Bivariate Interpolation Research Articles

Calculation and Analysis of Equilibrium Position of Aerostatic Bearings Based on Bivariate Interpolation Method

The solution of equilibrium positions is a critical component in the calculation of the dynamic characteristic coefficients of aerostatic bearings. The movement of the rotor in one direction leads to bidirectional variations in the air film force, resulting in low efficiency when using conventional calculation methods. It can even lead to iterative divergence if the initial value is improperly selected. This study concentrates on the orifice throttling aerostatic bearings and proposes a novel method called the bivariate interpolation method (BIM) to calculate the equilibrium position. The equilibrium equation for the rotor under the combined influence of air film forces, gravity, and external loads is established. A calculation program based on the finite difference method is developed to determine the equilibrium position. The process of solving the equilibrium position and the convergence is compared with the secant method and the search method. Furthermore, the variation trend of the equilibrium position and stiffness when the external loads changes are studied based on the BIM. Finally, the correctness of the BIM to solve the equilibrium position is proved by comparing it with the experiment results. The calculation results indicate that the BIM successfully resolves the problem of initial value selection and exhibits superior computational efficiency and accuracy. The equilibrium position initially moves away from the direction of the external load as the load increases, and then this gradually approaches the load direction. The main stiffness increases with increases in the external load, while the variation in cross stiffness depends on the direction of the external load.

Polyharmonic splines for interpolation over sun path

The optimization of solar tower power plants and other concentrating solar power (CSP) systems requires a customized approach for interpolation over the annual sun path. A good interpolation should predict accurately the efficiency of the system for different positions of the sun by using a small number of sampling points. This is very important to reduce the amount of simulations (e.g. ray tracing, cone optics, etc.), which are necessary to evaluate the annual performance for different CSP system configurations. From a mathematical point of view, the interpolation over sun path is an example of bivariate interpolation over a domain of irregular shape, and many methods are available to address it. In this paper, the advantages of meshless methods are investigated. The obtained results show that polyharmonic splines provide a good fit for the optical efficiency of the heliostat field of a solar tower system. The interpolated optical efficiency simplifies the calculation of annual energy reflected from the heliostat field to the receiver, and the result has an absolute error of 0.1% for less than 30 sampling points.

Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square [−1,1]2. And, we prove that the growth order of the Lebesgue constant is O((n + 2)2). This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square [−1,1]2, the growth order of which is O((lnn)2).

Bivariate Polynomial Coding for Efficient Distributed Matrix Multiplication

Coded computing is an effective technique to mitigate “stragglers” in large-scale and distributed matrix multiplication. In particular, univariate polynomial codes have been shown to be effective in straggler mitigation by making the computation time depend only on the fastest workers. However, these schemes completely ignore the work done by the straggling workers resulting in a waste of computational resources. To reduce the amount of work left unfinished at workers, one can further decompose the matrix multiplication task into smaller sub-tasks, and assign multiple sub-tasks to each worker, possibly heterogeneously, to better fit their particular storage and computation capacities. In this work, we propose a novel family of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bivariate polynomial codes</i> to efficiently exploit the work carried out by straggling workers. We show that bivariate polynomial codes bring significant advantages in terms of upload communication costs and storage efficiency, measured in terms of the number of sub-tasks that can be computed per worker. We propose two bivariate polynomial coding schemes. The first one exploits the fact that bivariate interpolation is always possible on a rectangular grid of evaluation points. We obtain such points at the cost of adding some redundant computations. For the second scheme, we relax the decoding constraints and require decodability for almost all choices of the evaluation points. We present interpolation sets satisfying such decodability conditions for certain storage configurations of workers. Our numerical results show that bivariate polynomial coding considerably reduces the average computation time of distributed matrix multiplication. We believe this work opens up a new class of previously unexplored coding schemes for efficient coded distributed computation.

THE CALCULUS OF BIVARIATE FRACTAL INTERPOLATION SURFACES

In this paper, we investigate partial integrals and partial derivatives of bivariate fractal interpolation functions (FIFs). We also prove that the mixed Riemann–Liouville fractional integral and derivative of order [Formula: see text], of bivariate FIFs are again bivariate interpolation functions corresponding to some iterated function system (IFS). Furthermore, we discuss the integral transforms and fractional order integral transforms of the bivariate FIFs.

A New Approach to Newton-Type Polynomial Interpolation with Parameters

Newton’s interpolation is a classical polynomial interpolation approach and plays a significant role in numerical analysis and image processing. The interpolation function of most classical approaches is unique to the given data. In this paper, univariate and bivariate parameterized Newton-type polynomial interpolation methods are introduced. In order to express the divided differences tables neatly, the multiplicity of the points can be adjusted by introducing new parameters. Our new polynomial interpolation can be constructed only based on divided differences with one or multiple parameters which satisfy the interpolation conditions. We discuss the interpolation algorithm, theorem, dual interpolation, and information matrix algorithm. Since the proposed novel interpolation functions are parametric, they are not unique to the interpolation data. Therefore, its value in the interpolant region can be adjusted under unaltered interpolant data through the parameter values. Our parameterized Newton-type polynomial interpolating functions have a simple and explicit mathematical representation, and the proposed algorithms are simple and easy to calculate. Various numerical examples are given to demonstrate the efficiency of our method.

A comparative assessment of different interpolation algorithms for prediction of GNSS/levelling geoid surface using scattered control data

The determination of point heights from a physical reference surface is still a challenge in geodetic and surveying applications today. Precise geoid models are used to obtain the physical point heights from the GNSS observations with height transformation. The precision of the geoid model mainly depends on the employed calculation method of the geoid as well as the accuracy, density, and distribution of the used data. This study provides a comprehensive comparison of four surface interpolation methods having different mathematical backgrounds in local geoid modeling, in the west of Turkey. In the tests, high accuracy GNSS/leveling data at the scattered control benchmarks on topography were used. The tested algorithms are multivariable polynomial regression (MPR) analysis with least-squares adjustment (LSA), stochastic based least-squares collocation (LSC), finite elements based bivariate (BIVAR) interpolation, learning-based wavelet neural networks (WNN) methods. Among these algorithms, BIVAR was applied for the first time in this study for local geoid modeling. Apart from its superiority for accuracy, this finite elements based algorithm is considerably successful than the other three methods in providing continuity of the surface model. The surface continuity is a crucial issue in local geoid modeling. In this regard, the results of this study make a significant contribution to the practical use of local geoids. The calculated local geoid models with interpolation techniques were also compared with a gravimetric geoid model in the area. In overall results, the BIVAR interpolation algorithm provided superior performance than the other tested geoid models. Hence, its use was highly recommended for local geoid modeling. The accuracy of the local geoid model calculated with the BIVAR method is 2.65 cm.

Accurate solutions of structured generalized Kronecker product linear systems

In this paper, we consider the generalized Kronecker product (GKP) linear system associated with a class of consecutive-rank-descending (CRD) matrices arising from bivariate interpolation problems. Relying on the sign sequences of CRD matrices, we show that the associated GKP linear system is accurately solved with an “ideal” componentwise forward error. In particular, a pleasantly small componentwise relative forward error is provided to illustrate that each component of the solution is computed to high relative accuracy. We then present the sign sequences of generalized Vandermonde matrices to show that the associated GKP linear system is accurately solved with the desired componentwise forward errors. Numerical experiments are performed to confirm the high relative accuracy.

Bivariate Interpolation in Triangular Form

The purpose of this paper is to derive lagrange interpolation formula for a single variable and two independent variables in triangular form. Firstly, single variable interpolation is derived and then two independent variable interpolation derived in triangular form. In this paper, we derived the formula of three points of fit approximation, six points of fit approximation and ten points of fit approximation with their examples respectively. And also derived the approximation using a general triangular form of points with examples.

Bivariate Interpolation in Rectangular Form

The purpose of this paper is to derive lagrange interpolation formula for a single variable and two independent variables. Firstly, single variable interpolation is derived and then two independent variable interpolation derived. Some practical problems are computed by virtue of these interpolation formula.

Bivariate Thiele-Like Rational Interpolation Continued Fractions with Parameters Based on Virtual Points

The interpolation of Thiele-type continued fractions is thought of as the traditional rational interpolation and plays a significant role in numerical analysis and image interpolation. Different to the classical method, a novel type of bivariate Thiele-like rational interpolation continued fractions with parameters is proposed to efficiently address the interpolation problem. Firstly, the multiplicity of the points is adjusted strategically. Secondly, bivariate Thiele-like rational interpolation continued fractions with parameters is developed. We also discuss the interpolant algorithm, theorem, and dual interpolation of the proposed interpolation method. Many interpolation functions can be gained through adjusting the parameter, which is flexible and convenient. We also demonstrate that the novel interpolation function can deal with the interpolation problems that inverse differences do not exist or that there are unattainable points appearing in classical Thiele-type continued fractions interpolation. Through the selection of proper parameters, the value of the interpolation function can be changed at any point in the interpolant region under unaltered interpolant data. Numerical examples are given to show that the developed methods achieve state-of-the-art performance.

The Chebyshev method for the implied volatility

The implied volatility is a crucial element in any financial toolbox, since it is used to both quote and hedge options as well as for model calibration. In contrast to the Black–Scholes formula, its inverse, the implied volatility is not explicitly avail- able, and numerical approximation is required. In this paper, we propose a bivariate interpolation of the implied volatility surface based on Chebyshev polynomials. This yields a closed-form approximation of the implied volatility, which is easy to implement and to maintain. We prove a subexponential error decay. This allows us to obtain an accuracy close to machine precision with polynomials of a low degree. We compare the performance of our chosen method in terms of runtime and accuracy with the most common reference methods. In contrast to existing interpolation methods, our method is able to compute the implied volatility for all relevant option data. We use numerical experiments to confirm this results in a considerable increase in efficiency, especially for large data sets.

Bivariate Lagrange interpolation based on Chebyshev points of the second kind

We give compact formulae of the Lagrange interpolation polynomials and cubature formulae based on the common zeros of product Chebyshev polynomials of the second kind. Further, for $$0<p\leq2$$ , we study the mean convergence of the Lagrange interpolation polynomials.

Constrained univariate and bivariate rational fractal interpolation

We study constrained natures of a new class of univariate and bivariate rational cubic fractal interpolation functions (RCFIFs). We derive the convergence results of the RCFIF towards an original function in In particular, when data lies (i) between two piecewise defined lines (ii) within a rectangle, we derive sufficient condition based on the restrictions of IFS parameters at fewer discretized values so that the corresponding RCFIF preserves the inherent property associated with constrained data. Using transfinite interpolation via blending functions, we extend constrained aspects to rational bivariate RCFIFs that lie above a piecewise plane and within a cuboid.

On Interpolation by Homogeneous Polynomials in ℝ2

In this paper, we study bivariate homogeneous interpolation polynomials. We show that the homogeneous Lagrange interpolation polynomial of a sufficiently smooth function converges to a homogeneous Hermite interpolation polynomial when the interpolation points coalesce.

Hidden variable bivariate fractal interpolation functions and errors on perturbations of function vertical scaling factors

In this paper, we present a construction of hidden variable bivariate fractal interpolation functions (HVBFIFs) with function vertical scaling factors and estimate errors of HVBFIFs on perturbation of the function vertical scaling factor. We construct HVBFIFs on the basis of the iterated function system (IFS) with function vertical scaling factors. The perturbation of the function vertical scaling factors in the IFS causes a change in the HVBFIF. An upper estimation of the errors between the original HVBFIF and the perturbed HVBFIF is given.

Ensemble models for data-poor assessment: accounting for uncertainty in life-history information

AbstractLength measurements from fishery catch can be used in data-limited assessments to estimate important population parameters to guide management, but results are highly sensitive to assumptions about biological information. Ideally, local life history studies inform biological parameters. In the absence of reliable local estimates, scientists and managers face the difficult task of agreeing on fixed values for life-history parameters, often leading to additional uncertainty unquantified in the assessment or indecision defaulting to status-quo management. We propose an ensemble approach for incorporating life history uncertainty into data-limited stock assessments. We develop multivariate distributions of growth, mortality, and maturity parameter values, then use bivariate interpolation and stacking as an ensemble learning algorithm to propagate uncertainty into length-based, data-limited stock assessment models. Simulation testing demonstrated that stacking across life history parameter values leads to improved interval coverage over simple model averaging or assuming the parameter distribution means when the true life-history parameter values are unknown. We then applied the stacking approach for a U.S. Caribbean stock where the Scientific and Statistical Committee did not accept the assessment due to uncertainty in life history parameters. Stacking can better characterize uncertainty in stock status whenever life-history parameters are unknown but likely parameter distributions are available.

Bivariate Hermite interpolation on the exponential curve

We study Hermite interpolation problems on the exponential curve $$y=e^x$$ in $$\mathbb {R}^2$$ . We construct some kind of regular Hermite interpolation schemes and investigate continuity properties of interpolation polynomials with respect to interpolation conditions.

A statistical analysis of seismo-ionospheric TEC anomalies before 63 Mw ≥ 5.0 earthquakes in Turkey during 2003–2016

In this study, pre-seismic and post-seismic total electron content (TEC) anomalies of 63 Mw ≥ 5.0 earthquakes in Turkey (36°–42°N, 26°–45°E) were statistically investigated. The largest earthquake that occurred in Turkey during 2003–2016 is the Mw 7.1 Van earthquake on October 23, 2011. The TEC data of epicenters is obtained from CODE-GIM using a simple 4-point bivariate interpolation. The anomalies of TEC variations were determined by using a quartile-based running median process. In order to validate GIM results, we used the GPS-TEC data of available four IGS stations within the size of the Van earthquake preparation area. The anomalies that are detected by GIM and GPS-TEC show a similar pattern. Accordingly, the results obtained with CODE-GIM are reliable. The statistical results show that there are not prominent earthquake precursors for Mw ≤ 6.0 earthquakes in Turkey.

On Further Study of Bivariate Polynomial Interpolation over Ortho-Triples

On Further Study of Bivariate Polynomial Interpolation over Ortho-Triples