including viscosity-dependent slip and hydrodynamic radius
No full textWe perform molecular dynamics simulations of liquid water at different temperatures and calculate the water viscosity, the translational and rotational water diffusivities in the laboratory frame as well as in the comoving molecular frame. Instead of interpreting the results as deviations from the Stokes-Einstein and Stokes-Einstein-Debye relations, we describe the translational and rotational diffusivities of water molecules by three models of increasing complexity that take the structural anisotropy of water into account on different levels. We first compare simulation results to analytical predictions for a no-slip sphere and a no-slip ellipsoid. We show that the no-slip sphere can approximate laboratory-frame isotropic translational and rotational diffusivities but fails to describe the anisotropic molecular-frame diffusivities. The no-slip ellipsoid can describe the translational anisotropic molecular-frame diffusivities exactly but fails to describe the translational and rotational anisotropic molecular-frame diffusivities simultaneously. Since an ellipsoidal model with slip boundary conditions is not analytically tractable, we define a heuristic spherical model with tensorial slip lengths and tensorial hydrodynamic radii. We show that this model simultaneously describes the laboratory-frame isotropic translational and rotational diffusivities, as well as, in a restricted viscosity range, the anisotropic molecular-frame diffusivities
Making Mathematical Research Data FAIR: Pathways to Improved Data Sharing
No full textThe sharing and citation of research data is becoming increasingly recognized as an essential building block in scientific research across various fields and disciplines. Sharing research data allows other researchers to reproduce results, replicate findings, and build on them. Ultimately, this will foster faster cycles in knowledge generation. Some disciplines, such as astronomy or bioinformatics, already have a long history of sharing data; many others do not. The current landscape of available systems for sharing research data is diverse. In this article, we conduct a detailed analysis of existing web-based systems, specifically focusing on mathematical research data
Generalized dimension truncation error analysis for high-dimensional numerical integration: lognormal setting and beyond
No full textPartial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi-Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to non-affine parametric operator equations, dimensionally-truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings
An inflated dynamic Laplacian to track the emergence and disappearance of semi-material coherent sets
No full textLagrangian methods continue to stand at the forefront of the analysis of time-dependent dynamical systems. Most Lagrangian methods have criteria that must be fulfilled by trajectories as they are followed throughout a given finite flow duration. This key strength of Lagrangian methods can also be a limitation in more complex evolving environments. It places a high importance on selecting a time window that produces useful results, and these results may vary significantly with changes in the flow duration. We show how to overcome this drawback in the tracking of coherent flow features. Finite-time coherent sets (FTCS) are material objects that strongly resist mixing in complicated nonlinear flows. Like other materially coherent objects, by definition they must retain their coherence properties throughout the specified flow duration. Recent work [Froyland and Koltai, CPAM, 2023] introduced the notion of semi-material FTCS, whereby a balance is struck between the material nature and the coherence properties of FTCS. This balance provides the flexibility for FTCS to come and go, merge and separate, or undergo other changes as the governing unsteady flow experiences dramatic shifts. The purpose of this work is to illustrate the utility of the inflated dynamic Laplacian introduced in [Froyland and Koltai, CPAM, 2023] in a range of dynamical systems that are challenging to analyse by standard Lagrangian means, and to provide an efficient meshfree numerical approach for the discretisation of the inflated dynamic Laplacian