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Centroid moment tensor (CMT) parameters of earthquakes are routinely estimated to gain information on structures and regional tectonics. However, for small earthquakes (M < 4), it is still challenging to determine CMTs due to the lack of high-quality waveform data. In this study, we propose to improve solutions for small earthquakes by incorporating multiple seismic data types in Bayesian joint inversion: polarities picked on broad-band signals, amplitude spectra for intermediate frequency bands (0.2–2.0 Hz), and waveforms at low frequencies (0.05–0.2 Hz). Both measurement and theory errors are accounted for by iterative estimation of non-Toeplitz covariance matrices, providing objective weightings for the different data types in the joint parameter estimation. Validity and applicability of the method are demonstrated using simulated and field data. Results demonstrate that combinations of data, such as a single high-quality waveform, a few amplitude spectra and many waveform polarities, are able to resolve CMT parameters to comparable quality as if many high-quality waveforms were available. Results of 10 induced seismic events that occurred in northeastern British Columbia, Canada, between January 2020 and February 2022 indicate predominantly strike-slip focal mechanisms with low non-double-couple components. These events appear to be located at shallow depths with short time duration, as expected for induced seismicity. These results are consistent with previous studies, indicating that this method reduces the dependence of source inversion on high-quality waveforms, and can provide resolution of CMT parameters for earthquakes as small as M L 1.6. [ABSTRACT FROM AUTHOR]

Determining the seismic moment tensor (MT) from the observed waveforms with available Earth's structure models is known as seismic waveform MT inversion. It remains challenging for small to moderate-size earthquakes at regional scales. First, because shallow isotropic (ISO) and compensated linear vector dipole (CLVD) components of MT radiate similar long-period waveforms at regional distances, an intrinsic ISO-CVLD ambiguity impedes resolving seismic sources at shallow depths within the Earth's crust. Secondly, regional scales usually bear 3-D structures; thus, inaccurate Earth's structure models can cause unreliable MT solutions but are rarely considered a theory error in the MT inversion. So far, only the error of the 1-D earth model (1-D structural error), apart from data errors, has been explicitly modelled in the source studies because of relatively inexpensive computation. Here, we utilize a hierarchical Bayesian MT inversion to address the above problems. Our approach takes advantage of affine-invariant ensemble samplers to explore the ISO-CLVD trade-off space thoroughly and effectively. Station-specific time-shifts are also searched for as free parameters to treat the structural errors along specific source–station paths (2-D structural errors). Synthetic experiments demonstrate the method's advantage in resolving the dominating ISO components. The explosive events conducted by the Democratic People's Republic of Korea (DPRK) are well-studied, and we use them to demonstrate highly similar source mechanisms, including dominating ISO and significant CLVD components. The recovered station-specific time-shifts from the blasts present a consistent pattern, which provides a better understanding of the azimuthal variation of Earth's 2-D structures surrounding the events' location. [ABSTRACT FROM AUTHOR]

In this work we study an inverse problem of recovering a space-time-dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian–Caputo fractional derivative of order |$\alpha \in (0,1)$| in time. The main technical challenges of both theoretical and numerical analyses lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results using a novel test function, which leads to a stability bound in |$L^2(0,T;L^2(\varOmega))$| under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for recovering the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal |$L^p$| regularity and a nonstandard energy argument. Under the positivity condition, we obtain a standard |$\ell ^2(L^2(\varOmega))$| error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples. [ABSTRACT FROM AUTHOR]

We study the full discretization of a general class of first- and second-order quasilinear wave-type problems with the implicit midpoint rule and a linearized variant thereof. Based on a proof by induction, we prove wellposedness and a rigorous error estimate for both schemes, combining energy techniques, inverse estimates and a linearized fixed-point iteration for the analysis of the nonlinear scheme. To confirm the relevance of the general framework, we derive novel error estimates for the full discretization of two prominent examples from nonlinear physics: the Westervelt equation and the Maxwell equations with Kerr nonlinearity. [ABSTRACT FROM AUTHOR]

The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the point-wise constraint |$|{\textbf{m}}|=1$|⁠. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal |$\textbf{L}^2$| error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition |$\tau =O(\epsilon _0 h)$| with some small |$\epsilon _0>0$|⁠. The analysis is based on more precise estimates of the extra error caused by the sphere projection in both |$\textbf{L}^2$| and |$\textbf{H}^1$| norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow and powers of mean and inverse mean curvature flow. Error estimates are proved for semidiscretizations and full discretizations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to nonlinear second-order parabolic evolution equations for the normal velocity and normal vector. A convergence proof is presented in the case of finite elements of polynomial degree at least 2 and backward difference formulae of orders 2 to 5. The error analysis combines stability estimates and consistency estimates to yield optimal-order |$H^1$| -norm error bounds for the computed surface position, velocity, normal vector, normal velocity and therefore for the mean curvature. The stability analysis is performed in the matrix–vector formulation and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results and also to report on monotone quantities, e.g. Hawking mass for inverse mean curvature flow, and complemented by experiments for nonconvex surfaces. [ABSTRACT FROM AUTHOR]

Gravity and magnetic data resolve the Earth with variable spatial resolution, and Earth structure exhibits both discontinuous and gradual features. Therefore, model parametrization complexity should be able to address such variability by locally adapting to the resolving power of the data. The reversible-jump Markov chain Monte Carlo (rjMcMC) algorithm provides variable spatial resolution that is consistent with data information. To address the prevalent non-uniqueness in joint inversion of potential field data, we use a novel spatial partitioning with nested Voronoi cells that is explored by rjMcMC sampling. The nested Voronoi parametrization partitions the subsurface in terms of rock types, such as sedimentary, salt and basement rocks. Therefore, meaningful prior information can be specified for each type which reduces non-uniqueness. We apply nonoverlapping prior distributions for density contrast and susceptibility between rock types. In addition, the choice of noise parametrization can lead to significant trade-offs with model resolution and complexity. We adopt an empirical estimation of full data covariance matrices that include theory and observational errors to account for spatially correlated noise. The method is applied to 2-D gravity and magnetic data to study salt and basement structures. We demonstrate that meaningful partitioning of the subsurface into sediment, salt, and basement structures is achieved by these advances without requiring regularization. Multiple simulated- and field-data examples are presented. Simulation results show clear delineation of salt and basement structures while resolving variable length scales. The field data show results that are consistent with observations made in the simulations. In particular, we resolve geologically plausible structures with varying length scales and clearly differentiate salt structure and basement topography. [ABSTRACT FROM AUTHOR]

We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree |$k$| for the pseudostress tensor and discontinuous piecewise polynomial elements of degree |$k$| for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters. [ABSTRACT FROM AUTHOR]

Evaluating the log-sum-exp function or the softmax function is a key step in many modern data science algorithms, notably in inference and classification. Because of the exponentials that these functions contain, the evaluation is prone to overflow and underflow, especially in low-precision arithmetic. Software implementations commonly use alternative formulas that avoid overflow and reduce the chance of harmful underflow, employing a shift or another rewriting. Although mathematically equivalent, these variants behave differently in floating-point arithmetic and shifting can introduce subtractive cancellation. We give rounding error analyses of different evaluation algorithms and interpret the error bounds using condition numbers for the functions. We conclude, based on the analysis and numerical experiments, that the shifted formulas are of similar accuracy to the unshifted ones, so can safely be used, but that a division-free variant of softmax can suffer from loss of accuracy. [ABSTRACT FROM AUTHOR]

We present an analysis of multilevel Monte Carlo (MLMC) techniques for the forward problem of uncertainty quantification for the radiative transport equation, when the coefficients (cross-sections) are heterogenous random fields. To do this we first give a new error analysis for the combined spatial and angular discretisation in the deterministic case, with error estimates that are explicit in the coefficients (and allow for very low regularity and jumps). This detailed error analysis is done for the one-dimensional space–one-dimensional angle slab-geometry case with classical diamond differencing. Under reasonable assumptions on the statistics of the coefficients, we then prove an error estimate for the random problem in a suitable Bochner space. Because the problem is not self-adjoint, stability can only be proved under a path-dependent mesh resolution condition. This means that, while the Bochner space error estimate is of order |$\mathcal{O}(h^{\eta })$| for some |$\eta $| where |$h$| is a (deterministically chosen) mesh diameter, smaller mesh sizes might be needed for some realisations. We also show that the expected cost for computing a typical quantity of interest remains of the same order as for a single sample. This leads to rigorous complexity estimates for Monte Carlo (MC) and MLMC: for particular linear solvers, the multilevel version gives up to two orders of magnitude improvement over MC. We provide numerical results supporting the theory. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of axisymmetric mean curvature flow with the help of linear finite elements. In the case of a closed genus-1 surface we derive optimal error bounds with respect to the |$L^2$| - and |$H^1$| -norms for a fully discrete approximation. We perform convergence experiments to confirm the theoretical results and also present numerical simulations for some genus-0 and genus-1 surfaces, including for the Angenent torus. [ABSTRACT FROM AUTHOR]

Centroid moment tensor (CMT) parameters can be estimated from seismic waveforms. Since these data indirectly observe the deformation process, CMTs are inferred as solutions to inverse problems which are generally underdetermined and require significant assumptions, including assumptions about data noise. Broadly speaking, we consider noise to include both theory and measurement errors, where theory errors are due to assumptions in the inverse problem and measurement errors are caused by the measurement process. While data errors are routinely included in parameter estimation for full CMTs, less attention has been paid to theory errors related to velocity-model uncertainties and how these affect the resulting moment-tensor (MT) uncertainties. Therefore, rigorous uncertainty quantification for CMTs may require theory-error estimation which becomes a problem of specifying noise models. Various noise models have been proposed, and these rely on several assumptions. All approaches quantify theory errors by estimating the covariance matrix of data residuals. However, this estimation can be based on explicit modelling, empirical estimation and/or ignore or include covariances. We quantitatively compare several approaches by presenting parameter and uncertainty estimates in nonlinear full CMT estimation for several simulated data sets and regional field data of the M l 4.4, 2015 June 13 Fox Creek, Canada, event. While our main focus is at regional distances, the tested approaches are general and implemented for arbitrary source model choice. These include known or unknown centroid locations, full MTs, deviatoric MTs and double-couple MTs. We demonstrate that velocity-model uncertainties can profoundly affect parameter estimation and that their inclusion leads to more realistic parameter uncertainty quantification. However, not all approaches perform equally well. Including theory errors by estimating non-stationary (non-Toeplitz) error covariance matrices via iterative schemes during Monte Carlo sampling performs best and is computationally most efficient. In general, including velocity-model uncertainties is most important in cases where velocity structure is poorly known. [ABSTRACT FROM AUTHOR]

The goal of the current study is the exploration of the standardization of grammatical errors through templates, a process that ultimately led to the development of a Grammar Checker for Modern Greek, an electronic tool for the automatic recognition and correction of grammatical errors. Moreover, the study investigates the utilization of such a tool in a classroom setting and its contribution to the teaching of mother tongue. The participants of the study were secondary school students and main results showed that errors they made in text production activities could be categorized as errors of mechanics, grammar, and usage through certain templates, which then lead to the development of a Grammar Checker for Modern Greek. Moreover, we also found other errors that could not be analysed and categorized using the theory of taxonomy of errors by Ho, i.e. style: forms—learned forms and semantic. Additionally, the participants highlighted that the specific tool has advantages for the teaching of mother tongue. Implications for the methodology of Grammar Checkers' development and their use in the teaching of mother tongue, generally, are discussed. [ABSTRACT FROM AUTHOR]

|$L^2$| norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments. [ABSTRACT FROM AUTHOR]

We present the first astrometry catalog from the Japanese VLBI (very long baseline interferometer) project VERA (VLBI Exploration of Radio Astrometry). We have compiled all the astrometry results from VERA, providing accurate trigonometric-annual-parallax and proper-motion measurements. In total, 99 maser sources are listed in the VERA catalog. Among them, 21 maser sources are newly reported, while the rest of the 78 sources are referred to in previously published results or those in preparation for forthcoming papers. The accuracy in the VERA astrometry is revisited and compared with that from the other VLBI astrometry projects such as BeSSeL (The Bar and Spiral Structure Legacy) Survey and GOBELINS (the Gould's Belt Distances Survey) with the VLBA (Very Long Baseline Array). We have confirmed that most of the astrometry results are consistent with each other, and the largest error sources are due to source structure of the maser features and their rapid variation, along with the systematic calibration errors and different analysis methods. Combined with the BeSSeL results, we estimate the up-to-date fundamental Galactic parameters of |$R_{0}=7.92\pm 0.16_{\rm {stat.}}\pm 0.3_{\rm {sys.}}\:$| kpc and |$\Omega _{\odot }=30.17\pm 0.27_{\rm {stat.}}\pm 0.3_{\rm {sys.}}\:$| km |$\:$| s |$^{-1}\:$| kpc |$^{-1}$|⁠ , where |$R_{0}$| and |$\Omega _{\odot }$| are the distance from the Sun to the Galactic center and the Sun's angular velocity of the Galactic circular rotation, respectively. [ABSTRACT FROM AUTHOR]

We propose stress formulation of the Stokes equations. The velocity |$u$| is approximated with |$H(\operatorname{div})$| -conforming finite elements providing exact mass conservation. While many standard methods use |$H^1$| -conforming spaces for the discrete velocity |$H(\operatorname{div})$| -conformity fits the considered variational formulation in this work. A new stress-like variable |$\sigma $| equalling the gradient of the velocity is set within a new function space |$H(\operatorname{curl} \operatorname{div})$|⁠. New matrix-valued finite elements having continuous 'normal-tangential' components are constructed to approximate functions in |$H(\operatorname{curl} \operatorname{div})$|⁠. An error analysis concludes with optimal rates of convergence for errors in |$u$| (measured in a discrete |$H^1$| -norm), errors in |$\sigma $| (measured in |$L^2$|⁠) and the pressure |$p$| (also measured in |$L^2$|⁠). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness , as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations. [ABSTRACT FROM AUTHOR]

The goal of this contribution is to investigate the expected improvement of temporal gravity field determination via a couple of high-low satellite-to-satellite tracking (HLSST) missions. The simulation system is firstly validated by determining monthly gravity field models with in situ GRACE GPS tracking data. The general consistency between the retrieved solutions and those developed by other official agencies indicates the good performance of our software. A 5-yr full-scale simulation is then performed using the full error sources including all error components. Analysis of each error component indicates that orbit error is the main contributor to the overall HLSST-derived gravity field model error. The noise level of monthly solution is therefore expected to reduce 90 per cent in terms of RMSE over ocean when the orbit accuracy improves for a magnitude of one order. As for the current HLSST mission consisting of a current GNSS receiver and an accelerometer (10−10 and 10−9 m s–2 noise for sensitive and non-sensitive axes), it is expected to observe monthly (or weekly) gravity solution at the spatial resolution of about 1300 km (or 2000 km). As for satellite constellations, a significant improvement is expected by adding the second satellite with the inclination of 70° and the third satellite with the inclination of 50°. The noise reduction in terms of cumulative geoid height error is approximately 51 per cent (or 62 per cent) when the observations of two (or three) HLSST missions are used. Moreover, the accuracy of weekly solution is expected to improve 40–70 per cent (or 27–59 per cent) for three (or two) HLSST missions when compared to one HLSST mission. Due to the low financial costs, it is worthy to build a satellite constellation of HLSST missions to fill the possible gaps between the dedicated temporal gravity field detecting missions. [ABSTRACT FROM AUTHOR]

Cosmic rays are a global source of ionization, and the ionization fraction represents a fundamental parameter in the interstellar medium. Ions couple to magnetic fields, and affect the chemistry and the dynamics of star-forming regions as well as planetary atmospheres. However, the cosmic ray ionization rate represents one of the bottlenecks for astrochemical models, and its determination is one of the most puzzling problems in astrophysics. While for diffuse clouds reasonable values have been provided from |${\mathrm{ H}_3}^+$| observations, for dense clouds, due to the lack of rotational transitions, this is not possible, and estimates are strongly biased by the employed model. We present here an analytical expression, obtained from first principles, to estimate the cosmic ray ionization rate from observational quantities. The theoretical predictions are validated with high-resolution 3D numerical simulations and applied to the well-known core L1544; we obtained an estimate of ζ2 ∼ 2–3 × 10−17 s−1. Our results and the analytical formulae provided represent the first model-independent robust tool to probe the cosmic ray ionization rate in the densest part of star-forming regions (on spatial scales of R ≤ 0.05 pc). An error analysis is presented to give statistical relevance to our study. [ABSTRACT FROM AUTHOR]

Two unfitted-mesh methods for a linear incompressible fluid/thin-walled structure interaction problem are introduced and analyzed. The spatial discretization is based on different variants of Nitsche's method with cut elements. The degree of fluid–solid splitting (semi-implicit or explicit) is given by the order in which the space and time discretizations are performed. The a priori stability and error analysis shows that strong coupling is avoided without compromising stability and accuracy. Numerical experiments with a benchmark illustrate the accuracy of the different methods proposed. [ABSTRACT FROM AUTHOR]

In this article we present an a posteriori error estimator for the spatial–stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the stochastic Galerkin method and for the spatial–temporal discretization of the stochastic Galerkin system a Runge–Kutta discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos (2016, Hyperbolic Conservation Laws in Continuum Physics , 4th edn., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, Berlin, Springer, pp. xxxviii+826), this leads to computable error bounds for the space–stochastic discretization error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretization. We conclude with some numerical examples confirming the theoretical findings. [ABSTRACT FROM AUTHOR]

We propose and analyze finite difference methods for solving the Klein–Gordon–Dirac (KGD) system. Due to the nonlinear coupling between the complex Dirac 'wave function' and the real Klein–Gordon field, it is a great challenge to design and analyze numerical methods for KGD. To overcome the difficulty induced by the nonlinearity, four implicit/semi-implicit/explicit finite difference time domain (FDTD) methods are presented, which are time symmetric or time reversible. By rigorous error estimates, the FDTD methods converge with second-order accuracy in both spatial and temporal discretizations, and numerical results in one dimension are reported to support our conclusion. The error analysis relies on the energy method, the special nonlinear structure in KGD and the mathematical induction. Thanks to tensor grids and discrete Sobolev inequalities, our approach and convergence results are valid in higher dimensions under minor modifications. [ABSTRACT FROM AUTHOR]

Epidemiologists are often confronted with datasets to analyse which contain measurement error due to, for instance, mistaken data entries, inaccurate recordings and measurement instrument or procedural errors. If the effect of measurement error is misjudged, the data analyses are hampered and the validity of the study's inferences may be affected. In this paper, we describe five myths that contribute to misjudgments about measurement error, regarding expected structure, impact and solutions to mitigate the problems resulting from mismeasurements. The aim is to clarify these measurement error misconceptions. We show that the influence of measurement error in an epidemiological data analysis can play out in ways that go beyond simple heuristics, such as heuristics about whether or not to expect attenuation of the effect estimates. Whereas we encourage epidemiologists to deliberate about the structure and potential impact of measurement error in their analyses, we also recommend exercising restraint when making claims about the magnitude or even direction of effect of measurement error if not accompanied by statistical measurement error corrections or quantitative bias analysis. Suggestions for alleviating the problems or investigating the structure and magnitude of measurement error are given. [ABSTRACT FROM AUTHOR]

We present a priori and a posteriori error analyses of a virtual element method (VEM) to approximate the vibration frequencies and modes of an elastic solid. We analyse a variational formulation relying only on the solid displacement and propose an |$H^{1}(\Omega)$| -conforming discretization by means of the VEM. Under standard assumptions on the computational domain, we show that the resulting scheme provides a correct approximation of the spectrum and prove an optimal–order error estimate for the eigenfunctions and a double order for the eigenvalues. Since the VEM has the advantage of using general polygonal meshes, which allows efficient implementation of mesh refinement strategies, we also introduce a residual-type a posteriori error estimator and prove its reliability and efficiency. We use the corresponding error estimator to drive an adaptive scheme. Finally, we report the results of a couple of numerical tests that allow us to assess the performance of this approach. [ABSTRACT FROM AUTHOR]

Linked data are increasingly being used for epidemiological research, to enhance primary research, and in planning, monitoring and evaluating public policy and services. Linkage error (missed links between records that relate to the same person or false links between unrelated records) can manifest in many ways: as missing data, measurement error and misclassification, unrepresentative sampling, or as a special combination of these that is specific to analysis of linked data: the merging and splitting of people that can occur when two hospital admission records are counted as one person admitted twice if linked and two people admitted once if not. Through these mechanisms, linkage error can ultimately lead to information bias and selection bias; so identifying relevant mechanisms is key in quantitative bias analysis. In this article we introduce five key concepts and a study classification system for identifying which mechanisms are relevant to any given analysis. We provide examples and discuss options for estimating parameters for bias analysis. This conceptual framework provides the 'links' between linkage error, information bias and selection bias, and lays the groundwork for quantitative bias analysis for linkage error. [ABSTRACT FROM AUTHOR]

This paper studies non inf-sup stable finite element approximations to the evolutionary Navier–Stokes equations. Several local projection stabilization (LPS) methods corresponding to different stabilization terms are analyzed, thereby separately studying the effects of the different stabilization terms. Error estimates are derived in which the constants are independent of inverse powers of the viscosity. For one of the methods, using velocity and pressure finite elements of degree |$l$|⁠ , it will be proved that the velocity error in |$L^\infty (0,T;L^2(\varOmega))$| decays with rate |$l+1/2$| in the case that |$\nu \le h$|⁠ , with |$\nu$| being the dimensionless viscosity and |$h$| being the mesh width. In the analysis of another method it was observed that the convective term can be bounded in an optimal way with the LPS stabilization of the pressure gradient. Numerical studies confirm the analytical results. [ABSTRACT FROM AUTHOR]

We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogue of the normal derivative as a Lagrange multiplier in the formulation of the problem. In order to obtain convergence orders for our scheme, regularity estimates are developed both for the solution and for its nonlocal derivative. The method we propose requires that, as meshes are refined, the discrete problems be solved in a family of domains of growing diameter. [ABSTRACT FROM AUTHOR]

We consider the a posteriori error analysis of fully discrete approximations of parabolic problems based on conforming |$hp$| -finite element methods in space and an arbitrary order discontinuous Galerkin method in time. Using an equilibrated flux reconstruction we present a posteriori error estimates yielding guaranteed upper bounds on the |$L^2(H^1)$| -norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes. It is known from the literature that the analysis of the efficiency of the estimators represents a significant challenge for |$L^2(H^1)$| -norm estimates. Here we show that the estimator is bounded by the |$L^2(H^1)$| -norm of the error plus the temporal jumps under the one-sided parabolic condition |$h^2 \lesssim \tau $|⁠. This result improves on earlier works that required stronger two-sided hypotheses such as |$h \simeq \tau $| or |$h^2\simeq \tau $|⁠ ; instead, our result now encompasses practically relevant cases for computations and allows for locally refined spatial meshes. The constants in our bounds are robust with respect to the mesh and time-step sizes, the spatial polynomial degrees and the refinement and coarsening between time steps, thereby removing any transition condition. [ABSTRACT FROM AUTHOR]

We introduce and analyze a hybridizable discontinuous Galerkin method for the gradient-velocity-pressure formulation of the Brinkman problem. We present an a priori error analysis of the method, showing optimal order of convergence of the error. We also introduce an a posteriori error estimator, of the residual type, which helps us to improve the quality of the numerical solution. We establish reliability and local efficiency of our estimator for the |$L^{2} $| -error of the velocity gradient and the pressure and the |$ H^{1} $| -error of the velocity, with constants written explicitly in terms of the physical parameters and independent of the size of the mesh. In particular, our results are also valid for the Stokes problem. Finally, we provide numerical experiments showing the quality of our adaptive scheme. [ABSTRACT FROM AUTHOR]

This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell's equations and finite elements with mass lumping for the acoustic wave equation. [ABSTRACT FROM AUTHOR]

Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues |$\{\lambda _i\}$| and eigenvectors |$\{\boldsymbol{u}_i\}$| of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size |$n$| to obtain empirical eigenvalues |$\{\tilde{\lambda }_i\}$| and eigenvectors |$\{\tilde{\boldsymbol{u}}_i\}$|⁠ , and therefore understanding the error so introduced is of central importance. We analyse eigenvector error |$\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$| while leveraging the assumption that the true covariance matrix having size |$p$| is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as |$p\to \infty $| to a spectral density |$\rho (\lambda)$| and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when |$p$| is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue |$\lambda $| and approximate the distribution of the expected square error |$r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$| across the matrix ensemble for all |$\boldsymbol{u}_i$| associated with |$\lambda _i=\lambda $|⁠. We find, for example, that for sufficiently large matrix size |$p$| and sample size |$n> p$|⁠ , the probability density of |$r$| scales as |$1/nr^2$|⁠. This power-law scaling implies that the eigenvector error is extremely heterogeneous—even if |$r$| is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments. [ABSTRACT FROM AUTHOR]

Proposes an exact Bayesian framework for examining the arbitrage pricing theory (APT) pricing restrictions. Framework of the APT restrictions and Bayesian inference; Application of the approach to portfolio returns grouped by both industry and size; Reduction in the pricing error; Risk measures in the APT model; Comparison of the market index with the APT factor.

Objective Competitive Queuing (CQ) models of memory for serial order comprise two layers: parallel planning where target items are activated and competitive choice where serial order is specified. The application of CQ models regarding healthy and pathological aging has received little attention. Method Participants included patients with Alzheimer's disease (AD; n = 26), vascular dementia (VaD; n = 29), and healthy controls (HC; n = 35). Memory for serial order in the visual domain was assessed using the Object Span Task, where participants briefly viewed then drew a sequence of four figures. Percent correct and total errors (omissions, intrusions, repetitions, transpositions) were computed for each serial position. Results Significant primacy effects were detected in each group. AD and VaD participants were less accurate and showed more omission and between-trial repetition errors than HC (HC < AD = VaD, p <.05). VaD participants produced more transposition and intrusion errors than the AD and HC groups (HC < AD < VaD, p <.05). A group × position interaction was significant for omissions (p <.05), with AD and VaD participants producing more omissions in later serial positions (SP1 < SP2 < SP3 < SP4, all p values <.05). Conclusions Analysis of accuracy and errors by serial position identified unique patterns of performance across groups that suggest involvement of distinct layers of response activation and selection. Serial order difficulties in AD may be due to weakened activation of task items affecting later serial positions, whereas poor performance in VaD may be due to weakened activation plus interference from extraneous stimuli at all serial positions. [ABSTRACT FROM AUTHOR]

We have analysed the differences in positions of 9081 matched sources between the Gaia Data Release 2 (DR2) and very long baseline interferometry (VLBI) catalogues. The median position uncertainty of matched sources in the VLBI catalogue is a factor of two larger than the median position uncertainty in Gaia DR2. There are 9 per cent matched sources with statistically significant offsets between both catalogues. We found that the reported positional errors should be rescaled by a factor of 1.3 for VLBI and 1.06 for Gaia and, in addition, the Gaia errors should be multiplied by the square root of chi squared per degree of freedom in order to best fit the normalized position differences to the Rayleigh distribution. We have established that the major contributor to statistically significant position offsets is the presence of optical jets. Among the sources for which the jet direction was determined, the position offsets are parallel to the jet directions for 62 per cent of the outliers. Among the matched sources with significant proper motion, the fraction of objects with proper motion directions parallel to jets is a factor of three greater than on average. Such sources have systematically higher chi squared per degree of freedom. We explain these proper motions as a manifestation of the source position jitter caused by flares, which we predicted earlier. Therefore, the assumption that quasars are fixed points, and thus that differential proper motions determined with respect to quasar photocentres can be regarded as absolute proper motions, should be treated with great caution. [ABSTRACT FROM AUTHOR]

We determine the total dynamical density in the solar neighbourhood using the Tycho–Gaia Astrometric Solution catalogue. Astrometric measurements of proper motion and parallax of stars inform us of both the stellar number density distribution and the velocity distribution of stars close to the plane. Assuming equilibrium, these distributions are interrelated through the local dynamical density. For the first time, we do a full joint fit of the velocity and stellar number density distributions while accounting for the astrometric error of individual stars, in the framework of Bayesian hierarchical modelling. We use a sample of stars whose distance extends to approximately 160 pc from the Sun. We find a local matter density of |$\rho _0 = 0.119^{+0.015}_{-0.012} {\rm {M}_{\odot }\, \rm {pc}^{-3}}$|⁠, where the result is presented as the median to the posterior distribution, plus/minus its 16th and 84th percentiles. We find the Sun's position above the Galactic plane to be |$z_\odot = 15.29^{+2.24}_{-2.16} \, \rm {pc}$|⁠, and the Sun's velocity perpendicular to the Galactic plane to be |${W}_\odot = 7.19^{+0.18}_{-0.18} \, \rm {km}\, \rm {s}^{-1}.$| [ABSTRACT FROM AUTHOR]

We propose a continuous testing framework to test the intensities of Poisson processes that allows a rigorous definition of the complete testing procedure, from an infinite number of hypotheses to joint error rates. Our work extends procedures based on scanning windows by controlling the familywise error rate and the false discovery rate in a non-asymptotic manner and in a continuous way. We introduce the p -value process on which the decision rule is based. Our method is applied in neuroscience via the standard homogeneity and two-sample tests. [ABSTRACT FROM AUTHOR]

Multiple types of data measured on a common set of subjects arise in many areas. Numerous empirical studies have found that integrative analysis of such data can result in better statistical performance in terms of prediction and feature selection. However, the advantages of integrative analysis have mostly been demonstrated empirically. In the context of two-class classification, we propose an integrative linear discriminant analysis method and establish a theoretical guarantee that it achieves a smaller classification error than running linear discriminant analysis on each data type individually. We address the issues of outliers and missing values, frequently encountered in integrative analysis, and illustrate our method through simulations and a neuroimaging study of Alzheimer's disease. [ABSTRACT FROM AUTHOR]

A posteriori error analysis in numerical partial differential equations aims at providing sufficiently accurate information about the distance of the numerically computed approximation to the true solution. Besides estimating the total error, a posteriori analysis should also provide information about its discretization and (inexact) algebraic computation parts. This issue has been addressed by many authors using different approaches. Historically, probably the first and practically very important approach is based on combination of the classical residual-based bound on the discretization error with the adaptive hierarchy of discretizations and computations that allow to incorporate, using various heuristic arguments, the algebraic error. Motivated by some recent publications, this text uses a complementary approach and examines subtleties of the (generalized) residual-based a posteriori error estimator for the total error that rigorously accounts for the algebraic part of the error. The aim is to show on the standard Poisson model problem example, which is used here as a case study, that a rigorous incorporation of the algebraic error represents an intriguing problem that is not yet completely resolved. That should be of concern in $$h$$ -adaptivity approaches, where the refinement of the mesh is determined using the residual-based a posteriori error estimator assuming Galerkin orthogonality. The commonly used terminology such as ‘guaranteed computable upper bounds’ should be in the presence of algebraic error cautiously examined. [ABSTRACT FROM AUTHOR]

Given an $$n$$ -by- $$n$$ large sparse matrix $$A$$ with low rank or with fast decaying singular values, we propose new algorithms for consecutively approximating the functions $$\varphi_{\ell}(A)$$ as well as their 2-norm condition numbers for $$\ell=0,1,\ldots,p$$. The key is to reduce the computation of $$\varphi_{\ell}$$ -functions of the large matrix to that of $$\varphi_{\ell+1}$$ -functions of some smaller matrices of size $$r$$ -by- $$r$$ , where $$r$$ is the numerical rank of $$A$$. For storage reasons, the new algorithms need to store only two $$n$$ -by- $$r$$ sparse matrices and some $$r$$ -by- $$r$$ matrices, rather than some $$n$$ -by- $$n$$ possibly dense matrices. Some error analysis is given. Numerical experiments illustrate the numerical behavior of our new algorithms and show the practical utility of new strategies. [ABSTRACT FROM AUTHOR]

We formulate a stabilized quasi-optimal Petrov–Galerkin method for singularly perturbed convection–diffusion problems based on the variational multiscale method. The stabilization is of Petrov–Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localization to local patch problems, which depend on the direction of the velocity field and the singular perturbation parameter, are rigorously justified. Under moderate assumptions, this stabilization guarantees stability and a quasi-optimal rate of convergence for arbitrary mesh Péclet numbers on fairly coarse meshes at the cost of additional inter-element communication. [ABSTRACT FROM AUTHOR]

It is well known the order reduction phenomenon which arises when exponential methods are used to integrate time-dependent initial boundary value problems, so that the classical order of these methods is reduced. In particular, this subject has been recently studied for Lie–Trotter and Strang exponential splitting methods, and the order observed in practice has been exactly calculated. In this article, a technique is suggested to avoid that order reduction. We deal directly with nonhomogeneous time-dependent boundary conditions, without having to reduce the problem to the homogeneous ones. We give a thorough error analysis of the full discretization and justify why the computational cost of the technique is negligible in comparison with the rest of the calculations of the method. Some numerical results for dimension splittings are shown, which corroborate that much more accuracy is achieved. [ABSTRACT FROM AUTHOR]

We performed an analysis of the main theoretical uncertainties that affect the radius of low- and very-low-mass stars predicted by current stellar models. We focused on stars in the mass range 0.1-1M ⊙, on both the zero-age main sequence (ZAMS) and on 1, 2, and 5 Gyr isochrones. First, we quantified the impact on the radius of the uncertainty of several quantities, namely the equation of state, radiative opacity, atmospheric models, convection efficiency, and initial chemical composition. Then, we computed the cumulative radius error stripe obtained by adding the radius variation due to all the analysed quantities. As a general trend, the radius uncertainty increases with the stellar mass. For ZAMS structures the cumulative error stripe of very-low-mass stars is about ±2 and ±3 per cent, while at larger masses it increases up to ±4 and ±5 per cent. The radius uncertainty gets larger and age dependent if isochrones are considered, reaching for M ~ 1M ⊙ about +12(-15) per cent at an age of 5Gyr. We also investigated the radius uncertainty at a fixed luminosity. In this case, the cumulative error stripe is the same for both ZAMS and isochrone models and it ranges from about ±4 to +7 and +9(-5) per cent. We also showed that the sole uncertainty on the chemical composition plays an important role in determining the radius error stripe, producing a radius variation that ranges between about ±1 and ±2 per cent on ZAMS models with fixed mass and about ±3 and ±5 per cent at a fixed luminosity. [ABSTRACT FROM AUTHOR]

We introduce a residual-based a posteriori error estimator for contact problems in two- and threedimensional linear elasticity, discretized with linear and quadratic finite elements and Nitsche's method. Efficiency and reliability of the estimator are proved under a saturation assumption. Numerical experiments illustrate the theoretical properties and the good performance of the estimator. [ABSTRACT FROM AUTHOR]

In this paper, we analyse a semidiscrete finite difference scheme for a conservation law driven by a homogeneous multiplicative Lévy noise. Thanks to BV estimates, we give a result of compactness for the sequence of approximate solutions generated by the finite difference scheme, and we prove that it converges to the unique entropy solution of the underlying problem as the spatial mesh size Δx ... 0. Moreover, we show that the expected value of the L¹ difference between the approximate solution and the unique entropy solution converges at a rate O(√Δx). [ABSTRACT FROM AUTHOR]

The purpose of this work is to illustrate how the theory of Muckenhoupt weights, Muckenhoupt-weighted Sobolev spaces and the corresponding weighted norm inequalities can be used in the analysis and discretization of partial-differential-equation-constrained optimization problems. We consider a linear quadratic constrained optimization problem where the state solves a nonuniformly elliptic equation, a problem where the cost involves pointwise observations of the state and one where the state has singular sources, e.g., point masses. For all the three examples, we propose and analyse numerical schemes and provide error estimates in two and three dimensions. While some of these problems might have been considered before in the literature, our approach allows for a simpler, Hilbert-space-based analysis and discretization and further generalizations. [ABSTRACT FROM AUTHOR]

Background. The purpose of the study was to explore the precision of an equation designed to estimate residual kidney urea clearance (KRU) from interdialytic urine collection data and pre-hemodialysis (HD) serum urea nitrogen (SUN) in different hemodialysis treatment schedules. Methods. The generalizability of the proposed equation was tested in 32 731 HD treatments where urine was collected prior to a dialysis session, mostly for 24 h but sometimes longer, in patients being dialyzed 1-4 times/week. Results. The residual kidney urea clearance estimating equation predicted a KRU that matched the one computed by formal modeling within 5% in>98% of sessions analyzed. The errors in estimated versus modeled KRU for interdialytic intervals (IDIs) of 2, 3, 4 and 7 days, were 1.661.5%, -0.461.6%, 0.961.6%, and 1.561.2%, respectively. Percent errors were similar for schedules of 1-4/week with the exception of urine collection during the 2-day interval of a 2:5-day twice-weekly schedule; here error averaged 5.061.2%. Use of the average of the SUN values at the start and end of the collection period overestimated modeled KRU by 11.364.5%, whereas an equation suggested by others underestimated modeled KRU by-9.963.4%. Conclusions. The equation tested predicts values for KRU that are similar to those obtained from formal urea kinetic modeling, with percent errors that only rarely exceed 5%. It gives relatively precise results for a wide range of HD treatment schedules, IDIs and urine collection periods. [ABSTRACT FROM AUTHOR]

We investigate and try to reduce the impacts on low-degree estimates of non-loading errors, that is, aliasing of unmodeled loading and Global Positioning System (GPS) draconitic year errors, to improve the sensitivity of GPS observations to the loading mass. ThreeGPS data sets, ITRF2008-GPS residuals, ITRF2014-GPS residuals and Jet Propulsion Laboratory (JPL)'s residuals, are used and compared in this paper. Results show that the aliasing signals in GPS displacements is an important error source, especially for inferring geocentre motion. The two International Terrestrial Reference Frame (ITRF)-GPS residuals generated in a two-step combination based on Helmert transformation show more complex aliasing errors than JPL's residuals produced in precise point positions mode. The seasonal variations of geocentre motion derived from JPL thus perform the best among all three solutions, while the higher degree coefficients from the two ITRF-GPS solutions do better. Compared with ITRF2008-GPS residuals, the aliasing errors are indeed reduced, and geocentre motion/ΔTC20 (degree-2 zonal coefficients in terms of surface mass density) are also much improved for ITRF2014-GPS residuals produced with a six-parameter transformation without scale parameter. Additional translation parameters should be included into ITRF2008-GPS residuals, or elseΔTC20 cannot be correctly obtained. The draconitic errors pose another obstacle to accurately studying the seasonal variations of surface loading using GPS data. The draconitic harmonics (first, second and third) are well extracted from ITRF2014-derived ΔTC20 and ΔTS21 (degree-2 and order-1 sine coefficients), even if the time span is not long enough to independently separate the seasonal variations and draconitic harmonics. These errors account for an increase of about 10 per cent in the annual amplitude of ITRF2014-derived ΔTC20 and ΔTS21. Removing the found draconitic errors brings ΔTC20 and ΔTS21 from GPS and Satellite Laser Ranging into closer agreement. [ABSTRACT FROM AUTHOR]

We suggest a staggered time integrator whose order of accuracy can arbitrarily be extended to solve the linear acoustic wave equation. A strategy to select the appropriate order of accuracy is also proposed based on the error analysis that quantitatively predicts the truncation error of the numerical solution. This strategy not only reduces the computational cost several times, but also allows us to flexibly set themodelling parameters such as the time step length, grid interval and P-wave speed. It is demonstrated that the proposed method can almost eliminate temporal dispersive errors during long term simulations regardless of the heterogeneity of the media and time step lengths. The method can also be successfully applied to the source problem with an absorbing boundary condition, which is frequently encountered in the practical usage for the imaging algorithms or the inverse problems. [ABSTRACT FROM AUTHOR]

In this work, we analyse a Crank-Nicolson type time-stepping scheme for the subdiffusion equation, which involves a Caputo fractional derivative of order α∈(0,1) in time. It hybridizes the backward Euler convolution quadrature with a θ-type method, with the parameter θ dependent on the fractional order α by θ=α/2 and naturally generalizes the classical Crank-Nicolson method. We develop essential initial corrections at the starting two steps for the Crank-Nicolson scheme and, together with the Galerkin finite element method in space, obtain a fully discrete scheme. The overall scheme is easy to implement and robust with respect to data regularity. A complete error analysis of the fully discrete scheme is provided, and a second-order accuracy in time is established for both smooth and nonsmooth problem data. Extensive numerical experiments are provided to illustrate its accuracy, efficiency and robustness, and a comparative study also indicates its competitive with existing schemes. [ABSTRACT FROM AUTHOR]

Discrete Hamiltonian variational integrators are derived from type II and type III generating functions for symplectic maps, and in this article, we establish a variational error analysis result that relates the order of accuracy of the associated numerical methods with the extent to which these generating functions approximate the exact discrete Hamiltonians. We also introduce the notion of an adjoint discrete Hamiltonian and relate it to the adjoint of the associated symplectic integrator. We show that when constructing discrete Lagrangians and discrete Hamiltonians using the same approximation method, this does not necessarily lead to the same symplectic integrator. Numerical experiments also indicate that the resonance behavior of variational integrators based on averaging methods depends on the type of generating functions used, and we relate this resonance behavior to the ill-posedness of the boundary-value problems used to define the exact discrete Lagrangian and exact discrete Hamiltonian. [ABSTRACT FROM AUTHOR]