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.

Two-level Bregmanized method for image interpolation with graph regularized sparse coding

A two-level Bregmanized method with graph regularized sparse coding （TBGSC） is presented for image interpolation. The outer-level Bregman iterative procedure enforces the observation data constraints, while the inner-level Bregmanized method devotes to dictionary updating and sparse represention of small overlapping image patches. The introduced constraint of graph regularized sparse coding can capture local image features effectively, and consequently enables accurate reconstruction from highly undersampled partial data. Furthermore, modified sparse coding and simple dictionary updating applied in the inner minimization make the proposed algorithm converge within a relatively small number of iterations. Experimental results demonstrate that the proposed algorithm can effectively reconstruct images and it outperforms the current state-of-the-art approaches in terms of visual comparisons and quantitative measures.

Solving Navier-Stokes equation by mixed interpolation method

The operator splitting method is used to deal with the Navier-Stokes equation, in which the physical process described by the equation is decomposed into two processes： a diffusion process and a convection process; and the finite element equation is established. The velocity field in the element is described by the shape function of the isoparametric element with nine nodes and the pressure field is described by the interpolation function of the four nodes at the vertex of the isoparametric element with nine nodes. The subroutine of the element and the integrated finite element code are generated by the Finite Element Program Generator （FEPG） successfully. The numerical simulation about the incompressible viscous liquid flowing over a cylinder is carded out. The solution agrees with the experimental results very well.

Improved absorbing boundary condition based on linear interpolation for ADI-FDTD method

With the linear interpolation method, an improved absorbing boundary condition（ABC）is introduced and derived, which is suitable for the alternating-direction-implicit finite- difference time-domain （ADI-FDTD） method. The reflection of the ABC caused by both the truncated error and the phase velocity error is analyzed. Based on the phase velocity estimation and the nonuniform cell, two methods are studied and then adopted to improve the performance of the ABC. A calculation case of a rectangular waveguide which is a typical dispersive transmission line is carried out using the ADI-FDTD method with the improved ABC for evaluation. According to the calculated case, the comparison is given between the reflection coefficients of the ABC with and without the velocity estimation and also the comparison between the reflection coefficients of the ABC with and without the nonuniform processing. The reflection variation of the ABC under different time steps is also analyzed and the acceptable worsening will not obscure the improvement on the absorption. Numerical results obviously show that efficient improvement on the absorbing performance of the ABC is achieved based on these methods for the ADI-FDTD.

A NEW DERIVATIVE FREE OPTIMIZATION METHOD BASED ON CONIC INTERPOLATION MODEL

In this paper, a new derivative free trust region method is developed based on the conic interpolation model for the unconstrained optimization. The conic interpolation model is built by means of the quadratic model function, the collinear scaling formula, quadratic approximation and interpolation. All the parameters in this model are determined by objective function interpolation condition. A new derivative free method is developed based upon this model and the global convergence of this new method is proved without any information on gradient.

A numerical method for one-dimensional nonlinear sine-Gordon equation using multiquadric quasi-interpolation

In this paper, we use a univariate multiquadric quasi-interpolation scheme to solve the one-dimensional nonlinear sine-Gordon equation that is related to many physical phenomena. We obtain a numerical scheme by using the derivative of the quasi-interpolation to approximate the spatial derivative and a difference scheme to approximate the temporal derivative. The advantage of the obtained scheme is that the algorithm is very simple so that it is very easy to implement. The results of numerical experiments are presented and compared with analytical solutions to confirm the good accuracy of the presented scheme.

A comparative study of spatial interpolation methods fordetermining fishery resources density in the Yellow Sea

Spatial interpolation is a common tool used in the study of fishery ecology, especially for the construction of ecosystem models. To develop an appropriate interpolation method of determining fishery resources density in the Yellow Sea, we tested four frequently used methods, including inverse distance weighted interpolation(IDW), global polynomial interpolation(GPI), local polynomial interpolation(LPI) and ordinary kriging(OK).A cross-validation diagnostic was used to analyze the efficacy of interpolation, and a visual examination was conducted to evaluate the spatial performance of the different methods. The results showed that the original data were not normally distributed. A log transformation was then used to make the data fit a normal distribution. During four survey periods, an exponential model was shown to be the best semivariogram model in August and October 2014, while data from January and May 2015 exhibited the pure nugget effect.Using a paired-samples t test, no significant differences(P>0.05) between predicted and observed data were found in all four of the interpolation methods during the four survey periods. Results of the cross-validation diagnostic demonstrated that OK performed the best in August 2014, while IDW performed better during the other three survey periods. The GPI and LPI methods had relatively poor interpolation results compared to IDW and OK. With respect to the spatial distribution, OK was balanced and was not as disconnected as IDW nor as overly smooth as GPI and LPI, although OK still produced a few 'bull's-eye' patterns in some areas.However, the degree of autocorrelation sometimes limits the application of OK. Thus, OK is highly recommended if data are spatially autocorrelated. With respect to feasibility and accuracy, we recommend IDW to be used as a routine interpolation method. IDW is more accurate than GPI and LPI and has a combination of desirable properties, such as easy accessibility and rapid processing.

A moving Kriging interpolation-based boundary node method for two-dimensional potential problems

In this paper, a meshfree boundary integral equation （BIE） method, called the moving Kriging interpolation- based boundary node method （MKIBNM）, is developed for solving two-dimensional potential problems. This study combines the DIE method with the moving Kriging interpolation to present a boundary-type meshfree method, and the corresponding formulae of the MKIBNM are derived. In the present method, the moving Kriging interpolation is applied instead of the traditional moving least-square approximation to overcome Kronecker＇s delta property, then the boundary conditions can be imposed directly and easily. To verify the accuracy and stability of the present formulation, three selected numerical examples are presented to demonstrate the efficiency of MKIBNM numerically.

A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation

Fractional diffusion equations have been the focus of modeling problems in hydrology, biology, viscoelasticity, physics, engineering, and other areas of applications. In this paper, a meshfree method based on the moving Kriging inter- polation is developed for a two-dimensional time-fractional diffusion equation. The shape function and its derivatives are obtained by the moving Kriging interpolation technique. For possessing the Kronecker delta property, this technique is very efficient in imposing the essential boundary conditions. The governing time-fractional diffusion equations are transformed into a standard weak formulation by the Galerkin method. It is then discretized into a meshfree system of time-dependent equations, which are solved by the standard central difference method. Numerical examples illustrating the applicability and effectiveness of the proposed method are presented and discussed in detail.

PREDICTION OF TEXTILE FABRIC REINFORCED COMPOSITE PROPERTIES BASED ON NODE INTERPOLATION CELL METHOD

Node interpolation cell method（NICM）is a micromechanics method employing the virtual displacement principle and the representative volume element（RVE）scheme to obtain the relationship between the global and the local strain.Mechanical properties of 2-D textile fabric reinforced ceramic matrix composites are predicted by NICM.Microstructures of 2-D woven and braided fabric reinforced composite are modeled by two kinds of RVE scheme.NICM is used to predict the macroscopic mechanical properties.The fill and warp yarns are simulated with cubic B-spline and their undulating forms are approximated by sinusoid.The effect of porosity on the fiber and matrix are considered as a reduction of elastic module.The connection of microstructure parameters and fiber volume fraction is modeled to investigate the reflection on the mechanical properties.The results predicted by NICM are compared with that by the finite element method（FEM）.The comparison shows that NICM is a valid and feasible method for predicting the mechanics properties of 2-D woven and braided fabric reinforced ceramic matrix composites.

Analysis and comparison of spatial interpolation methods for temperature data in Xinjiang Uygur Autonomous Region, China

Spatial interpolation methods are frequently used to estimate values of meteorological data in locations where they are not measured. However, very little research has been investigated the relative performance of different interpolation methods in meteorological data of Xinjiang Uygur Autonomous Region (Xinjiang). Actually, it has importantly practical significance to as far as possibly improve the accuracy of interpolation results for meteorological data, especially in mountainous Xinjiang. There- fore, this paper focuses on the performance of different spatial interpolation methods for monthly temperature data in Xinjiang. The daily observed data of temperature are collected from 38 meteorological stations for the period 1960- 2004. Inverse distance weighting (IDW), ordinary kriging (OK), temperature lapse rate method (TLR) and multiple linear regressions (MLR) are selected as interpolated methods. Two rasterized methods, multiple regression plus space residual error and directly interpolated observed temperature (DIOT) data, are used to analyze and compare the performance of these interpolation methods respectively. Moreover, cross-validation is used to evaluate the performance of different spatial interpolation methods. The results are as follows: 1) The method of DIOT is unsuitable for the study area in this paper. 2) It is important to process the observed data by local regression model before the spatial interpolation. 3) The MLR-IDW is the optimum spatial interpolation method for the monthly mean temperature based on cross-validation. For the authors, the reliability of results and the influence of measurement accuracy, density, distribution and spatial variability on the accuracy of the interpolation methods will be tested and analyzed in the future.

Regional Estimates of Evapotranspiration over Northern China Using a Remote-sensing-based Triangle Interpolation Method

Regional estimates of evapotranspiration （ET） are critical for a wide range of applications. Satellite remote sensing is a promising tool for obtaining reasonable ET spatial distribution data. However, there are at least two major problems that exist in the regional estimation of ET from remote sensing data. One is the conflicting requirements of simple data over a wide region, and accuracy of those data. The second is the lack of regional ET products that cover the entire region of northern China. In this study, we first retrieved the evaporative fraction （EF） by interpolating from the difference of day/night land surface temperature （AT） and the normalized difference vegetation index （NDVI） triangular-shaped scatter space. Then, ET was generated from EF and land surface meteorological data. The estimated eight-day EF and ET results were validated with 14 eddy covariance （EC） flux measurements in the growing season （July September） for the year 2008 over the study area. The estimated values agreed well with flux tower measurements, and this agreement was highly statistically significant for both EF and ET （p 〈0.01）, with the correlation coefficient for EF （R2=0.64） being relatively higher than for ET （R2---0.57）. Validation with EC-measured ET showed the mean RMSE and bias were 0.78 mm d-1 （22.03 W m-2） and 0.31 mm d-1 （8.86 W m-2）, respectively. The ET over the study area increased along a clear longitudinal gradient, which was probably controlled by the gradient of precipitation, green vegetation fractions, and the intensity of human activities. The satellite-based estimates adequately captured the spatial and seasonal structure of ET. Overall, our results demonstrate the potential of this simple but practical method for monitoring ET over regions with heterogeneous surface areas.

INTERPOLATION PERTURBATION METHOD FORSOLVING NONLINEAR PROBLEMS

In this paper, using the interpolation perturbation method. the author seeks tosolve several nonlinear problems. Numerical examples show that the method Df thispaper has good accuracy.

Hybrid N-order Lagrangian Interpolation Eulerian-Lagrangian Method for Salinity Calculation

The Eulerian？Lagrangian method（ELM） has been used by many ocean models as the solution of the advection equation,but the numerical error caused by interpolation imposes restriction on its accuracy.In the present study,hybrid N-order Lagrangian interpolation ELM（Li ELM） is put forward in which the N-order Lagrangian interpolation is used at first,then the lower order Lagrangian interpolation is applied in the points where the interpolation results are abnormally higher or lower.The calculation results of a step-shaped salinity advection model are analyzed,which show that higher order（N=3？8） Li ELM can reduce the mean numerical error of salinity calculation,but the numerical oscillation error is still significant.Even number order Li ELM makes larger numerical oscillation error than its adjacent odd number order Li ELM.Hybrid N-order Li ELM can remove numerical oscillation,and it significantly reduces the mean numerical error when N is even and the current is in fixed direction,while it makes less effect on mean numerical error when N is odd or the current direction changes periodically.Hybrid odd number order Li ELM makes less mean numerical error than its adjacent even number order Li ELM when the current is in the fixed direction,while the mean numerical error decreases as N increases when the current direction changes periodically,so odd number of N may be better for application.Among various types of Hybrid N-order Li ELM,the scheme reducing N-order directly to 1st-order may be the optimal for synthetic selection of accuracy and computational efficiency.

Comparing different spatial interpolation methods to predict the distribution of fishes:A case study of Coilia nasus in the Changjiang River Estuary

Spatial-temporal distribution of marine fishes is strongly influenced by environmental factors.To obtain a more continuous distribution of these variables usually measured by stationary sampling designs,spatial interpolation methods(SIMs)is usually used.However,different SIMs may obtain varied estimation values with significant differences,thus affecting the prediction of fish spatial distribution.In this study,different SIMs were used to obtain continuous environmental variables(water depth,water temperature,salinity,dissolved oxygen(DO),p H,chlorophyll a and chemical oxygen demand(COD))in the Changjiang River Estuary(CRE),including inverse distance weighted(IDW)interpolation,ordinary Kriging(OK)(semivariogram model:exponential(OKE),Gaussian(OKG)and spherical(OKS))and radial basis function(RBF)(regularized spline function(RS)and tension spline function(TS)).The accuracy and effect of SIMs were cross-validated,and two-stage generalized additive model(GAM)was used to predict the distribution of Coilia nasus from 2012 to 2014 in CRE.DO and COD were removed before model prediction due to their autocorrelation coefficient based on variance inflation factors analysis.Results showed that the estimated values of environmental variables obtained by the different SIMs differed(i.e.,mean values,range etc.).Cross-validation revealed that the most suitable SIMs of water depth and chlorophyll a was IDW,water temperature and salinity was RS,and p H was OKG.Further,different interpolation results affected the predicted spatial distribution of Coilia nasus in the CRE.The mean values of the predicted abundance were similar,but the differences between and among the maximum value were large.Studies showed that different SIMs can affect estimated values of the environmental variables in the CRE(especially salinity).These variations further suggest that the most applicable SIMs to each variable will also differ.Thus,it is necessary to take these potential impacts into consideration when studying the relationship between the spatial distribution of fishes and environmental changes in the CRE.

Comparison of Improved Meshless Interpolation Schemes for SPH Method and Accuracy Analysis

In the smoothed particle hydrodynamics (SPH) method, a meshless interpolation scheme is needed for the unknown function in order to discretize the governing equation.A particle approximation method has so far been used for this purpose.Traditional particle interpolation (TPI) is simple and easy to do, but its low accuracy has become an obstacle to its wider application.This can be seen in the cases of particle disorder arrangements and derivative calculations.There are many different methods to improve accuracy, with the moving least square (MLS) method one of the most important meshless interpolation methods.Unfortunately, it requires complex matrix computing and so is quite time-consuming.The authors developed a simpler scheme, called higher-order particle interpolation (HPI).This scheme can get more accurate derivatives than the MLS method, and its function value and derivatives can be obtained simultaneously.Although this scheme was developed for the SPH method, it has been found useful for other meshless methods.

An improved local radial point interpolation method for transient heat conduction analysis

The smoothing thin plate spline （STPS） interpolation using the penalty function method according to the optimization theory is presented to deal with transient heat conduction problems. The smooth conditions of the shape functions and derivatives can be satisfied so that the distortions hardly occur. Local weak forms are developed using the weighted residual method locally from the partial differential equations of the transient heat conduction. Here the Heaviside step function is used as the test function in each sub-domain to avoid the need for a domain integral. Essential boundary conditions can be implemented like the finite element method （FEM） as the shape functions possess the Kronecker delta property. The traditional two-point difference method is selected for the time discretization scheme. Three selected numerical examples are presented in this paper to demonstrate the availability and accuracy of the present approach comparing with the traditional thin plate spline （TPS） radial basis functions.

Comparison analysis of sampling methods to estimate regional precipitation based on the Kriging interpolation methods： A case of northwestern China

The accuracy of spatial interpolation of precipitation data is determined by the actual spatial variability of the precipitation, the interpolation method, and the distribution of observatories whose selections are particularly important. In this paper, three spatial sampling programs, including spatial random sampling, spatial stratified sampling, and spatial sandwich sampling, are used to analyze the data from meteorological stations of northwestern China. We compared the accuracy of ordinary Kriging interpolation methods on the basis of the sampling results. The error values of the regional annual pre-cipitation interpolation based on spatial sandwich sampling, including ME （0.1513）, RMSE （95.91）, ASE （101.84）, MSE （？0.0036）, and RMSSE （1.0397）, were optimal under the premise of abundant prior knowledge. The result of spatial stratified sampling was poor, and spatial random sampling was even worse. Spatial sandwich sampling was the best sampling method, which minimized the error of regional precipitation estimation. It had a higher degree of accuracy compared with the other two methods and a wider scope of application.

INTERPOLATION PERTURBATION METHOD FOR SOLVING THE BOUNDARY LAYER TYPE PROBLEMS

In this paper,on the basis of Ref.[1],the author studies the boundary value problems of the second-order differential equations,the highest order derivatives of which contain the small parameters.The numerical examples show that the calculating process of this method is quite simple and its accuracy is even higher than that of the multiple scales method.

Improved Ostrowski-Like Methods Based on Cubic Curve Interpolation

In this paper, we derive two higher order multipoint methods for solving nonlinear equations. The methodology is based on Ostrowski’s method and further developed by using cubic interpolation process. The adaptation of this strategy increases the order of Ostrowski’s method from four to eight and its efficiency index from 1.587 to 1.682. The methods are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified on the examples.