TY - JOUR A1 - Keller, Matthias A1 - Liu, Shiping A1 - Peyerimhoff, Norbert T1 - A note on eigenvalue bounds for non-compact manifolds JF - Mathematische Nachrichten N2 - 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. KW - Cheeger inequality KW - eigenvalues KW - Laplacian KW - negative curvature KW - Riemannian manifold Y1 - 2021 U6 - https://doi.org/10.1002/mana.201900209 SN - 0025-584X SN - 1522-2616 VL - 294 IS - 6 SP - 1134 EP - 1139 PB - Wiley-VCH CY - Weinheim ER - TY - JOUR A1 - Saynisch-Wagner, Jan A1 - Bärenzung, Julien A1 - Hornschild, Aaron A1 - Irrgang, Christopher A1 - Thomas, Maik T1 - Tide-induced magnetic signals and their errors derived from CHAMP and Swarm satellite magnetometer observations JF - Earth, planets and space : EPS N2 - 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. KW - Tides KW - Electromagnetic induction KW - Error covariance KW - Satellite magnetometer observations Y1 - 2021 U6 - https://doi.org/10.1186/s40623-021-01557-3 SN - 1880-5981 VL - 73 IS - 1 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Wormell, Caroline L. A1 - Reich, Sebastian T1 - Spectral convergence of diffusion maps BT - Improved error bounds and an alternative normalization JF - SIAM journal on numerical analysis / Society for Industrial and Applied Mathematics N2 - 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. KW - diffusion maps KW - graph Laplacian KW - Sinkhorn problem KW - kernel methods Y1 - 2021 U6 - https://doi.org/10.1137/20M1344093 SN - 0036-1429 SN - 1095-7170 VL - 59 IS - 3 SP - 1687 EP - 1734 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Chang, Der-Chen A1 - Khalil, Sara A1 - Schulze, Bert-Wolfgang T1 - Analysis on regular corner spaces JF - The journal of geometric analysis N2 - 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. KW - Boutet de Monvel's calculus KW - Pseudo-differential operators KW - Singular cones KW - Mellin symbols with values in the edge calculus KW - Parametrices of elliptic operators KW - Kegel space Y1 - 2021 U6 - https://doi.org/10.1007/s12220-021-00614-3 SN - 1050-6926 SN - 1559-002X VL - 31 IS - 9 SP - 9199 EP - 9240 PB - Springer CY - New York ER - TY - JOUR A1 - Rodríguez Zuluaga, Juan A1 - Stolle, Claudia A1 - Yamazaki, Yosuke A1 - Xiong, Chao A1 - England, Scott L. T1 - A synoptic-scale wavelike structure in the nighttime equatorial ionization anomaly JF - Earth and Space Science : ESS N2 - 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. KW - equatorial ionization anomaly KW - equatorial ionosphere KW - equatorial plasma bubbles KW - wave structure KW - forcing from below Y1 - 2021 U6 - https://doi.org/10.1029/2020EA001529 SN - 2333-5084 VL - 8 IS - 2 PB - American Geophysical Union CY - Malden, Mass. ER - TY - JOUR A1 - Eshghi, Nasim A1 - Mach, Thomas A1 - Reichel, Lothar T1 - New matrix function approximations and quadrature rules based on the Arnoldi process JF - Journal of computational and applied mathematics N2 - 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. KW - Arnoldi process KW - Matrix function approximation KW - Quadrature rule Y1 - 2021 U6 - https://doi.org/10.1016/j.cam.2021.113442 SN - 0377-0427 SN - 1879-1778 VL - 391 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Hethey, Christoph Philipp A1 - Hartung, Niklas A1 - Wangorsch, Gaby A1 - Weisser, Karin A1 - Huisinga, Wilhelm T1 - Physiology-based toxicokinetic modelling of aluminium in rat and man JF - Archives of toxicology : official journal of EUROTOX N2 - 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. KW - PBTK KW - Toxicokinetics KW - Al-26 KW - Aluminium Y1 - 2021 U6 - https://doi.org/10.1007/s00204-021-03107-y SN - 0340-5761 SN - 1432-0738 VL - 95 IS - 9 SP - 2977 EP - 3000 PB - Springer CY - Berlin ; Heidelberg ER - TY - JOUR A1 - Cvetković, Nada A1 - Conrad, Tim A1 - Lie, Han Cheng T1 - A convergent discretization method for transition path theory for diffusion processes JF - Multiscale modeling & simulation : a SIAM interdisciplinary journal N2 - 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. KW - ergodic diffusion processes KW - transition paths KW - rare events KW - Monte Carlo KW - methods Y1 - 2021 U6 - https://doi.org/10.1137/20M1329354 SN - 1540-3459 SN - 1540-3467 VL - 19 IS - 1 SP - 242 EP - 266 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Kempton, Mark A1 - Münch, Florentin A1 - Yau, Shing-Tung T1 - A homology vanishing theorem for graphs with positive curvature JF - Communications in analysis and geometry N2 - 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. Y1 - 2021 UR - https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2021/0029/0006/CAG-2021-0029-0006-a005.pdf U6 - https://doi.org/10.4310/CAG.2021.v29.n6.a5 SN - 1019-8385 SN - 1944-9992 VL - 29 IS - 6 SP - 1449 EP - 1473 PB - International Press of Boston CY - Somerville ER - TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Lie, Han Cheng A1 - Sullivan, Tim J. T1 - Gamma-convergence of Onsager-Machlup functionals BT - II. Infinite product measures on Banach spaces JF - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data N2 - 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. KW - Bayesian inverse problems KW - Gamma-convergence KW - maximum a posteriori KW - estimation KW - Onsager-Machlup functional KW - small ball probabilities KW - transition path theory Y1 - 2021 U6 - https://doi.org/10.1088/1361-6420/ac3f82 SN - 0266-5611 SN - 1361-6420 VL - 38 IS - 2 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Li, Han Cheng A1 - Sullivan, Tim J. T1 - Gamma-convergence of Onsager-Machlup functionals BT - I. With applications to maximum a posteriori estimation in Bayesian inverse problems JF - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data N2 - 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. KW - Bayesian inverse problems KW - Gamma-convergence KW - maximum a posteriori KW - estimation KW - Onsager-Machlup functional KW - small ball probabilities; KW - transition path theory Y1 - 2021 U6 - https://doi.org/10.1088/1361-6420/ac3f81 SN - 0266-5611 SN - 1361-6420 VL - 38 IS - 2 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Leung, Tsz Yan A1 - Leutbecher, Martin A1 - Reich, Sebastian A1 - Shepherd, Theodore G. T1 - Forecast verification BT - relating deterministic and probabilistic metrics JF - Quarterly journal of the Royal Meteorological Society N2 - 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. KW - CRPS KW - ensembles KW - idealised turbulence KW - NWP KW - RMSE KW - verification Y1 - 2021 U6 - https://doi.org/10.1002/qj.4120 SN - 0035-9009 SN - 1477-870X VL - 147 IS - 739 SP - 3124 EP - 3134 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Hartung, Niklas A1 - Wahl, Martin A1 - Rastogi, Abhishake A1 - Huisinga, Wilhelm T1 - Nonparametric goodness-of-fit testing for parametric covariate models in pharmacometric analyses JF - CPT: pharmacometrics & systems pharmacology N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1002/psp4.12614 SN - 2163-8306 VL - 10 IS - 6 SP - 564 EP - 576 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - De Oliveira Gomes, André A1 - Högele, Michael Anton T1 - The Kramers problem for SDEs driven by small, accelerated Lévy noise with exponentially light jumps JF - Stochastics and dynamics N2 - 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. KW - Freidlin-Wentzell theory KW - large deviations principle KW - accelerated small KW - noise Levy diffusions KW - first passage times KW - first exit location KW - strongly tempered stable Levy measure Y1 - 2021 U6 - https://doi.org/10.1142/S0219493721500192 SN - 0219-4937 SN - 1793-6799 VL - 21 IS - 04 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Schick, Thomas A1 - Seyedhosseini, Mehran T1 - On an index theorem of Chang, Weinberger and Yu JF - Münster journal of mathematics N2 - 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. Y1 - 2021 U6 - https://doi.org/10.17879/59019522628 SN - 1867-5778 SN - 1867-5786 VL - 14 IS - 1 SP - 123 EP - 154 PB - WWU, Fachbereich Mathematik und Informatik CY - Münster ER - TY - JOUR A1 - Cozzoni, Barbara A1 - Maibaum, Michael A1 - Hamm, Maximilian T1 - Thermal analysis and constraints for the MASCOT landing site selection on the asteroid Ryugu JF - Planetary and space science N2 - 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. KW - MASCOT KW - Thermal mathematical model KW - Landing site selection KW - Small KW - spacecraft operations Y1 - 2021 U6 - https://doi.org/10.1016/j.pss.2021.105286 SN - 0032-0633 SN - 1873-5088 VL - 205 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Reich, Sebastian A1 - Weissmann, Simon T1 - Fokker-Planck particle systems for Bayesian inference: computational approaches JF - SIAM ASA journal on uncertainty quantification N2 - 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. KW - Bayesian inverse problems KW - Fokker-Planck equation KW - gradient flow KW - affine KW - invariance KW - gradient-free sampling methods KW - localization Y1 - 2021 U6 - https://doi.org/10.1137/19M1303162 SN - 2166-2525 VL - 9 IS - 2 SP - 446 EP - 482 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Shlapunov, Alexander A1 - Tarchanov, Nikolaj Nikolaevič T1 - An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n JF - Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija N2 - 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. KW - Navier-Stokes equations KW - de Rham complex KW - open mapping theorem Y1 - 2021 U6 - https://doi.org/10.33048/semi.2021.18.108 SN - 1813-3304 VL - 18 IS - 2 SP - 1433 EP - 1466 PB - Institut Matematiki Imeni S. L. Soboleva CY - Novosibirsk ER - TY - JOUR A1 - Beckus, Siegfried A1 - Eliaz, Latif T1 - Eigenfunctions growth of R-limits on graphs JF - Journal of spectral theory / European Mathematical Society N2 - 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. KW - Essential spectrum KW - Schrodinger operators KW - graphs KW - right limits KW - generalized eigenfunctions Y1 - 2021 U6 - https://doi.org/10.4171/JST/389 SN - 1664-039X SN - 1664-0403 VL - 11 IS - 4 SP - 1895 EP - 1933 PB - EMS Press, an imprint of the European Mathematical Society - EMS - Publishing House GmbH, Institut für Mathematik, Technische Universität CY - Berlin ER - TY - JOUR A1 - Matzka, Jürgen A1 - Stolle, Claudia A1 - Yamazaki, Yosuke A1 - Bronkalla, Oliver A1 - Morschhauser, Achim T1 - The geomagnetic Kp index and derived indices of geomagnetic activity JF - Space weather : the international journal of research and applications N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1029/2020SW002641 SN - 1542-7390 VL - 19 IS - 5 PB - Wiley CY - New York ER -