Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

Suitable region of dynamic optimal interpolation for efficiently altimetry sea surface height mapping

The dynamic optimal interpolation(DOI)method is a technique based on quasi-geostrophic dynamics for merging multi-satellite altimeter along-track observations to generate gridded absolute dynamic topography(ADT).Compared with the linear optimal interpolation(LOI)method,the DOI method can improve the accuracy of gridded ADT locally but with low computational efficiency.Consequently,considering both computational efficiency and accuracy,the DOI method is more suitable to be used only for regional applications.In this study,we propose to evaluate the suitable region for applying the DOI method based on the correlation between the absolute value of the Jacobian operator of the geostrophic stream function and the improvement achieved by the DOI method.After verifying the LOI and DOI methods,the suitable region was investigated in three typical areas:the Gulf Stream(25°N-50°N,55°W-80°W),the Japanese Kuroshio(25°N-45°N,135°E-155°E),and the South China Sea(5°N-25°N,100°E-125°E).We propose to use the DOI method only in regions outside the equatorial region and where the absolute value of the Jacobian operator of the geostrophic stream function is higher than1×10^(-11).

Research on Interpolation Method for Missing Electricity Consumption Data

Missing value is one of the main factors that cause dirty data.Without high-quality data,there will be no reliable analysis results and precise decision-making.Therefore,the data warehouse needs to integrate high-quality data consistently.In the power system,the electricity consumption data of some large users cannot be normally collected resulting in missing data,which affects the calculation of power supply and eventually leads to a large error in the daily power line loss rate.For the problem of missing electricity consumption data,this study proposes a group method of data handling(GMDH)based data interpolation method in distribution power networks and applies it in the analysis of actually collected electricity data.First,the dependent and independent variables are defined from the original data,and the upper and lower limits of missing values are determined according to prior knowledge or existing data information.All missing data are randomly interpolated within the upper and lower limits.Then,the GMDH network is established to obtain the optimal complexity model,which is used to predict the missing data to replace the last imputed electricity consumption data.At last,this process is implemented iteratively until the missing values do not change.Under a relatively small noise level(α=0.25),the proposed approach achieves a maximum error of no more than 0.605%.Experimental findings demonstrate the efficacy and feasibility of the proposed approach,which realizes the transformation from incomplete data to complete data.Also,this proposed data interpolation approach provides a strong basis for the electricity theft diagnosis and metering fault analysis of electricity enterprises.

Integer multiple quantum image scaling based on NEQR and bicubic interpolation

As a branch of quantum image processing,quantum image scaling has been widely studied.However,most of the existing quantum image scaling algorithms are based on nearest-neighbor interpolation and bilinear interpolation,the quantum version of bicubic interpolation has not yet been studied.In this work,we present the first quantum image scaling scheme for bicubic interpolation based on the novel enhanced quantum representation(NEQR).Our scheme can realize synchronous enlargement and reduction of the image with the size of 2^(n)×2^(n) by integral multiple.Firstly,the image is represented by NEQR and the original image coordinates are obtained through multiple CNOT modules.Then,16 neighborhood pixels are obtained by quantum operation circuits,and the corresponding weights of these pixels are calculated by quantum arithmetic modules.Finally,a quantum matrix operation,instead of a classical convolution operation,is used to realize the sum of convolution of these pixels.Through simulation experiments and complexity analysis,we demonstrate that our scheme achieves exponential speedup over the classical bicubic interpolation algorithm,and has better effect than the quantum version of bilinear interpolation.

Comparison of Methods for Nitrate Interpolation in Wells in Aguascalientes, Mexico

The accuracy of interpolation models applied to groundwater depends, among other factors, on the interpolation method chosen. Therefore, it is necessary to compare different approaches. For this, different methods of interpolation of nitrate concentrations were contrasted in sixty-seven wells in an aquifer in Aguascalientes, Mexico. Four general interpolation methods were used in ArcGIS 10.5 to make the maps: IDW, Kriging, Natural Neighbor and Spline. In the modeling, only method type was varied. The input parameters (location, temporality, and nitrate concentration) were the same in the four interpolations;despite this, different maximum and minimum values were obtained for each interpolation method: for IDW, 0.2 to 22.0 mg/l, for Kriging, 3.5 to 16.5 mg/l, for Natural Neighbor, 0.3 to 21.7 mg/l and for Spline −30.8 to 37.2 mg/l. Finally, an assessment of the maps obtained was conducted by comparing them with the Official Mexican Standard (OMS), where 24 of the 67 wells were found outside the 10 mg/l that the OMS establishes as maximum permissible limit for human consumption. Taking as a starting point the measured values of nitrates (0.25 to 22.12 mg/l), as well as the spatial distribution of the interpolated values, it was determined that the Krigging method best fitted the data measured in the wells within the studied aquifer.

Comparison of Spatial Interpolation Methods of Precipitation Data in Central Macedonia, Greece

The purpose of this paper is to investigate the spatial interpolation of rainfall variability with deterministic and geostatic inspections in the Prefecture of Kilkis (Greece). The precipitation data where recorded from 12 meteorological stations in the Prefecture of Kilkis for 36 hydrological years (1973-2008). The cumulative monthly values of rainfall were studied on an annual and seasonal basis as well as during the arid-dry season. In the deterministic tests, the I.D.W. and R.B.F. checks were inspected, while in the geostatic tests, Ordinary Kriging and Universal Kriging respectively. The selection of the optimum method was made based on the least Root Mean Square Error (R.M.S.E.), as well as on the Mean Error (M.E.), as assessed by the cross validation analysis. The geostatical Kriging also considered the impact of isotropy and anisotropy across all time periods of data collection. Moreover, for Universal Kriging, the study explored spherical, exponential and Gaussian models in various combinations. Geostatistical techniques consistently demonstrated greater reliability than deterministic techniques across all time periods of data collection. Specifically, during the annual period, anisotropy was the prevailing characteristic in geostatistical techniques. Moreover, the results for the irrigation and seasonal periods were generally comparable, with few exceptions where isotropic methods yielded lower (R.M.S.E.) in some seasonal observations.

Quantum color image scaling based on bilinear interpolation

As a part of quantum image processing, quantum image scaling is a significant technology for the development of quantum computation. At present, most of the quantum image scaling schemes are based on grayscale images, with relatively little processing for color images. This paper proposes a quantum color image scaling scheme based on bilinear interpolation, which realizes the 2^(n_(1)) × 2^(n_(2)) quantum color image scaling. Firstly, the improved novel quantum representation of color digital images(INCQI) is employed to represent a 2^(n_(1)) × 2^(n_(2)) quantum color image, and the bilinear interpolation method for calculating pixel values of the interpolated image is presented. Then the quantum color image scaling-up and scaling-down circuits are designed by utilizing a series of quantum modules, and the complexity of the circuits is analyzed.Finally, the experimental simulation results of MATLAB based on the classical computer are given. The ultimate results demonstrate that the complexities of the scaling-up and scaling-down schemes are quadratic and linear, respectively, which are much lower than the cubic function and exponential function of other bilinear interpolation schemes.

OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES

This paper investigates the optimal Birkhoff interpolation and Birkhoff numbers of some function spaces in space L∞[-1,1]and weighted spaces Lp,ω[-1,1],1≤p<∞,with w being a continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal.We also show that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal when the function values of the two endpoints are included in the interpolation systems.

Missing interpolation model for wind power data based on the improved CEEMDAN method and generative adversarial interpolation network

Randomness and fluctuations in wind power output may cause changes in important parameters(e.g.,grid frequency and voltage),which in turn affect the stable operation of a power system.However,owing to external factors(such as weather),there are often various anomalies in wind power data,such as missing numerical values and unreasonable data.This significantly affects the accuracy of wind power generation predictions and operational decisions.Therefore,developing and applying reliable wind power interpolation methods is important for promoting the sustainable development of the wind power industry.In this study,the causes of abnormal data in wind power generation were first analyzed from a practical perspective.Second,an improved complete ensemble empirical mode decomposition with adaptive noise(ICEEMDAN)method with a generative adversarial interpolation network(GAIN)network was proposed to preprocess wind power generation and interpolate missing wind power generation sub-components.Finally,a complete wind power generation time series was reconstructed.Compared to traditional methods,the proposed ICEEMDAN-GAIN combination interpolation model has a higher interpolation accuracy and can effectively reduce the error impact caused by wind power generation sequence fluctuations.

Interpolation Technique for the Underwater DEM Generated by an Unmanned Surface Vessel

High-resolution underwater digital elevation models(DEMs)are important for water and soil conservation,hydrological analysis,and river channel dredging.In this work,the underwater topography of the Panjing River in Shanghai,China,was measured by an unmanned surface vessel.Five different interpolation methods were used to generate the underwater DEM and their precision and applicability for different underwater landforms were analyzed through cross-validation.The results showed that there was a positive correlation between the interpolation error and the terrain surface roughness.The five interpolation methods were all appropriate for the survey area,but their accuracy varied with different surface roughness.Based on the analysis results,an integrated approach was proposed to automatically select the appropriate interpolation method according to the different surface roughness in the surveying area.This approach improved the overall interpolation precision.The suggested technique provides a reference for the selection of interpolationmethods for underwater DEMdata.

Two Families of Multipoint Root-Solvers Using Inverse Interpolation with Memory

In this paper, a general family of derivative-free n + 1-point iterative methods using n + 1 evaluations of the function and a general family of n-point iterative methods using n evaluations of the function and only one evaluation of its derivative are constructed by the inverse interpolation with the memory on the previous step for solving the simple root of a nonlinear equation. The order and order of convergence of them are proved respectively. Finally, the proposed methods and the basins of attraction are demonstrated by the numerical examples.

Approximate solution of Volterra-Fredholm integral equations using generalized barycentric rational interpolant

It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.

The Comparison of Spatial Interpolation Methods on Temperature and Precipitation of Sanjiangyuan Area

In order to get the spatial grid data of monthly precipitation and monthly average temperature of Sanjiangyuan area, the Co-Kriging (COK) and thin plate smoothing splines(TPS) interpolation methods were applied by using the climate data during 1971-2000 of 58 meteorological stations around Qinghai Province and the 3 arc-second digital elevation model (DEM) data. The performance was evaluated by the smallest statistical errors by general cross validation (GCV). Root-mean-squared predicted errors (RMSE) and mean absolute errors (MAE) were used to compare the performance of the two methods. The results showed that: 1) After combing covariates into the models, both methods performed better; 2) The performance of TPS was significantly better than COK: for monthly average temperature, the RMSE derived from TPS was 69.48% higher than COK, as MAE increased by 70.56%. And for monthly precipitation, the RMSE derived from TPS was 28.07% higher than COK, as MAE increased by 29.06%.

Analysis of radial basis function interpolation approach

The radial basis function （RBF） interpolation approach proposed by Freedman is used to solve inverse problems encountered in well-logging and other petrophysical issues. The approach is to predict petrophysical properties in the laboratory on the basis of physical rock datasets, which include the formation factor, viscosity, permeability, and molecular composition. However, this approach does not consider the effect of spatial distribution of the calibration data on the interpolation result. This study proposes a new RBF interpolation approach based on the Freedman＇s RBF interpolation approach, by which the unit basis functions are uniformly populated in the space domain. The inverse results of the two approaches are comparatively analyzed by using our datasets. We determine that although the interpolation effects of the two approaches are equivalent, the new approach is more flexible and beneficial for reducing the number of basis functions when the database is large, resulting in simplification of the interpolation function expression. However, the predicted results of the central data are not sufficiently satisfied when the data clusters are far apart.

Application of the Delaunay triangulation interpolation in distortion XRII image

To alleviate the distortion of XRII X-ray image intensifier images in the C-arm CT computer tomography imaging system an algorithm based on the Delaunay triangulation interpolation is proposed.First the causes of the phenomenon the classical correction algorithms and the Delaunay triangulation interpolation are analyzed.Then the algorithm procedure is explained using flow charts and illustrations. Finally experiments are described to demonstrate its effectiveness and feasibility. Experimental results demonstrate that the Delaunay triangulation interpolation can have the following effects.In the case of the same center the root mean square distances RMSD and standard deviation STD between the corrected image with Delaunay triangulation interpolation and the ideal image are 5.760 4 ×10 -14 and 5.354 2 ×10 -14 respectively.They increase to 1.790 3 2.388 8 2.338 8 and 1.262 0 1.268 1 1.202 6 after applying the quartic polynomial model L1 and model L2 to the distorted images respectively.The RMSDs and STDs between the corrected image with the Delaunay triangulation interpolation and the ideal image are 2.489 × 10 -13 and 2.449 8 ×10 -13 when their centers do not coincide. When the quartic polynomial model L1 and model L2 are applied to the distorted images they are 1.770 3 2.388 8 2.338 8 and 1.269 9 1.268 1 1.202 6 respectively.

Aerodynamic Interpolation for Flight Vehicles with Aerodynamic Axial Asymmetry

Aim To study wind tunnel test data interpolation methods for flight vehicle with aerodynamic axial asymmetry. Methods For different body aerodynamic roll angles, proper wind tunnel test schemes were selected and trigonometric series were used for aerodynamic interpolation. Results and Conclusion A simple and effective scheme for wind tunnel test and an accurate aerodynamic interpolation method are developed with satisfactory results.

Direct interpolation algorithm for pen-cutting of sculptured surfaces

A direct interpolation algorithm for spatial curves on a surface is proposed for pen-cutting of sculptured surfaces. The algorithm can carry out the direct interpolation for projective curves lying on the sculptured surface. It is based on the geometric and kinetic relationships between drive curves and cutter-contact （C-C） curves. It evaluates the parameter of drive curves corresponding to interpolation points by the Taylor formula. Then it gains coordinates of interpolation points indirectly by inverse calculation. Finally, it generates the motion commands for machine tool controller. This method extends the locus-controlled function in the computer numerical control （CNC） system effectively and improves the efficiency for the numerical control （NC） machining of sculptured surfaces. The simulation shows that the proposed algorithm is feasible and practical. This algorithm can also be applied to the machining of the whole surface.

Interpolation of near offset using surface-related multiples

In this research, we present a seismic trace interpolation method which uses seismic data with surface-related multiples. It is different from conventional seismic data interpolation using information transformation or extrapolation of adjacent channels for reconstruction of missing seismic data. In this method there are two steps, first, we construct pseudo-primaries by cross-correlation of surface multiple data to extract the missing near- offset information in multiples, which are not displayed in the acquired seismic record. Second, we correct the pseudo-primaries by applying a Least-squares Matching Filter （LMF） and RMS amplitude correction method in time and space sliding windows. Then the corrected pseudo-primaries can be used to fill the data gaps. The method is easy to implement, without the need to separate multiples and primaries. It extracts the seismic information contained by multiples for filling missing traces. The method is suitable for seismic data with surfacerelated multiples.

On Triangle Interpolation Approximation of the Double Function

本文以两组不同的节点构造了一个组合型的二元三角插值多项式算子Lmn(f;x,y)，并且研究了二元连续周期函数对这个算子的收敛性及收敛阶的估计等问题。

Spatial interpolation method based on integrated RBF neural networks for estimating heavy metals in soil of a mountain region

A novel spatial interpolation method based on integrated radial basis function artificial neural networks （IRBFANNs） is proposed to provide accurate and stable predictions of heavy metals concentrations in soil at un- sampled sites in a mountain region. The IRBFANNs hybridize the advantages of the artificial neural networks and the neural networks integration approach. Three experimental projects under different sampling densities are carried out to study the performance of the proposed IRBFANNs-based interpolation method. This novel method is compared with six peer spatial interpolation methods based on the root mean square error and visual evaluation of the distribution maps of Mn elements. The experimental results show that the proposed method performs better in accuracy and stability. Moreover, the proposed method can provide more details in the spatial distribution maps than the compared interpolation methods in the cases of sparse sampling density.