Search

This study investigated how data series length and gaps in human kinematic data impact the accuracy of Lyapunov exponents (LyE) calculations with and without cubic spline interpolation. Kinematic time series were manipulated to create various data series lengths (28% and 100% of original) and gap durations (0.05–0.20 s). Longer gaps generally resulted in significantly higher LyE% error values in each plane in noninterpolated data. During cubic spline interpolation, only the 0.20-second gap in frontal plane data resulted in a significantly higher LyE% error. Data series length did not significantly affect LyE% error in noninterpolated data. During cubic spline interpolation, sagittal plane LyE% errors were significantly higher at shorter versus longer data series lengths. These findings suggest that not interpolating gaps in data could lead to erroneously high LyE values and mischaracterization of movement variability. When applying cubic spline, a long gap length (0.20 s) in the frontal plane or a short sagittal plane data series length (1000 data points) could also lead to erroneously high LyE values and mischaracterization of movement variability. These insights emphasize the necessity of detailed reporting on gap durations, data series lengths, and interpolation techniques when characterizing human movement variability using LyE values. [ABSTRACT FROM AUTHOR]

Grid is a free and open-source Python library for constructing numerical grids to integrate, interpolate, and differentiate functions (e.g., molecular properties), with a strong emphasis on facilitating these operations in computational chemistry and conceptual density functional theory. Although designed, maintained, and released as a stand-alone Python library, Grid was originally developed for molecular integration, interpolation, and solving the Poisson equation in the HORTON and ChemTools packages. Grid is designed to be easy to use, extend, and maintain; this is why we use Python and adopt many principles of modern software development, including comprehensive documentation, extensive testing, continuous integration/delivery protocols, and package management. We leverage popular scientific packages, such as NumPy and SciPy, to ensure high efficiency and optimized performance in grid development. This article is the official release note of the Grid library showcasing its unique functionality and scope. [ABSTRACT FROM AUTHOR]

Enhanced sampling algorithms are indispensable when working with highly disconnected multimodal distributions. An important application of these is the conformational exploration of particular internal degrees of freedom of molecular systems. However, despite the existence of many commonly used enhanced sampling algorithms to explore these internal motions, they often rely on system-dependent parameters, which negatively impact efficiency and reproducibility. Here, we present fully adaptive simulated tempering (FAST), a variation of the irreversible simulated tempering algorithm, which continuously optimizes the number, parameters, and weights of intermediate distributions to achieve maximally fast traversal over a space defined by the change in a predefined thermodynamic control variable such as temperature or an alchemical smoothing parameter. This work builds on a number of previously published methods, such as sequential Monte Carlo, and introduces a novel parameter optimization procedure that can, in principle, be used in any expanded ensemble algorithms. This method is validated by being applied on a number of different molecular systems with high torsional kinetic barriers. We also consider two different soft-core potentials during the interpolation procedure and compare their performance. We conclude that FAST is a highly efficient algorithm, which improves simulation reproducibility and can be successfully used in a variety of settings with the same initial hyperparameters. [ABSTRACT FROM AUTHOR]

The adiabatic connection interaction strength interpolation (ISI)-like method provides a high-level expression for the correlation energy, being, in principle, exact not only in the weak-interaction limit, where it recovers the second-order Görling–Levy perturbation term, but also in the strong-interaction limit that is described by the strictly correlated electron approach. In this work, we construct a genISI functional made accurate for the uniform electron gas, a solid-state physics paradigm that is a very difficult test for ISI-like correlation functionals. We assess the genISI functional for various jellium spheres with the number of electrons Z ≤ 912 and for the non-relativistic noble atoms with Z ≤ 290. For the jellium clusters, the genISI is remarkably accurate, while for the noble atoms, it shows a good performance, similar to other ISI-like methods. Then, the genISI functional can open the path using the ISI-like method in solid-state calculations. [ABSTRACT FROM AUTHOR]

Colloquially, quantitatively Hyper-Positive functions form a sub-family of the "Strictly Positive real" functions, where in addition state-space realization always exists and the limit at infinity is non-singular. In scalar terminology, such function maps the right-half of the complex plane into a sub-region of the right-half plane, which can be contained in a finite disk. Moreover, under inversion this disk is mapped onto itself. In [9] we showed that these functions appear in the "absolute stability" Lurie-type problems and characterized these families through their state-space realization, in the spirit of the Kalman-Yakubovich-Popov Lemma. Here, we present additional characterizations, i.e. as rational functions and through the corresponding kernels. Subsequently, a description of all solutions of the tangential Nevanlinna-Pick interpolation problem, in this framework, is presented. [ABSTRACT FROM AUTHOR]

Image interpolation is an important topic in the field of image processing. It is defined as the process of transforming low-resolution images into high-resolution ones using image processing methods. Recent studies on interpolation have shown that researchers are focusing on successful interpolation techniques that preserve edge information. Therefore, the edge detection phase plays a vital role in interpolation studies. However, these approaches typically rely on gradient-based linear computations for edge detection. On the other hand, non-linear structures that effectively simulate the human visual system have gained attention. In this study, a non-linear method was developed to detect edge information using a pixel similarity approach. Pixel similarity-based edge detection approach offers both lower computational complexity and more successful interpolation results compared to gradient-based approaches. 1D cubic interpolation was applied to the pixels identified as edges based on pixel similarity, while bicubic interpolation was applied to the remaining pixels. The algorithm was tested on 12 commonly used images and compared with various interpolation techniques. The results were evaluated using metrics such as SSIM and PSNR, as well as visual assessment. The experimental findings clearly demonstrated that the proposed method outperformed other approaches. Additionally, the method offers significant advantages, such as not requiring any parameters and having competitive computational cost. [ABSTRACT FROM AUTHOR]

Summary: In this paper, a generalized particle domain method (GPDM) is proposed and developed within the framework of the convected particle domain interpolation method. This new method generates particles directly from non‐uniform rational B‐spline (NURBS)‐based CAD file of a continuum body. The particle domain corresponds to a NURBS element even for trimmed elements of solids with complex geometries. The shape functions and the gradient of shape functions are evaluated using NURBS basis functions to map material properties between particles and grid nodes. It approves that this proposed GPDM can track the domain of particles accurately and avoid the issue of cell‐crossing instability. Several numerical examples are presented to demonstrate the high performance of this proposed new particle domain method. It is shown that the results obtained using the proposed GPDM are consistent with the experimental data reported in the literature. Further development of the generalized particle domain method can provide a link to the material point method and isogeometric analysis. [ABSTRACT FROM AUTHOR]

We study real interpolation, but instead of interpolating between Banach spaces, we interpolate between general functions taking values in [ 0 , ∞ ]. We show the equivalence of the mean method and the K -method and apply the general theory to interpolation between the norm on a Banach space and the set norm associated with an m-accretive operator on such a space. [ABSTRACT FROM AUTHOR]

Producing spatial transformations that are diffeomorphic is a key goal in deformable image registration. As a diffeomorphic transformation should have positive Jacobian determinant | J | everywhere, the number of pixels (2D) or voxels (3D) with | J | < 0 has been used to test for diffeomorphism and also to measure the irregularity of the transformation. For digital transformations, | J | is commonly approximated using a central difference, but this strategy can yield positive | J | 's for transformations that are clearly not diffeomorphic—even at the pixel or voxel resolution level. To show this, we first investigate the geometric meaning of different finite difference approximations of | J | . We show that to determine if a deformation is diffeomorphic for digital images, the use of any individual finite difference approximation of | J | is insufficient. We further demonstrate that for a 2D transformation, four unique finite difference approximations of | J | 's must be positive to ensure that the entire domain is invertible and free of folding at the pixel level. For a 3D transformation, ten unique finite differences approximations of | J | 's are required to be positive. Our proposed digital diffeomorphism criteria solves several errors inherent in the central difference approximation of | J | and accurately detects non-diffeomorphic digital transformations. The source code of this work is available at https://github.com/yihao6/digital_diffeomorphism. [ABSTRACT FROM AUTHOR]

We derive a family of interpolation estimates which improve Hardy's inequality and cover the Sobolev critical exponent. We also determine all optimizers among radial functions in the endpoint case and discuss open questions on nonrestricted optimizers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of trend function made from adaptive Fourier decomposition. Through applying a data preprocessing method, we achieve trend functions of the rational function type. The trend functions are divided into different levels to describe the tendency of the signal or time series in question. They are, in particular, smooth, and are free from the Gibbs phenomenon as usually introduced by the Fourier type methods. To justify the role of the trend function concept, an interpolation result is proved. An effective application of trend function in predicting stochastic time series is presented based on real COVID-19 outbreak data of China. The algorithm aspect is fully discussed. [ABSTRACT FROM AUTHOR]

We propose an efficient method with energy constraints for constructing a Catmull–Clark surface that interpolates a given mesh. We approximate the surface energy of Catmull–Clark surfaces near extraordinary points by summing their finite subpatches and then represent the energy of the subpatches as linear combinations of the vertices of control mesh. By minimizing the surface energy as a constraint, we generate a new control mesh whose limit surfaces interpolate a given mesh. Numerous examples and comparisons demonstrate that our method has the following characteristics: (1) The limit surfaces are fairer, reducing unnecessary undulations and having minimal surface energy, and (2) the approximation process is simple and intuitive, requiring only a small number of computational steps and avoiding complex parameterization processes. [ABSTRACT FROM AUTHOR]

Let {Xυ: υ ∈ ℤd} be i.i.d. random variables. Let S (π) = ∑ υ ∈ π X υ be the weight of a self-avoiding lattice path π. Let M n = max { S (π) : π has length n and starts from origin }. We are interested in the asymptotics of Mn as n → ∞. This model is closely related to the first passage percolation when the weights {Xυ: υ ∈ ℤd} are non-positive and it is closely related to the last passage percolation when the weights {Xυ, υ ∈ ℤd} are non-negative. For general weights, this model could be viewed as an interpolation between first passage models and last passage models. Besides, this model is also closely related to a variant of the position of right-most particles of branching random walks. Under the two assumptions that ∃ α > 0 , E (X 0 +) d (log + X 0 +) d + α < + ∞ and that E [ X 0 − ] < + ∞ , we prove that there exists a finite real number M such that Mn/n converges to a deterministic constant M in L1 as n tends to infinity. And under the stronger assumptions that ∃ α > 0 , E (X 0 +) d (log + X 0 +) d + α < + ∞ and that E [ (X 0 −) 4 ] < + ∞ , we prove that Mn/n converges to the same constant M almost surely as n tends to infinity. [ABSTRACT FROM AUTHOR]

This paper considers a new stabilized finite element method (FEM) of the Stokes equations based on Clément interpolation by the constrained quadrilateral nonconforming rotated Q 1 - Q 0 finite element pair. The stabilized term constructed in this method is quite different from those of the existing literature. This method not only has the same attractive computational properties as the conforming stabilized finite element method with local polynomial pressure projection, such as parameter-free, avoiding higher-order derivatives and edge-based date structure, but also guarantees the superclose and superconvergence results for the velocity and pressure. Finally, two numerical examples are supported to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

Artifact correction is a great challenge in cardiac imaging. During the correction of coronary tissue with motion‐induced artifacts, the spatial distribution of CT value not only shifts according to the motion vector field (MVF), but also shifts according to the volume change rate of the local voxels. However, the traditional interpolation method does not conserve the CT value during motion compensation. A new sample interpolation algorithm is developed based on the constraint of conservation of CT value before and after image deformation. This algorithm is modified on the existing interpolation algorithms and can be embedded into neural networks with deterministic back propagation. Comparative experimental results illustrate that the method can not only correct motion‐induced artifacts, but also ensure the conservation of CT value in the region of interest (ROI) area, so as to obtain corrected images with clinically recognized CT value. Both effectiveness and efficiency are proved in forward motion correction process and backward training steps in deep learning. Simultaneously, using the network to learn the MVF making this method more interpretable than the existing image‐based end‐to‐end deep learning method. [ABSTRACT FROM AUTHOR]

This paper proposes a method for obtaining adaptive and high-precision surface meshes directly from Industrial computed tomography (ICT) projection data. Firstly, an adaptive volume octree is recursively constructed from top to bottom using a two-stage geometric error metric function. The CT values and gradient values at the nodes are computed using the Feldkamp–Davis–Kress (FDK) reconstruction algorithm and its derivatives, achieving sub-voxel precision. Next, feature vertices are calculated based on Quadratic error functions (QEFs), and a dual mesh is constructed. Finally, Hermite interpolation is used to determine the iso-surface vertices, and the Convex Contouring lookup table is employed to accurately extract the iso-surface contours, resulting in high-precision and crack-free surface meshes. Experimental results show that the surface meshes generated by the proposed method exhibit superior dimensional accuracy, form and position accuracy, and surface model accuracy compared to traditional methods, and the dimensional accuracy has been enhanced by approximately 10–30%. [ABSTRACT FROM AUTHOR]

The periodic sinc function interpolation offers a compelling solution to address the issue of noise in the analysis of thermogravimetric analysis (TGA) data, thereby enhancing the outcomes of differential techniques such as the Friedman isoconversional method. In this study, we introduce a novel approach that leverages the periodic sinc function interpolation to directly obtain smooth reaction rates from TGA data, eliminating the reliance on numerical differentiation methods. The efficacy of this method has been confirmed through its application to noisy experimental data derived from the thermal decomposition of various polymers, showcasing its robustness. Readers are provided with the corresponding code for Gnu Octave, serving as a free alternative to MATLAB. Additionally, the activation energies calculated from the experimental data using both the Friedman method and periodic sinc function interpolation closely align with those determined by the integral Vyazovkin method, emphasizing the validity and reliability of this new approach. [ABSTRACT FROM AUTHOR]

This paper presents a comparison of numerical properties of the fixed-grid method of characteristics resulting from space and time linear interpolation used for the solution of transient flow in pipe equations. Modified equation analysis method is applied, in which the modified equations derived for the simplified linear version of the transient pipe flow equations provide explicitly the coefficients of numerical diffusion and dispersion generated by the method of characteristics. Through this approach the value of the numerical diffusion coefficient generated in the solution can be precisely estimated. Consequently, the effects resulting from the interpolation in time and space can be compared. The conclusions presented were confirmed by the results of numerical calculations. Practical Applications: If a valve is suddenly closed at the end of a pipeline in which water flows, a pressure wave will appear in it. This phenomenon, known as the water hammer, commonly occurs in water supply networks and in various industrial installations. The wave propagation process is described by a system of differential equations, which are solved using numerical methods. Numerical methods are approximate; they allow to solve the mathematical problems in which both the data and the results of calculations are given in the form of numbers. Numerical methods enable the calculation of approximate pressure and velocity values at selected cross-sections of the pipeline and at selected moments of time. The calculated values contain an error that depends on the solution method. This error determines the quality of the water hammer simulation results. There are a number of methods for numerical solution of water hammer equations. In this paper, the method of characteristics is used, which requires interpolating pressure and velocity values between selected points. The article shows that the generated numerical error depends on the applied technique of interpolation. Two techniques, interpolation in space and interpolation in time, are considered. [ABSTRACT FROM AUTHOR]

Various types of pixel candidates are available in the literature to replace impulse noise after effective detection. However, using them in the correct location and preserving the signal content, structural similarity, and image details is a task that draws attention, especially in a highly corrupted image. Non-linear Diffusion-based restoration is an efficient solution since it can iteratively update corrupted pixels without diffusing the edge. This work assigns the iterative weighted total variation diffusion technique only for the possibly noisy pixels in high noise ratio processing windows where the windows are pre-classified as low or high noise ratio by a custom CNN classifier. The work, called as CNN-based locally adapting filter (CNN-LAF), can achieve a high structural similarity of.9167 by maintaining a PSNR of 24.01 dB at a 0.8 noise ratio. [ABSTRACT FROM AUTHOR]

The main concern is the uncertainty in the real-world solid transportation problem. This study examines a supply, demand, and conveyance capacity-based multi-choice solid stochastic multi-objective transportation problem (MCSS-MOTP). Due to uncertainty, the concrete objective function coefficients of the proposed model are of multivariate type. Furthermore, the parameters of the constraints are treated as independent multivariate random variables with normal distribution. First, a Newton divided difference method-based interpolation polynomial is described that extends an interpolation polynomial using practical properties at non-negative integer nodes to deal with any multiple-choice parameter. Second, the probabilistic constraints are converted into precise ones utilizing a stochastic programming approach. In the end, ranking procedure was used to compare the existing approach with the old models. The proposed model’s applicability was confirmed using a numerical example. [ABSTRACT FROM AUTHOR]

The article focuses on enhancing contextual ranking models by proposing data augmentation methods to improve ranking performance effectively and robustly. It mentions by utilizing supervised and unsupervised augmentation schemes, along with contrastive losses adapted for ranking tasks, the study demonstrates significant performance improvements, particularly in sample efficiency and robustness across in-domain and out-of-domain benchmarks.

In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis. • A discrete analogue of an equality (1.3) by Grisvard [1] on H 2 functions is proved for the reduced rectangular Morley (RRM for short in the sequel) element functions. This discrete equality makes the RRM space usable for bi-Laplacian problems with inhomogeneous coefficients. • Based on piecewise quadratic polynomials, the RRM scheme is the lowest-degree finite element scheme for the inhomogeneous bi-Laplace problems. Compared to other kinds of methods, it does not need tuning parameter or using indirect differential operators. • As revealed by [3] , the RRM element space does not admit a locally-defined projective interpolator. In this paper, however, a locally-defined stable interpolator (not projective) is carefully constructed for the RRM element space, and an optimal approximation is proved rigorously on both convex and nonconvex domains. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study the superconvergence of nonconforming Rannacher-Turek finite element for elliptic interface problems under unfitted square meshes. In particular, we analyze its superclose property between the gradient of the numerical solution and the gradient of the interpolation of the exact solution. Moreover, we introduce a postprocessing interpolation operator which is applied to numerical solution, and we prove that the postprocessed gradient converges to the exact gradient with a superconvergent rate O (h 3 2 ). Finally, numerical results coincide with our theoretical analysis, and they show that the error estimates do not depend on the ratio of the discontinuous coefficients. [ABSTRACT FROM AUTHOR]

A modified wavelet interpolation Galerkin method (WIGM) with improved flexibility of nodal distribution is formulated and utilized to solve the Bratu's problem. Such a method treats the nonlinear term as an independent function and directly approximates it through the proposed wavelet interpolation, resulting in a low-cost discretization of the Bratu's equation with exponential nonlinearity. An efficient implementation of Newton's method to find the lower and upper branches of solution as well as the turning point is then presented. Numerical results demonstrate that the present WIGM can accurately estimate both two solution branches and turning point for the Bratu's problem. Compared with many other existing methods, the proposed WIGM also offers marked superiority in terms of accuracy when using the same node distribution. Moreover, a convergent solution to the resulting nonlinear discrete system obtained by the WIGM can be observed in a few iterations even when the initial guess significantly deviates from the exact solution. Finally, a highly accurate approximation of both two-branched solutions and the turning point for the two-dimensional Bratu's problem is obtained by the proposed WIGM. [ABSTRACT FROM AUTHOR]

Roundoff errors often disrupt interpolation methods for time-fractional equations, potentially causing suboptimal convergence or even failure. These issues primarily result from catastrophic cancellations. To address this, we introduce a novel framework for computing coefficients in standard and fast interpolation methods on nonuniform meshes. We propose δ -cancellation and associated threshold conditions to prevent such cancellations. If the thresholds aren't met, a Taylor expansion technique can be applied. Numerical experiments demonstrate our method's accuracy, on par with the Gauss–Kronrod quadrature, but significantly more efficient, allowing for extensive simulations with hundreds of thousands of time steps. [ABSTRACT FROM AUTHOR]

This paper proposes a Fourier-Legendre spectral method to find the minimizers of a variational problem, called \sigma _{2,p}-energy, in polar coordinates. Let {\mathbb {X}}\subset \mathbb {R}^n be a bounded Lipschitz domain and consider the energy functional (1.1) whose integrand is defined by {\mathbf {W}}(\nabla u(x))≔(\sigma _2(u))^{\frac {p}{2}}+\Phi (\det \nabla u) over an appropriate space of admissible maps, \mathcal {A}_p({\mathbb {X}}). Using Fourier and Legendre interpolation errors, we obtain an error estimate for the energy functional and prove a convergence theorem for the proposed method. Furthermore, we apply the gradient descent method to solve a nonlinear algebraic system which is obtained by discretizing the Euler-Lagrange equations. The numerical experiments are performed to demonstrate the accuracy and effectiveness of our method. [ABSTRACT FROM AUTHOR]

It has been proven that Graph Neural Networks focus more on the majority class instances and ignore minority class instances when the class distribution is imbalanced. To address the class imbalance problems on graphs, most of the existing approaches rely on the availability of minority nodes in the training set, which may be scarce in extremely imbalanced situations and lead to overfitting. To tackle this issue, this paper proposes a novel oversampling-based Neighbor imbalanced-aware Graph Neural Networks, abbreviated as Nia-GNNs. Specifically, we propose a novel interpolation method that selects interpolated minority nodes from the entire dataset according to their predicted labels and similarity. Meanwhile, a class-wise interpolation ratio is applied to prevent the generation of out-of-domain nodes. Additionally, the generated minority nodes are inserted into the neighbor of minority nodes according to their neighbor distribution to balance the graph both neighborly and globally. Numerous experiments on different imbalanced datasets demonstrate the superiority of our method in classifying imbalanced nodes. [ABSTRACT FROM AUTHOR]

Summary: Weighted compact nonlinear schemes (WCNSs) are a popular family of high‐resolution shock‐capturing schemes for simulating compressible flows, of which the nonlinear interpolation procedure is dominant for the performance. In this work, a simplified weighting strategy is introduced for the nonlinear interpolation procedure. Firstly, an equivalent weighting formulation of WCNS is presented by explicitly including the whole‐point stencil into the set of candidate stencils. Secondly, motivated by the reorganization of WCNS, the WCNS‐CU6 scheme is achieved in a more straightforward way. Thirdly, by introducing a TENO selection procedure in the framework of WCNS‐CU6‐Simplified, a TCNS6‐Simplified scheme is proposed, the resolution of which is comparable with the excellent TENO6 scheme, while the computational cost is much lower. The simplified schemes exhibit more outstanding, at least comparable, fidelity than the original schemes, however, with superior characteristics in terms of efficiency and simplicity. A variety of benchmark test problems are studied to demonstrate the behaviour of the simplified weighting strategy. [ABSTRACT FROM AUTHOR]

A displacement‐only finite‐element formulation of strain‐gradient models requires elements with C1$$ {C}^1 $$ continuous interpolation. Mixed formulations have been proposed to allow the use of more common C0$$ {C}^0 $$ element shape functions. These mixed formulations are based on the interpolation of two different fields, displacement and some kind of displacement gradient, with the relation between the two fields enforced using either Lagrange multipliers or penalty methods. All elements proposed in the literature for such formulations use a distinct set of degrees of freedom to discretise each field. In this work, we introduce for the first time shared degrees of freedom, that lead to a mixed formulation with a significantly better numerical performance. We describe how this novel mixed formulation can be derived, present individual elements implementing this, and discuss the significance of the results. [ABSTRACT FROM AUTHOR]

Eight spatial interpolation models were used to map the spatial distribution of precipitation in the southwestern sector of the Iberian Peninsula (22330 km2) over 40 years (1980/1981–2019/2020). Rainfall data from 103 meteorological stations were used to generate the interpolation models, namely inverse distance weight (IDW) with 6, 12 and 24 points, regression spline (RS), thin spline (TS), universal kriging with spherical and Gaussian variogram (UK_Sphe and UK_gauss, respectively) and multilinear regression (MR), based on physiographic and geographic variables. Furthermore, 32 rainfall stations were used to assess the performance of the previous methods through 7 statistical metrics Ordinary Least Squares (OLS), Root Mean Square Error (RMSE), Normalized root mean square error (NRMSE), Coefficient of determination (R2), Nash-Sutcliffe efficiency coefficient (NSE), Mean Absolute Error (Bias and MAE). Based on these metrics, UK_gauss and IDW_6 provided the best adjustments, whereas MR presented the highest errors. All methods were suitable to predict the spatial distribution of rainfall, but adjustments are conditioned by the features of the study area, gauge density and gauge spatial distribution. [ABSTRACT FROM AUTHOR]

Due to various unavoidable reasons or gross error elimination, missing data inevitably exist in global navigation satellite system (GNSS) position time series, which may result in many analysis methods not being applicable. Typically, interpolating the missing data is a crucial preprocessing step before analyzing the time series. The conventional methods for filling missing data do not consider the influence of adjacent stations. In this work, an improved Gaussian process (GP) approach is developed to fill the missing data of GNSS time series, in which the time series of adjacent stations are applied to construct impact factors, together with a comparison of the conventional GP and the commonly used cubic spline methods. For the simulation experiments, the root mean square error (RMSE), mean absolute error (MAE) and correlation coefficient (R) are adopted to evaluate the performance of the improved GP. The results show that the filled missing data of the improved GP are closer to the true values than those of the conventional GP and cubic spline methods, regardless of the missing percentages ranging from 5 to 30%, with an interval of 5%. Specifically, the mean relative RMSE and MAE improvements for the improved GP with respect to the conventional GP are 21.2%, 21.3% and 8.3% and 12.7%, 16.2% and 11.01% for the North (N), East (E) and Up (U) components, respectively. In the real experiment, eight GNSS stations are analyzed using improved GP, together with conventional GP and a cubic spline. The results indicate that the first three principal components (PCs) of the improved GP can perverse 98.3%, 99.8% and 77.0% of the total variance for the N, E and U components, respectively. This value is obviously higher than those of the conventional GP and cubic spline. Therefore, we can conclude that the improved GP can better fill in the missing data in GNSS position time series than the conventional GP and cubic spline because of the impacts of adjacent stations. [ABSTRACT FROM AUTHOR]

ABSTRACT With the widespread worldwide adoption of location‐based service technologies, accurate and reliable driving trajectories have become crucial. However, because of the inherent deficiencies of sensor devices, accurate road matching results may not always be obtained directly from trajectory data, which poses a challenge for many location and trajectory based services. Existing map‐matching techniques mainly focus on high‐sampling‐rate trajectory data while paying relatively less attention to low‐frequency trajectory data. Low‐sampling‐rate trajectory data have greater matching difficulties than high‐sampling‐rate data owing to the limited available information. Moreover, in the case of signal loss or interference, the accuracy of map‐matching algorithms can decrease significantly for low‐sampling‐rate data. To achieve accurate map‐matching results for low‐sampling‐rate trajectory data, this study proposes a map‐matching algorithm based on probability interpolation. First, the trajectory data are denoised to eliminate redundant trajectory points. Second, the concept of the probability truth value is introduced to handle the relationship between the interpolated virtual points and actual sampled trajectory points accurately. A higher probability truth value indicates a higher confidence level of the interpolation. Third, the denoised trajectory data are interpolated and a probability truth value is assigned based on the interpolation accuracy. Finally, a comprehensive probability composed of the probability truth value, emission probability, and transition probability is used to determine the correctly matched road segments. Experimental results on real trajectory datasets demonstrated that the proposed algorithm outperformed several advanced algorithms in terms of accuracy and performance. [ABSTRACT FROM AUTHOR]

We prove error estimates for the Raviart-Thomas interpolation in weighted L 2 -norm for functions in appropriate weighted Sobolev spaces. These results allow us to obtain a priori error estimates in the fractional order case for mixed approximations of degenerate elliptic problems. [ABSTRACT FROM AUTHOR]

An increasingly common viewpoint is that protein dynamics datasets reside in a nonlinear subspace of low conformational energy. Ideal data analysis tools should therefore account for such nonlinear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich mathematical structure to account for a wide range of geometries that can be modeled after an energy landscape. Second, many standard data analysis tools developed for data in Euclidean space can be generalized to Riemannian manifolds. In the context of protein dynamics, a conceptual challenge comes from the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium-sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop. [ABSTRACT FROM AUTHOR]

The small, two-verse interpolation of Ezekiel 47:22-23 is widely considered one of the most inclusive texts in the Hebrew Bible. Its instruction regarding the גרים allowed for full inclusion into the community that Ezekiel believed to be the true Israel. However, interpretations that view these verses as hyper-inclusive often view the גרים as foreigners. Such a view is largely based on understanding the term גר along linguistic rather than anthropological lines. This paper will explore the use of the term גר in both Ezekiel and the Holiness School to better understand the group envisioned by the author of Ezekiel 47:22-23. In so doing, I will demonstrate that these verses should not be understood simply along a binary of inclusive versus exclusive, but rather as part of a larger project of identity reformulation that occurred as part of a trauma process around the Babylonian exile. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we discuss a novel model reduction framework for linear structured dynamical systems. The transfer functions of these systems are assumed to have a special structure, for example, coming from second‐order linear systems or time‐delay systems, and they may also have parameter dependencies. Firstly, we investigate the connection between classic interpolation‐based model reduction methods with the reachability and observability subspaces of linear structured parametric systems. We show that if enough interpolation points are taken, the projection matrices of interpolation‐based model reduction encode these subspaces. Consequently, we are able to identify the dominant reachable and observable subspaces of the underlying system. Based on this, we propose a new model reduction algorithm combining these features and leading to reduced‐order systems. Furthermore, we discuss computational aspects of the approach and its applicability to a large‐scale setting. We illustrate the efficiency of the proposed approach with several numerical large‐scale benchmark examples. [ABSTRACT FROM AUTHOR]

Machine learning has been widely used to guide the search for new materials by learning the patterns underlying available data. However, the pure prediction-oriented search is often biased to interpolation due to the limited data in a large unexplored space. Here we present a sampling framework towards extrapolation, that integrates unsupervised clustering, interpretable analysis, and similarity evaluation to sample target candidates with improved properties from a vast search space. Using the design of superalloys with improved γ ′ -phase solvus temperature ( T γ ′ ) as a model case, we start with sparse data, and by a few experiments, we find nine new superalloys with chemistries distinct to those in the training data. Three of them show improved T γ ′ by about 50 °C, a large enhancement for superalloys. Moreover, we find two features characterizing mismatch in atomic size and mixing enthalpy linearly effect. This work demonstrates the capability of unsupervised learning to search for new materials when limited data is available. [ABSTRACT FROM AUTHOR]

In this paper, we designed and analyzed a weak Galerkin finite element method on layer adapted meshes for solving the time-dependent convection-dominated problems. Error estimates for semi-discrete and fully-discrete schemes were presented, and the optimal order of uniform convergence has been obtained. A special interpolation was delicately designed based on the structures of the designed method and layer-adapted meshes. We provided various numerical examples to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

Geospatial interpolation plays a pivotal role in spatial analysis because it provides high-quality data support for various spatiotemporal data mining (STDM) tasks. However, statistical methods, such as kriging, face challenges in dealing with complex geo-big data. Additionally, deep-learning-based methods, despite their exceptional performance, suffer from limitations, such as poor interpretability. To harness the complementary advantages of these statistical methods and deep learning approaches, this study proposes a novel geospatial artificial intelligence (GeoAI) framework called deep kriging neural network (DKNN). The primary contribution lies in the development of an asymmetric encoder-decoder structure, which includes a deep-learning-based spatial encoder and a geostatistics-based kriging decoder. The spatial encoder consists of three specialized neural networks, whereas the kriging decoder relies on the proposed unified kriging system. During forward propagation, the kriging decoder leverage messages from the spatial encoder to generate interpolation weights for prediction. Conversely, during backward propagation, the kriging decoder guides the spatial encoder in learning interpretable knowledge. Experiments were conducted using both synthetic and practical datasets. The results demonstrate an average improvement of 20.18% in MAE, 25.04% in RMSE and 24.06% in MAPE when compared to the best-performing baseline method. Furthermore, these results confirm the superior interpretability of our DKNN framework. [ABSTRACT FROM AUTHOR]

In the fixed time-step electromagnetic transient (EMT)-type program, an interpolation process is applied to deal with switching events. The interpolation method frequently reduces the algorithm's accuracy when dealing with power electronics. In this study, we use the Butcher tableau to analyze the defects of linear interpolation. Then, based on the theories of Runge–Kutta integration, we propose two three-stage diagonally implicit Runge–Kutta (3S-DIRK) algorithms combined with the trapezoidal rule (TR) and backward Euler (BE), respectively, with TR-3S-DIRK and BE2-3S-DIRK for the interpolation and synchronization processes. The proposed numerical integral interpolation scheme has second-order accuracy and does not produce spurious oscillations due to the size change in the time step. The proposed method is compared with the critical damping adjustment method (CDA) and the trapezoidal method, showing that it does not produce spurious numerical oscillations or first-order errors. [ABSTRACT FROM AUTHOR]

Unlike uniform linear arrays (ULAs), coprime arrays require fewer physical sensors yet provide higher degrees of freedom (DOF) and larger array apertures. However, due to the existence of "holes" in the differential co-array, the target detection performance deteriorates, especially in adaptive beamforming. To address these challenges, this paper proposes an efficient and robust adaptive beamforming algorithm leveraging coprime array interpolation. The algorithm eliminates unwanted signals and uses the Gauss–Legendre quadrature method to reconstruct an Interference-plus-Noise Covariance Matrix (INCM), thereby obtaining the beamforming coefficients. Unlike previous techniques, we utilize a virtual interpolated ULA to expand the aperture, enabling the acquisition of a high-dimensional covariance matrix. Additionally, a projection matrix is constructed to eliminate unwanted signals from the received data, greatly enhancing the accuracy of INCM reconstruction. To address the high computational complexity of integral operations used in most INCM reconstruction algorithms, we propose an approximation based on the Gauss–Legendre quadrature, which reduces the computational load while maintaining accuracy. This algorithm avoids the array aperture loss caused by using only the ULA segment in the difference co-array and improves the accuracy of INCM reconstruction. Simulation and experimental results show that the performance of the proposed algorithm is superior to the compared beamformers and is closer to the optimal beamformer in various scenarios. [ABSTRACT FROM AUTHOR]

Working with pushbroom imagery in photogrammetry and remote sensing presents a fundamental challenge in object-to-image space transformation. For this transformation, accurate estimation of Exterior Orientation Parameters (EOPs) for each scanline is required. To tackle this challenge, Best Scanline Search or Determination (BSS/BSD) methods have been developed. However, the current BSS/BSD methods are not efficient for real-time applications due to their complex procedures and interpolations. This paper introduces a new non-iterative BSD method specifically designed for line-type pushbroom images. The method involves simulating a pair of sets of points, Simulated Control Points (SCOPs), and Simulated Check Points (SCPs), to train and test a Multilayer Perceptron (MLP) model. The model establishes a strong relationship between object and image spaces, enabling a direct transformation and determination of best scanlines. This proposed method does not rely on the Collinearity Equation (CE) or iterative search. After training, the MLP model is applied to the SCPs for accuracy assessment. The proposed method is tested on ten images with diverse landscapes captured by eight sensors, exploiting five million SCPs per image for statistical assessments. The Root Mean Square Error (RMSE) values range between 0.001 and 0.015 pixels across ten images, demonstrating the capability of achieving the desired sub-pixel accuracy within a few seconds. The proposed method is compared with conventional and state-of-the-art BSS/BSD methods, indicating its higher applicability regarding accuracy and computational efficiency. These results position the proposed BSD method as a practical solution for transforming object-to-image space, especially for real-time applications. [ABSTRACT FROM AUTHOR]

Let M$\mathcal {M}$ be a semifinite von Neumann algebra equipped with an increasing filtration (Mn)n⩾1$(\mathcal {M}_n)_{n\geqslant 1}$ of (semifinite) von Neumann subalgebras of M$\mathcal {M}$. For 1⩽p⩽∞$1\leqslant p \leqslant \infty$, let Hpc(M)$\mathcal {H}_p^c(\mathcal {M})$ denote the noncommutative column martingale Hardy space constructed from column square functions associated with the filtration (Mn)n⩾1$(\mathcal {M}_n)_{n\geqslant 1}$ and the index p$p$. We prove the following real interpolation identity: If 0<θ<1$0<\theta <1$ and 1/p=1−θ$1/p=1-\theta$, then H1c(M),H∞c(M)θ,p=Hpc(M).$$\begin{equation*} \hspace*{48pt}{\left(\mathcal {H}_1^c(\mathcal {M}), \mathcal {H}_\infty ^c(\mathcal {M})\right)}_{\theta,p}=\mathcal {H}_p^c(\mathcal {M}).\hspace*{-48pt} \end{equation*}$$This is new even for classical martingale Hardy spaces as it is previously known only under the assumption that the filtration is regular. We also obtain an analogous result for noncommutative column martingale Orlicz–Hardy spaces. [ABSTRACT FROM AUTHOR]

We study multiple orthogonal polynomials exploiting their explicit determinantal representation in terms of moments. Our reasoning follows that applied to solve the Hermite–Padé approximation and interpolation problems. We also study families of multiple orthogonal polynomials obtained by variation of the measures known from the theory of discrete‐time Toda lattice equations. We present determinantal proofs of certain fundamental results of the theory, obtained earlier by other authors in a different setting. We also derive quadratic identities satisfied by the polynomials, which are new elements of the theory. Resulting equations allow to present multiple orthogonal polynomials within the theory of integrable systems. [ABSTRACT FROM AUTHOR]

Groundwater is a valuable and universally distributed resource on Earth. Understanding the spatial and temporal dynamics of groundwater is of utmost importance for effective management. Normally, groundwater levels are recorded at arbitrary points, but groundwater modeling requires interpolating the measured values at specific grid nodes. This study aims to dentify and evaluate the geographical variations of groundwater levels in Türkiye using three geostatistical interpolation techniques. Data from 355 groundwater wells from 1970 to 2019 were used for this purpose. In addition, an investigation of changes in annual average temperature and precipitation was conducted for two different time periods: 1985–2000 and 2001–2016. The results show an increase in the annual average temperature in Türkiye by 0.82 °C during the reference period (1985–2000). Despite regional differences in the precipitation regime, the average annual precipitation in Türkiye has not changed significantly overall. Especially in the Meriç-Ergene, Konya Closed (Konya Kapalı), and Euphrates-Tigris basins, a significant decrease in groundwater levels was observed, even though this decrease is less than 100 m in some wells. After a comprehensive analysis of all these data, possible explanations for the changes in groundwater levels were considered. [ABSTRACT FROM AUTHOR]

This paper reports a comparison of different spatial and statistical models to predict the concentrations of the particulate matter (PM10), measured by the environmental stations of the Community of Madrid (Spain). Three methods were compared: Inverse Distance Weighting (IDW), Ordinary Kriging (OK) and Empirical Bayesian Kriging (EBK). The most accurate spatial interpolation method was the EBK. An interpolation map obtained by applying the EBK geostatistical method is presented to identify the areas with the highest pollution of PM10.] [ABSTRACT FROM AUTHOR]