The purpose of this paper is to present the class of atomic basis functions(ABFs)which are of exponential type and are denoted by EFupn(x,ω).While ABFs of the algebraic type are already represented in the numerical m...The purpose of this paper is to present the class of atomic basis functions(ABFs)which are of exponential type and are denoted by EFupn(x,ω).While ABFs of the algebraic type are already represented in the numerical modeling of various problems inmathematical physics and computationalmechanics,ABFs of the exponential type have not yet been sufficiently researched.These functions,unlike the ABFs of the algebraic type Fupn(x),contain the tension parameterω,which gives them additional approximation properties.Exponential monomials up to the nth degree can be described exactly by the linear combination of the functions EFupn(x,ω).The function EFupn for n=0 is called the“mother”ABF of the exponential type,i.e.,EFup0(x,ω)≡Eup(x,ω).In other words,the functions EFupn(x,ω)are elements of the linear vector space EUPn and retain all the properties of their“mother”function Eup(x,ω).Thus,this paper,in terms of its content and purpose,can be understood as a sequel of the article by Brajcic Kurbasa et al.,which shows the basic properties and application of the basis function Eup(x,ω).This paper presents,in an analogous way,the development and application of the exponential basis functions EFupn(x,ω).Here,for the first time,expressions for calculating the values of the functions EFupn(x,ω)and their derivatives are given in a form suitable for application in numerical analyses,which is shown in the verification examples of the approximations of known functions.展开更多
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
The shallow-water temperature profile is typically parameterized using a few empirical orthogonal function(EOF)coefficients.However,when the experimental area is poorly known or highly variable,the adaptability of the...The shallow-water temperature profile is typically parameterized using a few empirical orthogonal function(EOF)coefficients.However,when the experimental area is poorly known or highly variable,the adaptability of the EOFs will be significantly reduced.In this study,a new set of basis functions,generated by combining the internal-wave eigenmodes with the average temperature gradient,is developed for characterizing the temperature perturbations.Temperature profiles recorded by a thermistor chain in the South China Sea in 2015 are processed and analyzed.Compared to the EOFs,the new set of basis functions has higher reconstruction accuracy and adaptability;it is also more stable in ocean regions that have internal waves.展开更多
For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basis function (?),existence and characterization theorems and a variational principle are proved. Examples include (?)(r)=r^b,Duchon'...For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basis function (?),existence and characterization theorems and a variational principle are proved. Examples include (?)(r)=r^b,Duchon's thin-plate splines,Hardy's multiquadrics,and inverse multiquadrics.展开更多
In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to in...In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to interpolate the implicit level set function of the evolution equation with a high level of accuracy and smoothness. Then, the original initial value problem is discretized into an interpolation problem. Accordingly, the evolution equation is converted into a set of coupled ordinary differential equations, and a smooth evolution can be retained. Compared with finite difference scheme based level set approaches, the complex and costly re-initialization procedure is unnecessary. Numerical examples are also given to show the efficiency of the method.展开更多
Solving large radial basis function (RBF) interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficult...Solving large radial basis function (RBF) interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.展开更多
The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to impl...The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to implement and mathematically simple.In this paper,the commonly⁃used multiquadric RBF,conical RBF,and Gaussian RBF were applied to solve boundary value problems which are governed by partial differential equations with variable coefficients.Numerical results were provided to show the good performance of the three RBFs as numerical tools for a wide range of problems.It is shown that the conical RBF numerical results were more stable than the other two radial basis functions.From the comparison of three commonly⁃used RBFs,one may obtain the best numerical solutions for boundary value problems.展开更多
Ensemble simulations, which use multiple short independent trajectories from dispersive initial conformations, rather than a single long trajectory as used in traditional simulations, are expected to sample complex sy...Ensemble simulations, which use multiple short independent trajectories from dispersive initial conformations, rather than a single long trajectory as used in traditional simulations, are expected to sample complex systems such as biomolecules much more efficiently. The re-weighted ensemble dynamics(RED) is designed to combine these short trajectories to reconstruct the global equilibrium distribution. In the RED, a number of conformational functions, named as basis functions,are applied to relate these trajectories to each other, then a detailed-balance-based linear equation is built, whose solution provides the weights of these trajectories in equilibrium distribution. Thus, the sufficient and efficient selection of basis functions is critical to the practical application of RED. Here, we review and present a few possible ways to generally construct basis functions for applying the RED in complex molecular systems. Especially, for systems with less priori knowledge, we could generally use the root mean squared deviation(RMSD) among conformations to split the whole conformational space into a set of cells, then use the RMSD-based-cell functions as basis functions. We demonstrate the application of the RED in typical systems, including a two-dimensional toy model, the lattice Potts model, and a short peptide system. The results indicate that the RED with the constructions of basis functions not only more efficiently sample the complex systems, but also provide a general way to understand the metastable structure of conformational space.展开更多
The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equ...The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equations for constructing G3 splines can be presented independently of geometric shape parameters' values. It makes the equation's solving easier. It is also known that the general form of the G3spline basis function is given in the first time. Its geometric construction method is presented.展开更多
This study evaluates the effectiveness of a new technique that transforms doma in integrals into boundary integrals that is applicable to the boundary element method.Si mulations were conducted in which two-dimensiona...This study evaluates the effectiveness of a new technique that transforms doma in integrals into boundary integrals that is applicable to the boundary element method.Si mulations were conducted in which two-dimensional surfaces were approximated by inter polation using radial basis functions with full and compact supports.Examples involving Poisson’s equation are presented using the boundary element method and the proposed te chnique with compact radial basis functions.The advantages and the disadvantages are e xamined through simulations.The effects of internal poles,the boundary mesh refinemen t and the value for the support of the radial basis functions on performance are assessed.展开更多
The present work describes the application of the method of fundamental solutions (MFS) along with the analog equation method (AEM) and radial basis function (RBF) approximation for solving the 2D isotropic and ...The present work describes the application of the method of fundamental solutions (MFS) along with the analog equation method (AEM) and radial basis function (RBF) approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson's equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition (SVD) technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric (MQ) are fully analyzed to provide useful reference.展开更多
Based on our previous study,the accuracy of derivatives of interpolating functions are usually very poor near the boundary of domain when Compactly Supported Radial Basis Functions (CSRBFs)are used,so that it could re...Based on our previous study,the accuracy of derivatives of interpolating functions are usually very poor near the boundary of domain when Compactly Supported Radial Basis Functions (CSRBFs)are used,so that it could result in significant error in solving partial differential equations with Neumann boundary conditions.To overcome this drawback,the Consistent Compactly Supported Radial Basis Functions(CCSRBFs)are developed,which satisfy the predetermined consistency con- ditions.Meshless method based on point collocation with CCSRBFs is developed for solving partial differential equations.Numerical studies show that the proposed method improves the accuracy of approximation significantly.展开更多
The Radial Basis Functions Neural Network (RBFNN) is used to establish the model of a response system through the input and output data of the system. The synchronization between a drive system and the response syst...The Radial Basis Functions Neural Network (RBFNN) is used to establish the model of a response system through the input and output data of the system. The synchronization between a drive system and the response system can be implemented by employing the RBFNN model and state feedback control. In this case, the exact mathematical model, which is the precondition for the conventional method, is unnecessary for implementing synchronization. The effect of the model error is investigated and a corresponding theorem is developed. The effect of the parameter perturbations and the measurement noise is investigated through simulations. The simulation results under different conditions show the effectiveness of the method.展开更多
We use Radial Basis Functions (RBFs) to reconstruct smooth surfaces from 3D scattered data. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improveme...We use Radial Basis Functions (RBFs) to reconstruct smooth surfaces from 3D scattered data. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improvements on the methods of surface reconstruction with radial basis functions. A sparse approximation set of scattered data is constructed by reducing the number of interpolating points on the surface. We present an adaptive method for finding the off-surface normal points. The order of the equation decreases greatly as the number of the off-surface constraints reduces gradually. Experimental results are provided to illustrate that the proposed method is robust and may draw beautiful graphics.展开更多
A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable ...A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.展开更多
A new method based on angular momentum theory was proposed to construct the basis functions of the irreducible representations(IRs) of point groups. The transformation coefficients, i. e., coefficients S, are the com...A new method based on angular momentum theory was proposed to construct the basis functions of the irreducible representations(IRs) of point groups. The transformation coefficients, i. e., coefficients S, are the components of the eigenvectors of some Hermitian matrices, and can be made as real numbers for all pure rotation point groups. The general formula for coefficient S was deduced, and applied to constructing the basis functions of single-valued irreducible representations of icosahedral group from the spherical harmonics with angular momentum j≤7.展开更多
Background An important purpose of orthodontic treatment is to gain the harmonic soft tissue profile. This article describes a novel way to build patient-specific models of facial soft tissues by transforming a standa...Background An important purpose of orthodontic treatment is to gain the harmonic soft tissue profile. This article describes a novel way to build patient-specific models of facial soft tissues by transforming a standard finite element (FE) model into one that has two stages: a first transformation and a second transformation, so as to evaluate the facial soft tissue changes after orthodontic treatment for individual patients. Methods The radial basis functions (RBFs) interpolation method was used to transform the standard FE model into a patient-specific one based on landmark points. A combined strategy for selecting landmark points was developed in this study: manually for the first transformation and automatically for the second transformation. Four typical patients were chosen to validate the effectiveness of this transformation method. Results The results showed good similarity between the transformed FE models and the computed tomography (CT) models. The absolute values of average deviations were in the range of 0.375-0.700 mm at the lip-mouth region after the first transformation, and they decreased to a range of 0.116-0.286 mm after the second transformation. Conclusions The modeling results show that the second transformation resulted in enhanced accuracy compared to the first transformation. Because of these results, a third transformation is usually not necessary.展开更多
This paper is concerned with stable solutions of time domain integral equation (TDIE) methods for transient scattering problems with 3D conducting objects. We use the quadratic B-spline function as temporal basis fu...This paper is concerned with stable solutions of time domain integral equation (TDIE) methods for transient scattering problems with 3D conducting objects. We use the quadratic B-spline function as temporal basis functions, which permits both the induced currents and induced charges to be properly approximated in terms of completeness. Because the B-spline function has the least support width among all polynomial basis functions of the same order, the resulting system matrices seem to be the sparsest. The TDIE formula-tions using induced electric polarizations as unknown function are adopted and justified. Numerical results demonstrate that the proposed approach is accurate and efficient, and no late-time instability is observed.展开更多
We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes,which can provide a good balance between the numerical accuracy and computational cost.The mu...We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes,which can provide a good balance between the numerical accuracy and computational cost.The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions.The multiscale basis functions have abilities to capture originally perturbed information in the local problem,as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes,where the layer-adapted meshes are generated by a given parameter.Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L^(2)norm and first order convergence in the energy norm on graded meshes,which is independent ofε.In contrast with the conventional methods,our method is much more accurate and effective.展开更多
The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matri...The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation(VCE) and minimum standard deviation(MSTD) approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the firstorder Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.展开更多
基金supported through Project KK.01.1.1.02.0027a project co-financed by the Croatian Government and the European Union through the European Regional Development Fund-the Competitiveness and Cohesion Operational Programme.
文摘The purpose of this paper is to present the class of atomic basis functions(ABFs)which are of exponential type and are denoted by EFupn(x,ω).While ABFs of the algebraic type are already represented in the numerical modeling of various problems inmathematical physics and computationalmechanics,ABFs of the exponential type have not yet been sufficiently researched.These functions,unlike the ABFs of the algebraic type Fupn(x),contain the tension parameterω,which gives them additional approximation properties.Exponential monomials up to the nth degree can be described exactly by the linear combination of the functions EFupn(x,ω).The function EFupn for n=0 is called the“mother”ABF of the exponential type,i.e.,EFup0(x,ω)≡Eup(x,ω).In other words,the functions EFupn(x,ω)are elements of the linear vector space EUPn and retain all the properties of their“mother”function Eup(x,ω).Thus,this paper,in terms of its content and purpose,can be understood as a sequel of the article by Brajcic Kurbasa et al.,which shows the basic properties and application of the basis function Eup(x,ω).This paper presents,in an analogous way,the development and application of the exponential basis functions EFupn(x,ω).Here,for the first time,expressions for calculating the values of the functions EFupn(x,ω)and their derivatives are given in a form suitable for application in numerical analyses,which is shown in the verification examples of the approximations of known functions.
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
基金The Natural Science Foundation of Shandong Province of China under contract Nos ZR2022MA051 and ZR2020MA090the Fund of China Postdoctoral Science Foundation under contract No.2020M670891+1 种基金the Shandong University of Science and Technology Research Fund under contract No.2019TDJH103the Talent Introduction Plan for Youth Innovation Team in Universities of Shandong Province(Innovation Team of Satellite Positioning and Navigation).
文摘The shallow-water temperature profile is typically parameterized using a few empirical orthogonal function(EOF)coefficients.However,when the experimental area is poorly known or highly variable,the adaptability of the EOFs will be significantly reduced.In this study,a new set of basis functions,generated by combining the internal-wave eigenmodes with the average temperature gradient,is developed for characterizing the temperature perturbations.Temperature profiles recorded by a thermistor chain in the South China Sea in 2015 are processed and analyzed.Compared to the EOFs,the new set of basis functions has higher reconstruction accuracy and adaptability;it is also more stable in ocean regions that have internal waves.
文摘For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basis function (?),existence and characterization theorems and a variational principle are proved. Examples include (?)(r)=r^b,Duchon's thin-plate splines,Hardy's multiquadrics,and inverse multiquadrics.
基金Project supported by the National Natural Science Foundation of China (Grant No.11101454)the Educational Commission Foundation of Chongqing City,China (Grant No.KJ130626)the Program of Innovation Team Project in University of Chongqing City,China (Grant No.KJTD201308)
文摘In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to interpolate the implicit level set function of the evolution equation with a high level of accuracy and smoothness. Then, the original initial value problem is discretized into an interpolation problem. Accordingly, the evolution equation is converted into a set of coupled ordinary differential equations, and a smooth evolution can be retained. Compared with finite difference scheme based level set approaches, the complex and costly re-initialization procedure is unnecessary. Numerical examples are also given to show the efficiency of the method.
基金Supported by National Natural Science Youth Foundation (10401021).
文摘Solving large radial basis function (RBF) interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.
基金the Natural Science Foundation of Anhui Province(Grant No.1908085QA09)the University Natural Science Research Project of Anhui Province(KJ2019A0591).
文摘The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to implement and mathematically simple.In this paper,the commonly⁃used multiquadric RBF,conical RBF,and Gaussian RBF were applied to solve boundary value problems which are governed by partial differential equations with variable coefficients.Numerical results were provided to show the good performance of the three RBFs as numerical tools for a wide range of problems.It is shown that the conical RBF numerical results were more stable than the other two radial basis functions.From the comparison of three commonly⁃used RBFs,one may obtain the best numerical solutions for boundary value problems.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175250)
文摘Ensemble simulations, which use multiple short independent trajectories from dispersive initial conformations, rather than a single long trajectory as used in traditional simulations, are expected to sample complex systems such as biomolecules much more efficiently. The re-weighted ensemble dynamics(RED) is designed to combine these short trajectories to reconstruct the global equilibrium distribution. In the RED, a number of conformational functions, named as basis functions,are applied to relate these trajectories to each other, then a detailed-balance-based linear equation is built, whose solution provides the weights of these trajectories in equilibrium distribution. Thus, the sufficient and efficient selection of basis functions is critical to the practical application of RED. Here, we review and present a few possible ways to generally construct basis functions for applying the RED in complex molecular systems. Especially, for systems with less priori knowledge, we could generally use the root mean squared deviation(RMSD) among conformations to split the whole conformational space into a set of cells, then use the RMSD-based-cell functions as basis functions. We demonstrate the application of the RED in typical systems, including a two-dimensional toy model, the lattice Potts model, and a short peptide system. The results indicate that the RED with the constructions of basis functions not only more efficiently sample the complex systems, but also provide a general way to understand the metastable structure of conformational space.
基金Supported by National Natural Science Foundation of China(Grants 61100129)Open Program of Key Laboratory of Intelligent Information Processing,Institute of Computing Technology,Chinese Academy of Sciences(IIP2014-7)
文摘The explicit expression of the G3 basis function is presented in this paper. It is derived by constructing the conversion matrix between G3 basis function and Brzier representation. After the matrix decomposition, equations for constructing G3 splines can be presented independently of geometric shape parameters' values. It makes the equation's solving easier. It is also known that the general form of the G3spline basis function is given in the first time. Its geometric construction method is presented.
文摘This study evaluates the effectiveness of a new technique that transforms doma in integrals into boundary integrals that is applicable to the boundary element method.Si mulations were conducted in which two-dimensional surfaces were approximated by inter polation using radial basis functions with full and compact supports.Examples involving Poisson’s equation are presented using the boundary element method and the proposed te chnique with compact radial basis functions.The advantages and the disadvantages are e xamined through simulations.The effects of internal poles,the boundary mesh refinemen t and the value for the support of the radial basis functions on performance are assessed.
文摘The present work describes the application of the method of fundamental solutions (MFS) along with the analog equation method (AEM) and radial basis function (RBF) approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson's equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition (SVD) technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric (MQ) are fully analyzed to provide useful reference.
基金The project supported by the National Natural Science Foundation of China (10172052)
文摘Based on our previous study,the accuracy of derivatives of interpolating functions are usually very poor near the boundary of domain when Compactly Supported Radial Basis Functions (CSRBFs)are used,so that it could result in significant error in solving partial differential equations with Neumann boundary conditions.To overcome this drawback,the Consistent Compactly Supported Radial Basis Functions(CCSRBFs)are developed,which satisfy the predetermined consistency con- ditions.Meshless method based on point collocation with CCSRBFs is developed for solving partial differential equations.Numerical studies show that the proposed method improves the accuracy of approximation significantly.
基金This project was supported in part by the Science Foundation of Shanxi Province (2003F028)China Postdoctoral Science Foundation (20060390318).
文摘The Radial Basis Functions Neural Network (RBFNN) is used to establish the model of a response system through the input and output data of the system. The synchronization between a drive system and the response system can be implemented by employing the RBFNN model and state feedback control. In this case, the exact mathematical model, which is the precondition for the conventional method, is unnecessary for implementing synchronization. The effect of the model error is investigated and a corresponding theorem is developed. The effect of the parameter perturbations and the measurement noise is investigated through simulations. The simulation results under different conditions show the effectiveness of the method.
文摘We use Radial Basis Functions (RBFs) to reconstruct smooth surfaces from 3D scattered data. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improvements on the methods of surface reconstruction with radial basis functions. A sparse approximation set of scattered data is constructed by reducing the number of interpolating points on the surface. We present an adaptive method for finding the off-surface normal points. The order of the equation decreases greatly as the number of the off-surface constraints reduces gradually. Experimental results are provided to illustrate that the proposed method is robust and may draw beautiful graphics.
文摘A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.
文摘A new method based on angular momentum theory was proposed to construct the basis functions of the irreducible representations(IRs) of point groups. The transformation coefficients, i. e., coefficients S, are the components of the eigenvectors of some Hermitian matrices, and can be made as real numbers for all pure rotation point groups. The general formula for coefficient S was deduced, and applied to constructing the basis functions of single-valued irreducible representations of icosahedral group from the spherical harmonics with angular momentum j≤7.
基金This study was supported by grants from the National Natural Science Foundation of China
文摘Background An important purpose of orthodontic treatment is to gain the harmonic soft tissue profile. This article describes a novel way to build patient-specific models of facial soft tissues by transforming a standard finite element (FE) model into one that has two stages: a first transformation and a second transformation, so as to evaluate the facial soft tissue changes after orthodontic treatment for individual patients. Methods The radial basis functions (RBFs) interpolation method was used to transform the standard FE model into a patient-specific one based on landmark points. A combined strategy for selecting landmark points was developed in this study: manually for the first transformation and automatically for the second transformation. Four typical patients were chosen to validate the effectiveness of this transformation method. Results The results showed good similarity between the transformed FE models and the computed tomography (CT) models. The absolute values of average deviations were in the range of 0.375-0.700 mm at the lip-mouth region after the first transformation, and they decreased to a range of 0.116-0.286 mm after the second transformation. Conclusions The modeling results show that the second transformation resulted in enhanced accuracy compared to the first transformation. Because of these results, a third transformation is usually not necessary.
文摘This paper is concerned with stable solutions of time domain integral equation (TDIE) methods for transient scattering problems with 3D conducting objects. We use the quadratic B-spline function as temporal basis functions, which permits both the induced currents and induced charges to be properly approximated in terms of completeness. Because the B-spline function has the least support width among all polynomial basis functions of the same order, the resulting system matrices seem to be the sparsest. The TDIE formula-tions using induced electric polarizations as unknown function are adopted and justified. Numerical results demonstrate that the proposed approach is accurate and efficient, and no late-time instability is observed.
基金National Natural Science Foundation of China(Grant No.11301462)University Science Research Project of Jiangsu Province(Grant No.13KJB110030)Yangzhou University Overseas Study Program and New Century Talent Project to Shan Jiang。
文摘We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes,which can provide a good balance between the numerical accuracy and computational cost.The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions.The multiscale basis functions have abilities to capture originally perturbed information in the local problem,as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes,where the layer-adapted meshes are generated by a given parameter.Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L^(2)norm and first order convergence in the energy norm on graded meshes,which is independent ofε.In contrast with the conventional methods,our method is much more accurate and effective.
基金supported by the National Natural Science Foundation of China (Nos.41374023,41131067,41474019)the National 973 Project of China (No.2013CB733302)+2 种基金the China Postdoctoral Science Foundation (No.2016M602301)the Key Laboratory of Geospace Envi-ronment and Geodesy,Ministry of Education,Wuhan University (No.15-02-08)the State Scholarship Fund from Chinese Scholarship Council (No.201306270014)
文摘The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation(VCE) and minimum standard deviation(MSTD) approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the firstorder Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.