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.

An Efficient Reliability-Based Optimization Method Utilizing High-Dimensional Model Representation and Weight-Point Estimation Method

The objective of reliability-based design optimization(RBDO)is to minimize the optimization objective while satisfying the corresponding reliability requirements.However,the nested loop characteristic reduces the efficiency of RBDO algorithm,which hinders their application to high-dimensional engineering problems.To address these issues,this paper proposes an efficient decoupled RBDO method combining high dimensional model representation(HDMR)and the weight-point estimation method(WPEM).First,we decouple the RBDO model using HDMR and WPEM.Second,Lagrange interpolation is used to approximate a univariate function.Finally,based on the results of the first two steps,the original nested loop reliability optimization model is completely transformed into a deterministic design optimization model that can be solved by a series of mature constrained optimization methods without any additional calculations.Two numerical examples of a planar 10-bar structure and an aviation hydraulic piping system with 28 design variables are analyzed to illustrate the performance and practicability of the proposed method.

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.

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.

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.

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

By introducing the dimensional splitting(DS)method into the multiscale interpolating element-free Galerkin(VMIEFG)method,a dimension-splitting multiscale interpolating element-free Galerkin(DS-VMIEFG)method is proposed for three-dimensional(3D)singular perturbed convection-diffusion(SPCD)problems.In the DSVMIEFG method,the 3D problem is decomposed into a series of 2D problems by the DS method,and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method.The improved interpolation-type moving least squares(IIMLS)method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems.The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems.The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes.For extremely small singular diffusion coefficients,the numerical solution will avoid numerical oscillation and has high computational stability.

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.

EXTENSION AND APPLICATION OF NEWTON'S METHOD IN NONLINEAR OSCILLATION THEORY

In this paper we suggest and prove that Newton's method may calculate the asymptotic analytic periodic solution of strong and weak nonlinear nonautonomous systems, so that a new analytic method is offered for studying strong and weak nonlinear oscillation systems. On the strength of the need of our method, we discuss the existence and calculation of the periodic solution of the second order nonhomogeneous linear periodic system. Besides, we investigate the application of Newton's method to quasi-linear systems. The periodic solution of Duffing equation is calculated by means of our 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.

Finite Element Method Computations of the Acoustics of the Human Head Based on the Projection Based Interpolation Data

In this paper we present the Projection Based Interpolation (PBI) technique for construction of continuous approximation of MRI scan data of the human head. We utilize the result of the PBI algorithm to perform three dimensional (3D) Finite Element Method (FEM) simulations of the acoustics of the human head. The computational problem is a multi-physics problem modeled as acoustics coupled with linear elasticity. The computational grid contains tetrahedral finite elements with the number of equations and polynomial orders of approximation varying locally on finite element edges, faces, and interiors. We utilize our own out-of-core parallel direct solver for the solution of this multi-physics problem. The solver minimizes the memory usage by dumping out all local systems from all nodes of the entire elimination tree during the elimination phase.

A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.