Interpolation Error Research Articles

Error-constant estimation under the maximum norm for linear Lagrange interpolation

For the linear Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite-element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the space decomposition using the Fujino–Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Numerical results for the lower and upper bounds of the interpolation error constant for triangles of various types are presented to verify the efficiency of the proposed method.

The Shape Parameter in the Shifted Surface Spline—An Easily Accessible Approach

In this paper, we present an easily accessible approach to finding a suitable shape parameter in the shifted surface spline for function interpolation. We aim at helping more readers, including mathematicians and non-mathematicians, to use our method to solve practical problems. Hence, some highly complicated mathematical theorems and definitions are avoided. The major requirement, as in our previous approach, that the data points should be evenly spaced in the domain is also relaxed. This means that the data points are purely scattered without restrictions. The drawback is that the shape parameter thus obtained is not exactly the same as the theoretically predicted optimal value, which can always be achieved by using our previous rigorous approach. However, experiments show that the gap is quite small and the final interpolation errors thus obtained are satisfactory.

Current meter methodology for discharge measurement in circular pipe

ABSTRACT One of the main methods used for determining the discharge of a high head hydropower plant equipped with a circular penstock is the current meter method. The article presents a simple, but safe way to increase the accuracy of discharge measurement by means of the current meter method based on a new methodology for location velocity measuring points in the measuring section of a circular conduit. The proposed methodology utilizes the interdependence between the admissible interpolation and the radial distribution of the measuring points for diminishing the coincidence error, adopting the interpolation law proposed by Winternitz for this goal, a law that is both simple and whose calculation error is low. At the same time, the new methodology envisages increasing the number of intervals the admissible interpolating function domain is divided into through the measuring point lay out in a circular measuring section along a spiral considering that the interpolation error is inversely proportional to this number of intervals.

Kalman-Filter-Based, Dynamic 3-D Shape Reconstruction for Steerable Needles With Fiber Bragg Gratings in Multicore Fibers

Steerable needles are a promising technology to provide safe deployment of tools through complex anatomy in minimally invasive surgery, including tumor-related diagnoses and therapies. For the 3-D localization of these instruments in soft tissue, fiber Bragg gratings (FBGs) based reconstruction methods have gained in popularity because of the inherent advantages of optical fibers in a clinical setting, such as flexibility, immunity to electromagnetic interference, nontoxicity, and the absence of line-of-sight issues. However, methods proposed thus far focus on shape reconstruction of the steerable needle itself, where accuracy is susceptible to errors in interpolation and curve fitting methods used to estimate the curvature vectors along the needle. In this study, we propose reconstructing the shape of the path created by the steerable needle tip based on the follow-the-leader nature of many of its variants. By assuming that the path made by the tip is equivalent to the shape of the needle, this novel approach paves the way for shape reconstruction through a single set of FBGs at the needle tip, which provides curvature information about every section of the path during navigation. We propose a Kalman-filter-based sensor fusion method to update the curvature information about the sections as they are continually estimated during the insertion process. The proposed method is validated through simulation, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">in vitro</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ex vivo</i> experiments employing a programmable bevel-tip steerable needle (PBN). The results show clinically acceptable accuracy, with 2.87-mm mean PBN tip position error, and a standard deviation of 1.63 mm for a 120-mm 3-D insertion.

Calibration of reference torque transducer in one direction and use of its cubic coefficients in both directions with improved interpolation error

Calibration of reference torque transducer in one direction and use of its cubic coefficients in both directions with improved interpolation error

Consistency analysis and bias elimination of the Instrumental-Variable-based State Variable Filter method

In this paper, we analyse the consistency of the Instrumental-Variable-based State Variable Filter (IVSVF) estimator by taking into account the intersample behaviour of the input and output signals. It is found that when only sampled input and output data are available for estimation, the IVSVF estimator is not consistent for any fixed sampling period due to the interpolation error that arises from constructing the filtered output. A Bias-Eliminated IVSVF (BEIVSVF) estimator is then proposed and shown to be consistent. The theoretical results developed in the paper are also discussed from a practical standpoint. Simulations are performed to verify the performance of the proposed method as well as to support the theoretical results.

An improved method for real-time PPP timing and time transfer with broadcast ephemerides

Due to the problem of network communication, it is difficult to ensure the reliability of real-time precise point positioning (PPP) timing/time transfer with real-time precise products. With the continued reduction in the signal-in-space range error, the performance and feasibility of GPS and Galileo PPP timing/time transfer with broadcast ephemeris were analyzed for the first time in real time. Then, we present a smoothing method and time-series decomposition method to reduce the noise and the interpolation error for GPS and Galileo PPP timing\ ime transfer with broadcast ephemeris in real time. The results show that GPS or Galileo PPP timing with broadcast ephemeris can achieve a 4 × 1 × 10−14 level at 15 360 s in the current state. The accuracy is about (0.46–0.81) ns and (0.44–0.61) ns for GPS and Galileo PPP time transfer. The frequency stability is at about 7.0 × 1 × 10−14 and 5.0 × 1 × 10−14 levels at 15 360 s for GPS and Galileo PPP. It is important to note that by applying our approach, the maximum improvement in frequency stability for GPS and Galileo PPP timing/time transfer is up to 99%. Furthermore, the average accuracy of GPS or Galileo PPP time transfer can achieve approximately 0.3 ns, which is an improvement of up to 67.3% compared to the traditional method.

Adaptive Sampling for Interpolation of Reduced-Order Aeroelastic Systems

A new strategy for the interpolation of parametric reduced-order models of dynamic aeroelastic systems is introduced. Its aim is to accelerate the numerical exploration of geometrically nonlinear aeroelastic systems over large design spaces or multiple flight conditions. The parametric reduced-order models are obtained from high-dimensional models by Krylov-subspace projection. They are subsequently interpolated to acquire realizations inexpensively anywhere in the parameter space, where having the state space as opposed to interpolating an output metric permits the use of linear analysis tools. The interpolation scheme is heavily conditioned by the available training points; thus, a novel methodology based on adaptive sampling and a combinatorial use of the available true-systems knowledge is presented, whereby we reuse all known data of the true models as different combinations of training and testing data to build statistical surrogates of the interpolation error across the parameter space and refine the sampling in those regions that need it. This minimizes the number of costly to evaluate function calls and ensures that parameter space regions are sampled according to the underlying system dynamics. The initial implementation of this adaptive sampling strategy is demonstrated on a very flexible wing with a complex stability envelope.

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

We present a class of orthogonal functions on infinite domain based on Jacobi polynomials. These functions are generated by applying a tanh transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo‐derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced tanh Jacobi functions, we develop an efficient spectral tanh Jacobi collocation method for the numerical simulation of nonlinear Schrödinger equations on the infinite domain without using artificial boundary conditions. The applicability and accuracy of the solution method are demonstrated by two numerical examples for solving the nonlinear Schrödinger equation and the nonlinear Ginzburg–Landau equation.

Three‐Bits Linear Interpolation Error Signal Based ECG Beat Classification

About three million people die from sudden cardiac arrest (SCA) or infarction each year. It is the biggest killer on the planet and it can happen to anyone, anywhere, and anytime. Early and effective diagnosis from ECG signals can help save a large part of these SCAs. Premature ventricular contraction (PVC) detection and classification was performed with different methods. A new method of automatic classification of PVC based on a three-bit linear interpolation error signal (LIES) is proposed in this paper. We start by a QRS detection, a nonlinear transformation and then a sliding window. This method was tested on normal and abnormal beats obtained from MIT-BIH and AHA databases. The proposed algorithm with a sensitivity 94.8% and a specificity was 98.6%. is accurate. This algorithm is also automatic, robust, and fast. © 2022 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Improved multi-dimensional dynamic programming energy management strategy for a vehicle power-split hybrid powertrain

Dynamic programming is a widely used algorithm to solve the optimal control problem for hybrid electric vehicle in the whole driving scenario, but the problems of interpolation error and dimensional disaster have not been completely solved. This study proposes an adaptive adjustment method to solve the interpolation error problem, in which the solution without theoretical error can be obtained at the cost of a tiny driving cycle accuracy. Furthermore, a dynamic equivalent consumption factor calculation method is proposed to analyze the change of energy storage quality in the battery pack and the instantaneous equivalent consumption of the motors. The effectiveness of the improved algorithm is verified on a passenger car with hybrid powertrain using planetary gear mechanism. The results show that the established multi-dimensional dynamic programming algorithm has a relatively stable energy-saving ability. The maximum gap of fuel consumption rate is 3.2% in the four test driving cycles. Besides, there is only a maximum difference of 0.29‱ between the original driving cycles and the new cycles generated by the adaptive adjustment method. And obvious dynamic equivalent consumption factor changes of 15.7% in WLTC-class3, 10.3% in CLTC-P, 11.8% in FTP75 and 11.0% in NEDC are observed through the proposed calculation method.

Interpolation and stability estimates for edge and face virtual elements of general order

We develop interpolation error estimates for general order standard and serendipity edge and face virtual elements in two and three dimensions. Contextually, we investigate the stability properties of the associated [Formula: see text] discrete bilinear forms. These results are fundamental tools in the analysis of general order virtual elements, e.g. for electromagnetic problems.

Virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media

In this paper, we propose a conforming virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media. By using [Formula: see text]-coercivity theory, the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove the spectral convergence of the discrete scheme and the optimal a priori error estimates for the discrete eigenvalues and eigenfunctions. The virtual element method has great flexibility in handling polygonal meshes, which motivates us to construct a fully computable a posteriori error estimator for the virtual element method. Then the upper bound of the approximation error is derived from the residual equation and the inf-sup condition. In turn, the related lower bound is established by using the bubble function strategy. Finally, we provide numerical examples to verify the theoretical results, including the optimal convergence of the virtual element scheme on uniformly refined meshes and the efficiency of the estimator on adaptively refined meshes.

A Library of Synthetic X-Ray Spectra for Fitting Tidal Disruption Events

We present a tabulated version of our slim-disk model for fitting tidal disruption events (TDEs). We create a synthetic X-ray spectral library by ray-tracing stationary general relativistic slim disks and including gravitational redshift, Doppler, and lensing effects self-consistently. We introduce the library to reduce computational expense and increase access for fitting future events. Fitting requires interpolation between the library spectra; the interpolation error in the synthetic flux is generally <10% (it can rise to 40% when the disk is nearly edge-on). We fit the X-ray spectra of the TDEs ASASSN-14li and ASASSN-15oi, successfully reproducing our earlier constraints on black hole mass M • and spin a • from full on-the-fly ray-tracing. We use the library to fit mock observational data to explore the degeneracies among parameters, finding that (1) spectra from a hotter thermal disk and edge-on inclination angle offer tighter constraints on M • and a •; (2) the constraining power of spectra on M • and a • increases as a power law with the number of X-ray counts, and the index of the power law is higher for hotter thermal disk spectra; (3) multiepoch X-ray spectra partially break the degeneracy between M • and a •; (4) the time-dependent level of X-ray absorption can be constrained from spectral fitting. The tabulated model and slim-disk model are available at https://doi.org/10.25739/hfhz-xn60.

A metric tensor approach to data assimilation with adaptive moving meshes

Adaptive moving spatial meshes are useful for solving physical models given by time-dependent partial differential equations. However, special consideration must be given when combining adaptive meshing procedures with ensemble-based data assimilation (DA) techniques. In particular, we focus on the case where each ensemble member evolves independently upon its own mesh and is interpolated to a common mesh for the DA update. This paper outlines a framework to develop time-dependent reference meshes using locations of observations and the metric tensors (MTs) or monitor functions that define the spatial meshes of the ensemble members. We develop a time-dependent spatial localization scheme based on the metric tensor (MT localization). We also explore how adaptive moving mesh techniques can control and inform the placement of mesh points to concentrate near the location of observations, reducing the error of observation interpolation. This is especially beneficial when we have observations in locations that would otherwise have a sparse spatial discretization. We illustrate the utility of our results using discontinuous Galerkin (DG) approximations of 1D and 2D inviscid Burgers equations. The numerical results show that the MT localization scheme compares favorably with standard Gaspari-Cohn localization techniques. In problems where the observations are sparse, the choice of common mesh has a direct impact on DA performance. The numerical results also demonstrate the advantage of DG-based interpolation over linear interpolation for the 2D inviscid Burgers equation.

ALGORITHM AND SOFTWARE TOOLS FOR IMPROVING INTERPOLATION ACCURACY BASED ON CUBIC SPLINE

In this work, the construction of the cubic spline for the continuous function defined in the segment, as well as the approximation algorithm of the interpolation process and the software for determining the interpolation error were developed. We used cubic splines of the third degree in this work. In the spline definition formula, the value of the coefficient is expressed by the distance between the nodes of the function and the distance between the nodes. The spline-functions method was improved based on the result of analytical analysis to study the methods of approximating functions and functional connections obtained from experience

Logarithmic Jacobi collocation method for Caputo–Hadamard fractional differential equations

We introduce a class of orthogonal functions associated with integral and fractional differential equations with a logarithmic kernel. These functions are generated by applying a log transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo-derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced logarithmic Jacobi functions, we develop an efficient spectral logarithmic Jacobi collocation method for the integrated form of the Caputo–Hadamard fractional nonlinear differential equations. To demonstrate the proposed approach's spectral accuracy, an error estimate is derived, which is then confirmed by numerical results.

Some Estimates for Virtual Element Methods in Three Dimensions

Abstract Some estimates on virtual element methods (VEMs), including inverse inequalities, norm equivalence, and interpolation error estimates, are developed for star-shaped polyhedral meshes whose faces admit virtual regular and quasi-uniform triangulations. This mesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are carried out in a straightforward way.

Direction-of-arrival estimation for circular partial coprime array via nuclear norm optimization model after coprime interpolation

This paper introduces a special nonlinear generalized coprime array, namely circular partial coprime array (CPCA). The antennas of CPCA are distributed on a circle, but after the antenna positions of CPCA are orthogonally projected to a certain baseline, the antenna spacing satisfies the coprime spacing characteristic of linear coprime array (LCA). Because the CPCA signal cannot be equivalent to the virtual difference coarray signal of LCA after vectoring the covariance matrix of incoming signals, the existing direction-of-arrival (DOA) estimation ideas of LCA model will be invalid in CPCA model. To make full use of the coprime information of CPCA model, the coprime interpolation method which can transform the steering manifolds of CPCA into the steering manifolds of expected LCA (ELCA) will be proposed. To alleviate the negative impact of interpolation error caused by coprime interpolation method on DOA estimation accuracy of CPCA model, a sparse Hermitian Toeplitz matrix (HTM) related to the spatial smoothed matrix will be recovered by the nuclear norm optimization model, where recovering the sparse part of HTM is equivalent to fitting the virtual hole data. Therefore, the recovered HTM can improve the virtual degree of freedom (DOF) of ELCA, and the high-precise DOA estimation can be obtained by combining the recovered HTM and MUSIC spectrum. In addition, the influence of inevitable interpolation error on the CPCA signal model is quantitatively analysed. Finally, simulation experiments verify the effectiveness and compatibility of the coprime interpolation method, and show the estimation performance advantages of the proposed method in CPCA model.

Jacobi collocation method and smoothing transformation for numerical solution of neutral nonlinear weakly singular Fredholm integro-differential equations

In this work, we consider Jacobi collocation method for the numerical solution of neutral nonlinear weakly singular Fredholm integro-differential equations. We propose conditions that under them the equation has a unique solution. Since the second derivative of the solution is singular at the end points of the interval, we have to use a smoothing transformation to improve the smoothness of the solution. Then, we use Jacobi points systems to collocate the solution and also Gauss quadrature rules to approximate the integrals. For the error analysis, we define integral and quadrature operators and using a result for the interpolation error of functions with limited regularity, we derive an error bound for the numerical solution. The proposed method is applied on some numerical examples and the errors are reported for different parameters of the smoothing transformation.