Local Interpolation Research Articles

Competence Coefficients Calculation Method of Participants in Group Decision-Making for Selecting the Best Alternative with the Multivariate of the Result

The problem of obtaining the best alternative using decision-making methods based on the experience of specialists and mathematical calculations is considered in the article. Group decision-making is appropriate for solving this problem. However, it can lead to the selection of several best alternatives (multivariate of the result). Accounting for competence will prioritize the decision of more competent participants and eliminate the emergence of several best alternatives in the process of group decision-making. The problem of determining the competence coefficients for participants in group decision-making has been formulated. The selection of the best alternative with the multivariate of the result is provided in the problem. A method for solving the problem has been developed. It involves discretizing the range of input variables and refining the competence coefficients values of group decision-making participants in it to select the best alternative, either by the majority principle or with the decision-maker’s involvement. Further calculation of the competence coefficients for participants in group decision-making is carried out using local linear interpolation of the refined competence coefficient at surrounding points from the discretized range. The use of the proposed method for solving the problem is considered using the example of group decision-making according to the main types of the majoritarian principle for selecting an electrodeposition variant. The results show that the proposed method for calculating the competence coefficients of participants in group decision-making through local linear interpolation is the most effective for selecting the best alternative with a multivariate result based on the relative majority.

The Application of a Laser-Printed Miniature Five-Hole Probe in the End-Wall Flow Measurement of a Multistage Axial Compressor

To make measurement of end-wall flow between blade rows in a compact multistage configuration possible, a miniature L-shaped five-hole probe was employed in this paper. This compact tip structure, realized by laser-printing instead of the conventional machining technique, reduces the blockage effect of this intrusive measurement on the flow and ensures high spatial resolution. The zonal method is introduced to extend the usable flow angle range up to 60 degrees. A local least-squares interpolation technique is utilized to acquire flow angle and static/total pressure. In order to improve accuracy for the points located at the sector boundary, the overlap region method is included in the interpolation. Additional test data indicate that the maximum error in flow angle is nearly within 1 degree, and the maximum errors of total pressure and static pressure are 0.56% and 1.9% respectively. The application in a low-speed multistage axial compressor indicates that the zonal method can decrease the number of points exceeding the measurable flow range and is of great significance for the end-wall flow measurement, especially for the near-stall condition. Compared with the traditional method, the proportion of available data for the near-stall state measurement was increased by 18% by using the zonal method.

Spline manipulations for empirical mode decomposition (EMD) on bounded intervals and beyond

Empirical mode decomposition (EMD), introduced by N.E. Huang et al. in 1998, is perhaps the most popular data-driven computational scheme for the decomposition of a non-stationary signal or time series f(t), with time-domain R:=(−∞,∞), into finitely many oscillatory components {f1(t),⋯,fK(t)}, called intrinsic mode functions (IMFs), and some “almost monotone” remainder r(t), called the trend of f(t). The core of EMD is the iterative “sifting process” applied to each function mk−1(t) to compute fk(t), for k=1,⋯,K, where m0(t):=f(t) and mk(t):=mk−1(t)−fk(t), with trend r(t):=mK(t). For the computation of each IMF, the sifting process depends on cubic spline interpolation of the local maxima and local minima for computing the upper and lower envelopes, respectively, and on subtracting the mean of the two envelopes from the result of the previous iterative step. Since it is not feasible to search for all local extrema in the entire time-domain (−∞,∞), implementation of the sifting process is commonly performed on some desired truncated bounded interval [a,b]. The main objective of this paper is to introduce and develop four “cubic spline manipulation engines”, called “quasi-interpolation (QI)”, “enhanced quasi-interpolation (EQI)”, “local interpolation (LI)”, and “improved global interpolation (IGI)” cubic spline manipulation engines, in order to significantly improve the performance of EMD on the truncated time-domains with minimal boundary artifacts, computational efficiency, accuracy, and consistency. Introduction and construction of the “fundamental quasi-interpolation” (FQI) splines as basis functions of the QI manipulation engine eliminates the need of matrix inversion for computing (global) cubic spline interpolation, since the local maximum values and local minimum values are used as coefficients of their FQI spline series representations, respectively. For the EQI spline manipulation engine, the FQI functions are formulated in terms of the same cubic B-spline basis for both the upper and lower envelopes; and for the LI spline manipulation engine, the “cubic spline blending” operation is applied to further modify the FQI splines to enable true cubic spline interpolation by “correcting the approximate interpolation error” of the EQI engine. As a consequence, the EQI and LI manipulation engines have the common property that in computing the means of the upper and lower envelopes, the only computation is averaging the B-spline coefficients, instead of computing the upper and lower envelopes separately. Furthermore, fast cubic spline pre-processing of the given f(t) is also introduced to assure numerical stability in the computation of the Hilbert transform of the first IMF f1(t) on the truncated time-domain. The theory, along with methods and explicit formulas, developed in this paper are intended for other applications beyond EMD.

Local Cr interpolations on simplicial grids in any dimension

This paper presents a construction of local Cr interpolations on simplicial grids in Rd by using piecewise polynomials of degree k≥2dr+1. To this end, an interpolative decomposition of the set of corresponding multi-indices is introduced. It is shown that the piecewise polynomial interpolation is of Cr continuity provided that the corresponding function is of C2d−1r continuity.

A new analytical approach for the local radial point interpolation discretisation in space and applications to high-order in time schemes for two-dimensional fractional PDEs

In weak-form formulations of the local radial point interpolation method (LRPIM) for the solution of partial differential equations, space discretisation matrices have most often been obtained entirely through numerical integration. This work introduces a novel approach which derives closed-form expressions for obtaining the entries of the discretisation matrix for the solution of two-dimensional time-fractional diffusion problems. This analytical approach also yields a closed-form formula for the approximation of the Laplacian. These techniques are then applied for developing LRPIM-based numerical algorithms. Since the exact solutions usually have unbounded first-order time derivatives at time zero, a graded mesh is employed for a high-order in time approximation of the Caputo derivative. The analytical shape functions are used to develop a weak-form algorithm and a strong-form algorithm is developed using the analytical approximation of the Laplacian. We demonstrate that computed solutions obtained using the weak-form and strong-form algorithms have the accuracy levels consistent with the theoretical accuracy in space. An appropriate choice of the mesh grading parameter yields a high-order convergence rate in time. The unconditional stability and convergence of a LRPIM strong form algorithm on a uniform temporal mesh is established under the assumption that the exact solution has sufficient regularity.

Temporal and Spatial Analysis of Rainfall and Evapotranspiration in Erbil Plain and the Peripheral Areas

Since analysis of spatial events is region specific and cannot be generalized, local studies are of vital importance to identify the most accurate interpolation methods. Furthermore, the trend analysis of climatic parameters at different time scales is helpful for making a better climate change adaptation and mitigation plan to overcome water scarcity. Accordingly, the current study was proposed to detect trends in rainfall and evapotranspiration at monthly and annual time scales over the Erbil plain and the surrounding area using parametric and non-parametric tests. Moreover, four deterministic and five geostatistical methods were evaluated for searching the best interpolation method to generate a continuous surface for the indicated climatic variables. The results revealed that the majority of data sets are categorized useful class and serially independent. Furthermore, it was found both rainfall and potential evapotranspiration have a mix of upward and downward trends and most of them are insignificant at 5% level of significance. Further, the interpolation analysis indicated that local polynomial interpolation (LPI) method was proven to be best interpolator for generating continuous surfaces for rainfall and ETo over the area under study followed by the empirical Bayesian Kriging method.

Ensemble-based assimilation of wave model predictions: Contrasting the impact of assimilation in nearshore and offshore forecasting at different distances from assimilated data

Accurate wave predictions are crucial for many marine-related activities, including offshore operations, navigation, and marine renewable energy. To improve the forecast performance, this study presents a data assimilation system that utilizes the WAVEWATCH III wave model and the local Ensemble Optimal Interpolation scheme to assimilate significant wave heights data from HY-2B altimeter in the China Seas. The effects of assimilation are analyzed in both coastal and offshore regions at different distances from assimilated satellite data points during a forecast period. The results show that systematic bias in wave forecasting for both offshore and nearshore locations can be effectively reduced through the use of data assimilation, with localization employed to minimize the influence of random errors. Nearshore forecasting benefits from assimilation to a greater extent than offshore forecasting, with a more enduring improvement observed. Additionally, the decline in forecast improvement with increasing distance from assimilated data highlights the necessity of assimilating a greater quantity of observational satellite data. Furthermore, careful consideration should be given to the size of the ensemble when using a specific localization radius, as the improvement in skill may stagnate with larger ensembles. Finally, assimilating additional nearshore satellite data does not negatively impact the predictive precision.

Static aeroelasticity analysis of a rotor blade using a Gauss-Seidel fluid-structure interaction method

The present study introduces a Gauss-Seidel fluid-structure interaction (FSI) method including the flow solver, structural statics solver and a fast data transfer technique, for the research of structural deformation and flow field variation of rotor blades under the combined influence of steady aerodynamic and centrifugal forces. The FSI method is illustrated and validated by the static aeroelasticity analysis of a transonic compressor rotor blade, NASA Rotor 37. An improved local interpolation with data reduction (LIWDR) algorithm is introduced for fast data transfer on the fluid-solid interface of blade. The results of FSI calculation of NASA Rotor 37 show that when compared with the radial basis function (RBF) based interpolation algorithm, LIWDR meets the interpolation accuracy requirements, while the calculation cost can be greatly improved. The data transmission time is only about 1% of that of RBF. Moreover, the iteration step of steady flow computation within one single FSI has little impact on the converged aerodynamic and structural results. The aerodynamic load-caused deformation accounts for nearly 50% of the total. The effects of blade deformation on the variations of aerodynamic performance are given, demonstrating that when static aeroelasticity is taken into account, the choke mass flow rate increases and the peak adiabatic efficiency slightly decreases. The impact mechanisms on performance variations are presented in detail.

Polyharmonic splines interpolation on scattered data in 2D and 3D with applications

Data interpolation is a fundamental problem in many applied mathematics and scientific computing fields. This paper introduces a modified implicit local radial basis function interpolation method for scattered data using polyharmonic splines (PS) with a low degree of polynomial basis. This is an improvement to the original method proposed in 2015 by Yao et al.. In the original approach, only radial basis functions (RBFs) with shape parameters, such as multiquadrics (MQ), inverse multiquadrics (IMQ), Gaussian, and Matern RBF are used. The authors claimed that the conditionally positive definite RBFs such as polyharmonic splines r2nlnr and r2n+1 “failed to produce acceptable results”.In this paper, we verified that when polyharmonic splines together with a polynomial basis is used on the interpolation scheme, high-order accuracy and excellent conditioning of the global sparse systems are gained without selecting a shape parameter. The scheme predicts functions’ values at a set of discrete evaluation points, through a global sparse linear system. Compared to standard implementation, computational efficiency is achieved by using parallel computing. Applications of the proposed algorithms to 2D and 3D benchmark functions on uniformly distributed random points, the Halton quasi-points on regular or Stanford bunny shape domains, and an image interpolation problem confirmed the effectiveness of the method. We also compared the algorithms with other methods available in the literature to show the superiority of using PS augmented with a polynomial basis. High accuracy can be easily achieved by increasing the order of polyharmonic splines or the number of points in local domains, when small order of polynomials are used in the basis. MATLAB code for the 3D bunny example is shared on MATLAB Central File Exchange (Yao, 2023).

Look Up Table Method for Five Hole Probe Data Reduction

A lookup method for extracting flow field information from five hole probe measurements is presented. The look up table method consists of two step interpolation. In the first step, the calibration space is divided into a large number of small intervals of yaw and pitch coefficients (0.1, 0.05 or 0.01) and the required quantities are interpolated using local cubic spline interpolation technique. The interpolated values are stored in a large square matrix. In the second step, the measured values of yaw and pitch coefficients are used to determine yaw and pitch angles by linear interpolation. Same procedure is used to determine total and static pressure coefficients. Additional data taken during the calibration is used for validating lookup table method and spline interpolation method. The errors from the look up table method, particularly with the interval of 0.01 are about the same or lower than those obtained from the spline interpolation method. The computational times for lookup table method are found to be lower than that for the spline interpolation method. The lookup table method seems to be attractive when a large amount of five hole probe data is to be processed and in cases where online data processing is required.

GM(1,1)-Based Weighted K-Nearest Neighbor Algorithm for Indoor Localization

Along with the IoT technology, the importance of indoor positioning is increasing, but the accuracy of the traditional fingerprint positioning algorithm is negatively affected by the complex indoor environment. This issue of low indoor spatial geolocation localization accuracy when the signal is collected away from the present stage occurs due to the signal instability of the iBeacon in the traditional fingerprint localization algorithm, which generates a variety of factors such as object blocking and reflection, multipath effect, etc., as well as the scarcity of reference fingerprint data points. In response, this study proposes an inverse distance-weighted optimization WKNN algorithm for indoor localization based on the GM(1,1) model. By implementing GM(1,1) model pre-process leveling, the original fingerprint library was reconstructed into a large-capacity fingerprint database using the inverse distance-weighted interpolation method. The local inverse distance-weighted interpolation was used for interpolation, combined with the WKNN algorithm to complete the coordinate solution in real time. This effectively solved the issue of low localization accuracy caused by the large fluctuation of the received signal strength (RSS) sampling measurement data and the existence of few reference fingerprint datapoints in the fingerprint database. The results show that this algorithm reduced the average positioning error by 5.9% compared with ordinary kriging (OK) interpolation leveling and reduced the average positioning error by 18.2% compared with the indoor spatial location accuracy of the original fingerprint database, which can effectively improve the positioning accuracy and provide technical support for indoor location and navigation services.

High-Performance Evaluation of the Interpolations and Anterpolations in the GPU-Accelerated Massively Parallel MLFMA

This communication investigates high-performance computation schemes for local Lagrange interpolation and anterpolation operations in the parallel graphics processing unit (GPU)-accelerated distributed-memory multilevel fast multipole algorithm (MLFMA). Two ELLPACK format-based schemes, namely, block ELLPACK (ELL-B) and hybrid compressed sparse column (CSC)-block ELLPACK (CSC-ELL-B), are proposed for the evaluation of interpolation and anterpolation operations, respectively, which ensure high computational throughput for GPU calculation. Optimization using the GPU hierarchical memory architecture, the mechanism of the stream and the CPU/GPU asynchronous computation pattern are employed to further improve the overall performance. The proposed schemes are proven to be an order of magnitude faster than the conventional schemes for aggregation/disaggregation operations. For an aircraft model involving over 10 billion unknowns, the iteration time is reduced by over half, which is remarkable progress in the development of GPU-accelerated parallelization of MLFMA.

The Application of Splines of the Seventh Order Approximation to the Solution of Integral Fredholm Equations

There are various numerical methods for solving integral equations. Among the new numerical methods, methods based on splines and spline wavelets should be noted. Local interpolation splines of a low order of approximation have proved themselves well in solving differential and integral equations. In this paper, we consider the construction of a numerical solution to the Fredholm integral equation of the second kind using spline approximations of the seventh order of approximation. The support of the basis spline of the seventh order of approximation occupies seven grid intervals. We apply various modifications of the basis splines of the seventh order of approximation at the beginning, the middle, and at the end of the integration interval. It is assumed that the solution of the integral equation is sufficiently smooth. The advantages of using splines of the seventh order of approximation include the use of a small number of grid nodes to achieve the required error of approximation. Numerical examples of the application of spline approximations of the seventh order for solving integral equations are given.

Comparison between different spatial interpolation methods for the development of sediment distribution maps in coastal areas

Sediment grain size and its spatial distribution is a very important aspect for many applications and processes that occur in the coastal zone. One of these is coastal erosion which is strongly dependent on sediment distribution and transportation. To highlight this fact, surficial coastal sediments were collected from a densely populated coastal zone in Western Greece, which suffers extensive erosion, and grain size distribution was thoroughly analysed, to predict the spatial distribution of the median grain size diameter (D50) and produce sediment distribution maps. Four different geostatistical interpolation techniques (Ordinary Kriging, Simple Kriging, Empirical Bayesian Kriging and Universal Kriging) and three deterministic (Radial Basis Function, Local Polynomial Interpolation, and Inverse Distance Weighting) were employed for the construction of the respective surficial sediment distribution maps with the use of GIS. Moreover, a comparative study between the deterministic and geostatistical approaches was applied and the performance of each interpolation method was evaluated using cross-validation and estimating the Pearson Corellation and the coefficient of determination (R2). The best interpolation technique for this research proved to be the Ordinary Kriging for the shoreline materials and the Empirical Bayesian Kriging (EBK) for the seabed materials since both had the lowest prediction errors and the highest R2.

A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates

In this study, two-dimensional (2D) Berger equation arising in deflection of thin plates is solved numerically. A local meshfree collocation technique based on radial point interpolation is developed for the Berger equation. As radial basis function thin plate splines are used to avoid selecting optimal shape parameter. The method is used to solve both linear and non-linear Berger equation on regular and irregular geometries. Obtained results during numerical simulations, are compared with different methods exist in literatrue such as method of fundamental solutions, local Kansa’s method and Pascal polynomial based meshless method. The comparisons affirm the accuracy and reliability of the suggested method.

Evaluation of interpolation methods for the prediction of urban methane concentrations

Having data about atmospheric concentrations in an entire urban area is difficult, hence interpolation methods are helpful. Their choice will depend on minimising the error. In this work, two deterministic (Inverse Distance Weight and Local Polynomial Interpolation) and two stochastic methods (Simple and Ordinary Kriging) were applied to predict seasonal and annual atmospheric methane (CH4) concentration means. Two sampling networks were designed in an intermediate city, covering a wide variety of urban densities, with different sampling site numbers. The main objective was to find the interpolation model that best predicts CH4 concentration and to analyse if the network's expansion improves the metric errors - the mean error (ME) and the root-mean-square error (RMSE). The ME values were close to zero in all cases, and the stochastic methods had the smallest RMSE for both networks. Besides, adding more sampling sites improved up to 50% of the RMSE values. Finally, an integrated map was obtained incorporating all the best interpolation models, which gave a difference of <4% between the measured and the estimated CH4 concentration. This type of study is helpful to evaluate the design of a sampling network, the territorial planning and future installations of CH4 sources.

Application of radial basis functions in strain analysis of digital image correlation

AbstractDigital image correlation systems provide the motion of material points on the surface of a specimen or a component. Commonly, local interpolation concepts are chosen to determine the in‐plane strains or principal stretches. However, this approach yields discontinuous fields. In this paper, a global interpolation concept, particular, radial basis functions is considered to determine the displacement and in‐plane strain field and to overcome the underlying problem of discontinuities. Since there might be boundary effects, i.e. there are larger errors in the approximation of the strains in comparison to analytical verification examples, multiquadric are applied as particular radial basis function. It turns out that the error becomes smaller at the edges of the domain of interest, although the condition number of the resulting systems of equations in the applied interpolation scheme, might be critical. However, the studies at additively manufactured parts lead to the conclusion that the approach is promising for real‐world applications.

Comparing Deterministic and Stochastic Methods in Geospatial Analysis of Groundwater Fluoride Concentration

Dental and skeletal fluorosis caused by consuming high-fluoride groundwater has been reported over several decades globally. Prediction maps to estimate the fluoride contaminated area rely on interpolation methods. This study presents a comparison of the accuracy of nine spatial interpolation methods in predicting the fluoride in groundwater. Leave-one-out cross-validation (LOOCV), hold-out validation and validation with an independent dataset were used to assess the precision of the interpolation methods. This is the first study on fluoride with a large dataset (N = 13,585) applied at the regional level in India. Our findings showed that the inverse distance weighted (IDW) algorithm outperformed other methods in terms of less discrepancy between measured and predicted fluoride. IDW and local polynomial interpolation (LPI) were the only methods to predict contaminated areas (fluoride &gt; 1.5 mg/L). However, the area estimated by the typical assessment of the percentage of unsuitable samples was much higher (6.1%) compared to that estimated by IDW (0.2%) and LPI (0.2%). LOOCV provided viable results than the other two validation methods. Interpolation methods are accompanied with uncertainty which are regulated by the sample size, sample density, sample distribution, minimum and maximum measured concentrations, smoothing and border effects. Drawing a comparison among variegated interpolation methods capturing a wide range of prediction uncertainty is suggested rather than relying on one method exclusively. The high-fluoride areas identified in this study can be used by the Government in planning remediation actions.

Triquintic interpolation in three dimensions

This paper develops a local triquintic interpolation method for a function defined over a unit cube mesh element. The resulting polynomial interpolant is isotropic and has global C2 continuity. The triquintic polynomial coefficients are related to the function and its derivatives at the eight corners of a mesh element by an invertible square matrix of rank 216. The proposed method is an extension of a previously developed tricubic interpolation scheme with global C1 continuity.

A high-order generalized differential quadrature method with lattice Boltzmann flux solver for simulating incompressible flows

This paper presents a high-order generalized differential quadrature method with lattice Boltzmann flux solver (LBFS-GDQ) for simulating incompressible isothermal flows. In this method, high-order polynomials are adopted to approximate both the solution and fluxes globally across the computational domain. Solution derivatives and flux divergence are conveniently computed by the GDQ method. At the interior solution points, the viscous and inviscid fluxes are evaluated simultaneously via LBFS. Treatments to prevent the global accuracy from being contaminated by the streaming error of LBFS are proposed and studied, including the choice for the local streaming spacing and interpolation methods for the local reconstruction. The present method inherits the advantages of both GDQ and LBFS, i.e., global spectral accuracy, direct evolution of macroscopic variables, and convenient implementation of boundary conditions. Numerical experiments with a wide selection of incompressible flow problems confirm the excellent accuracy, efficiency, and flexibility of the proposed method.