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 ...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.展开更多
Geologic surface approximation is profoundly affected by the presence, density and location of scattered geologic input data. Many studies have recognized the importance of utilizing varied sources of information when...Geologic surface approximation is profoundly affected by the presence, density and location of scattered geologic input data. Many studies have recognized the importance of utilizing varied sources of information when reconstructing a surface. This paper presents an improved geologic surface approximation method using a multiquadric function and borehole data. Additional information, i.e., inequality elevation and dip-strikes data extracted from outcrops or mining faces, is introduced in the form of physical constraints that control local changes in the estimated surface. Commonly accepted hypothesis states that geologic surfaces can be approximated to any desired degree of exactness by the summation of regular, mathematically defined, surfaces: in particular displaced quadric forms. The coefficients of the multiquadric functions are traditionally found by a least squares method. The addition of physical constraints in this work makes such an approach into a non-deterministic polynomial time problem. Hence we propose an objective function that represents the quality of the estimated surface and that includes the additional constraints by incorporation of a penalty function. Maximizing the smoothness of the estimated surface and its fitness to the additional constraints then allows the coefficients of the multiquadric function to be obtained by iterative methods. This method was implemented and demonstrated using data collected from the 81'st coal mining area of the Huaibei Coal Group.展开更多
This article introduces a fastmeshless algorithm for the numerical solution nonlinear partial differential equations(PDE)by Radial Basis Functions(RBFs)approximation connected with the Total Variation(TV)-basedminimiz...This article introduces a fastmeshless algorithm for the numerical solution nonlinear partial differential equations(PDE)by Radial Basis Functions(RBFs)approximation connected with the Total Variation(TV)-basedminimization functional and to show its application to image denoising containing multiplicative noise.These capabilities used within the proposed algorithm have not only the quality of image denoising,edge preservation but also the property of minimization of staircase effect which results in blocky effects in the images.It is worth mentioning that the recommended method can be easily employed for nonlinear problems due to the lack of dependence on a mesh or integration procedure.The numerical investigations and corresponding examples prove the effectiveness of the recommended algorithm regarding the robustness and visual improvement as well as peak-signal-to-noise ratio(PSNR),signal-to-noise ratio(SNR),and structural similarity index(SSIM)corresponded to the current conventional TV-based schemes.展开更多
Based on the existing continuous borehole strain observation,the multiquadric function fitting method was used to deal with time series data. The impact of difference kernel function parameters was discussed to obtain...Based on the existing continuous borehole strain observation,the multiquadric function fitting method was used to deal with time series data. The impact of difference kernel function parameters was discussed to obtain a valuable fitting result,from which the physical connotation of the original data and its possible applications were analyzed.Meanwhile,a brief comparison was made between the results of multiquadric function fitting and polynomial fitting.展开更多
The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than ba...The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.展开更多
The electric inversion technique reconstructs the subsurface medium distribution from acquired data.On the basis of electric inversion,objects buried under the earth or seabed,such as pipelines and unexploded ordnance...The electric inversion technique reconstructs the subsurface medium distribution from acquired data.On the basis of electric inversion,objects buried under the earth or seabed,such as pipelines and unexploded ordnance,are detected and located in a contactless manner.However,the process of accurately reconstructing the shape of the target object is challenging because electric inversion is a nonlinear and ill-posed problem.In this work,we present an inverse multiquadric(IMQ)regularization method based on the level set function for reconstructing buried pipelines.In the case of locating underwater objects,the unknown inversion area is split into two parts,the background and the pipeline with known conductivity.The geometry of the pipeline is represented based on the level set function for achieving a noiseless inversion image.To obtain a binary image,the IMQ is used as the regularization term,which‘pushes’the level set function away from 0.We also provide an appropriate method to select the bandwidth and regularization parameters for the IMQ regularization term,resulting in reconstructed images with sharp edges.The simulation results and analysis show that the proposed method performs better than classical inversion methods.展开更多
Price prediction plays a crucial role in portfolio selection (PS). However, most price prediction strategies only make a single prediction and do not have efficient mechanisms to make a comprehensive price prediction....Price prediction plays a crucial role in portfolio selection (PS). However, most price prediction strategies only make a single prediction and do not have efficient mechanisms to make a comprehensive price prediction. Here, we propose a comprehensive price prediction (CPP) system based on inverse multiquadrics (IMQ) radial basis function. First, the novel radial basis function (RBF) system based on IMQ function rather than traditional Gaussian (GA) function is proposed and centers on multiple price prediction strategies, aiming at improving the efficiency and robustness of price prediction. Under the novel RBF system, we then create a portfolio update strategy based on kernel and trace operator. To assess the system performance, extensive experiments are performed based on 4 data sets from different real-world financial markets. Interestingly, the experimental results reveal that the novel RBF system effectively realizes the integration of different strategies and CPP system outperforms other systems in investing performance and risk control, even considering a certain degree of transaction costs. Besides, CPP can calculate quickly, making it applicable for large-scale and time-limited financial market.展开更多
We solve numerically an eigenvalue elliptic partial differential equation(PDE)ranging from two to six dimensions using the generalized multiquadric(GMQ)radial basis functions(RBFs).Two discretization methods are em-pl...We solve numerically an eigenvalue elliptic partial differential equation(PDE)ranging from two to six dimensions using the generalized multiquadric(GMQ)radial basis functions(RBFs).Two discretization methods are em-ployed.The first method is similar to the classic mesh-based discretization method requiring n centers per dimension or a total ndpoints.The second method is based upon n randomly generated points in dℜrequiring far fewer data centers than the classic mesh method.Instead of having a crisp boundary,we form a“fuzzy”boundary.Using the analytic or numerical in-verse interior and boundary operators,we find the local and global minima and maxima to cull discretization points.We also find that the GMQ-RBF“flatness”can be controlled by increasing the GMQ exponential,β.We per-form a search to find the smallest root mean squared error(RMSE)by varying the exponent,the maximum,the minimum range of the GMQ shape parame-ter,and boundary influence,with the exponential having the most influence.Because the GMQ-RBFs are essentially nonlinear,it follows that the starting point of the simple search influences the end result.The optimal algorithm for high dimensional PDEs is still the subject of much research and may wait for the common place availability of massively parallel quantum computers for even higher dimensional PDEs and integral equations(IEs).展开更多
本文针对积分值条件下的拟插值问题,提出了一种基于Multiquadric (MQ)函数的新型高精度数值逼近方法。作为一类条件正定径向基函数,MQ函数凭借其指数级收敛特性在拟插值理论中具有重要的应用价值。现有的MQ拟插值方法主要基于函数值,在...本文针对积分值条件下的拟插值问题,提出了一种基于Multiquadric (MQ)函数的新型高精度数值逼近方法。作为一类条件正定径向基函数,MQ函数凭借其指数级收敛特性在拟插值理论中具有重要的应用价值。现有的MQ拟插值方法主要基于函数值,在实际应用中,函数信息经常以连续区间上的积分值形式呈现,本文重点解决仅知积分值条件下的构造问题。具体地,首先基于积分值的线性组合实现对节点处函数值及二阶导数值的逼近,进而结合利用函数值与二阶导数信息的拟插值方法,构造出新型的高精度积分值型MQ拟插值算子并推导了相应的误差估计表达式。数值实验结果表明,该方法有较好的逼近效果且其数值收敛阶与理论分析是吻合的,验证了所提算法的有效性。This paper proposes a novel high-precision numerical approximation method for quasi-interpolation problems under integral value conditions, utilizing Multiquadric (MQ) functions. As a class of conditionally positive definite radial basis functions, MQ functions hold significant application value in quasi-interpolation theory due to their exponential convergence properties. Existing MQ quasi-interpolation methods primarily rely on function values;however, in practical scenarios, functional information is often presented in the form of integral values over continuous intervals. This work focuses on addressing the construction of quasi-interpolation operators under the condition of known integral values. Specifically, we first approximate the function values and second-order derivative values at nodes through linear combinations of integral values. Subsequently, by integrating a quasi-interpolation framework that incorporates both function values and second-order derivative information, a novel high-precision integral-value-based MQ quasi-interpolation operator is constructed, accompanied by derived error estimation formulas. Numerical experiments demonstrate the favorable approximation performance of the proposed method, with the numerical convergence order aligning well with theoretical analyses, thereby validating the effectiveness of the algorithm.展开更多
Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical me...Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical method is the divided difference method.However,it has been shown strongly unstable in practice.Actually,it can only be used to simulate the lower order derivatives in applications.To simulate the high order derivatives,this paper suggests a new method using multiquadric quasi-interpolation.The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method.Moreover,some numerical examples are presented to confirm the theoretical results.Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method.This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.展开更多
In this paper, by using multivariate divided differences to approximate the partial derivative and superposition, we extend the multivariate quasi-interpolation scheme based on dimension-splitting technique which can ...In this paper, by using multivariate divided differences to approximate the partial derivative and superposition, we extend the multivariate quasi-interpolation scheme based on dimension-splitting technique which can reproduce linear polynomials to the scheme quadric polynomials. Furthermore, we give the approximation error of the modified scheme. Our multivariate multiquadric quasi-interpolation scheme only requires information of lo- cation points but not that of the derivatives of approximated function. Finally, numerical experiments demonstrate that the approximation rate of our scheme is significantly im- proved which is consistent with the theoretical results.展开更多
Quasi-interpolation is very useful in the study of approximation theory and its applications,since it can yield solutions directly without the need to solve any linear system of equations.Based on the good performance...Quasi-interpolation is very useful in the study of approximation theory and its applications,since it can yield solutions directly without the need to solve any linear system of equations.Based on the good performance,Chen and Wu presented a kind of multiquadric （MQ） quasi-interpolation,which is generalized from the L D operator,and used it to solve hyperbolic conservation laws and Burgers’ equation.In this paper,a numerical scheme is presented based on Chen and Wu’s method for solving the Korteweg-de Vries （KdV） equation.The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative,and the forward divided difference to approximate the temporal derivative,where the spatial derivative is approximated by the derivative of the generalized L D quasi-interpolation operator.The algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.展开更多
Multiquadric (MQ)函数作为径向基函数的一种,其作为核函数可逼近任何光滑函数,被广泛应用在拟插值的研究中,现有的MQ拟插值大部分都是以离散函数值为已知条件,而在实际应用中,积分值作为已知条件也比较常见,为了让MQ拟插值得到更广泛...Multiquadric (MQ)函数作为径向基函数的一种,其作为核函数可逼近任何光滑函数,被广泛应用在拟插值的研究中,现有的MQ拟插值大部分都是以离散函数值为已知条件,而在实际应用中,积分值作为已知条件也比较常见,为了让MQ拟插值得到更广泛的应用,本文提出了一种新的基于积分值的MQ拟插值算子。首先利用连续区间上积分值的线性组合来对节点处的导数值进行逼近,然后根据已有的MQ拟插值进一步得到新的积分值型MQ拟插值算子,并给出了误差估计。最后通过数值实验展示了本文构造的积分值型MQ拟插值算子的逼近效果,说明了该方法的可行性和有效性。As a kind of radial basis function, Multiquadric (MQ) function, as a kernel function, can approximate any smooth function, and is widely used in the research of quasi-interpolation. Most of the existing MQ quasi-interpolation is based on the known condition of the discrete function value, and in practical applications, the integral value is also a common-known condition. In order to make MQ quasi-interpolation more widely used, a new MQ quasi-interpolation operator based on integral value is proposed in this paper. First, the derivative value at the node is approximated by the linear combination of integral values on the continuous interval, and then a new integral value MQ quasi-interpolation operator is obtained according to the existing MQ quasi-interpolation, and the error estimate is given. Finally, the approximation effect of the integral-valued MQ quasi-interpolation operator constructed in this paper is demonstrated by numerical experiments, and the feasibility and effectiveness of the proposed method are demonstrated.展开更多
基金supported by the State Key Development Program for Basic Research of China (Grant No 2006CB303102)Science and Technology Commission of Shanghai Municipality,China (Grant No 09DZ2272900)
文摘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.
基金provided by the National Science and Technology Major Project of China (Nos.2009ZX05039-004 and 2009ZX 05039-002)the National Natural Science Foundation of China (Nos.40771167 and 70621001)
文摘Geologic surface approximation is profoundly affected by the presence, density and location of scattered geologic input data. Many studies have recognized the importance of utilizing varied sources of information when reconstructing a surface. This paper presents an improved geologic surface approximation method using a multiquadric function and borehole data. Additional information, i.e., inequality elevation and dip-strikes data extracted from outcrops or mining faces, is introduced in the form of physical constraints that control local changes in the estimated surface. Commonly accepted hypothesis states that geologic surfaces can be approximated to any desired degree of exactness by the summation of regular, mathematically defined, surfaces: in particular displaced quadric forms. The coefficients of the multiquadric functions are traditionally found by a least squares method. The addition of physical constraints in this work makes such an approach into a non-deterministic polynomial time problem. Hence we propose an objective function that represents the quality of the estimated surface and that includes the additional constraints by incorporation of a penalty function. Maximizing the smoothness of the estimated surface and its fitness to the additional constraints then allows the coefficients of the multiquadric function to be obtained by iterative methods. This method was implemented and demonstrated using data collected from the 81'st coal mining area of the Huaibei Coal Group.
文摘This article introduces a fastmeshless algorithm for the numerical solution nonlinear partial differential equations(PDE)by Radial Basis Functions(RBFs)approximation connected with the Total Variation(TV)-basedminimization functional and to show its application to image denoising containing multiplicative noise.These capabilities used within the proposed algorithm have not only the quality of image denoising,edge preservation but also the property of minimization of staircase effect which results in blocky effects in the images.It is worth mentioning that the recommended method can be easily employed for nonlinear problems due to the lack of dependence on a mesh or integration procedure.The numerical investigations and corresponding examples prove the effectiveness of the recommended algorithm regarding the robustness and visual improvement as well as peak-signal-to-noise ratio(PSNR),signal-to-noise ratio(SNR),and structural similarity index(SSIM)corresponded to the current conventional TV-based schemes.
基金sponsored by the Annual Earthquake Tracking Task,CEA(2017010214)
文摘Based on the existing continuous borehole strain observation,the multiquadric function fitting method was used to deal with time series data. The impact of difference kernel function parameters was discussed to obtain a valuable fitting result,from which the physical connotation of the original data and its possible applications were analyzed.Meanwhile,a brief comparison was made between the results of multiquadric function fitting and polynomial fitting.
基金Supported by the 973program-2006CB303102SGST 09DZ 2272900NSFC No.11026089
文摘The aim of this survey paper is to propose a new concept ＂generator＂. In fact, generator is a single function that can generate the basis as well as the whole function space. It is a more fundamental concept than basis. Various properties of generator are also discussed. Moreover, a special generator named multiquadric function is introduced. Based on the multiquadric generator, the multiquadric quasi-interpolation scheme is constructed, and furthermore, the properties of this kind of quasi-interpolation are discussed to show its better capacity and stability in approximating the high order derivatives.
基金supported by the National Natural Sci-ence Foundation of China(No.52101383)the Fundamen-tal Research Funds for the Central Universities(No.3072021CF0802)+3 种基金the Key Laboratory of Advanced Marine Communication and Information Technology,Ministry of Industry and Information Technology(No.AMCIT2101-02)the Sino-Russian Cooperation Fund of Harbin Engi-neering University(No.2021HEUCRF006)the Ministry of Science and Higher Education of the Russian Federation(No.075-15-2020-934)the International Science&Technology Cooperation Program of China(No.2014DF R10240).
文摘The electric inversion technique reconstructs the subsurface medium distribution from acquired data.On the basis of electric inversion,objects buried under the earth or seabed,such as pipelines and unexploded ordnance,are detected and located in a contactless manner.However,the process of accurately reconstructing the shape of the target object is challenging because electric inversion is a nonlinear and ill-posed problem.In this work,we present an inverse multiquadric(IMQ)regularization method based on the level set function for reconstructing buried pipelines.In the case of locating underwater objects,the unknown inversion area is split into two parts,the background and the pipeline with known conductivity.The geometry of the pipeline is represented based on the level set function for achieving a noiseless inversion image.To obtain a binary image,the IMQ is used as the regularization term,which‘pushes’the level set function away from 0.We also provide an appropriate method to select the bandwidth and regularization parameters for the IMQ regularization term,resulting in reconstructed images with sharp edges.The simulation results and analysis show that the proposed method performs better than classical inversion methods.
文摘Price prediction plays a crucial role in portfolio selection (PS). However, most price prediction strategies only make a single prediction and do not have efficient mechanisms to make a comprehensive price prediction. Here, we propose a comprehensive price prediction (CPP) system based on inverse multiquadrics (IMQ) radial basis function. First, the novel radial basis function (RBF) system based on IMQ function rather than traditional Gaussian (GA) function is proposed and centers on multiple price prediction strategies, aiming at improving the efficiency and robustness of price prediction. Under the novel RBF system, we then create a portfolio update strategy based on kernel and trace operator. To assess the system performance, extensive experiments are performed based on 4 data sets from different real-world financial markets. Interestingly, the experimental results reveal that the novel RBF system effectively realizes the integration of different strategies and CPP system outperforms other systems in investing performance and risk control, even considering a certain degree of transaction costs. Besides, CPP can calculate quickly, making it applicable for large-scale and time-limited financial market.
文摘We solve numerically an eigenvalue elliptic partial differential equation(PDE)ranging from two to six dimensions using the generalized multiquadric(GMQ)radial basis functions(RBFs).Two discretization methods are em-ployed.The first method is similar to the classic mesh-based discretization method requiring n centers per dimension or a total ndpoints.The second method is based upon n randomly generated points in dℜrequiring far fewer data centers than the classic mesh method.Instead of having a crisp boundary,we form a“fuzzy”boundary.Using the analytic or numerical in-verse interior and boundary operators,we find the local and global minima and maxima to cull discretization points.We also find that the GMQ-RBF“flatness”can be controlled by increasing the GMQ exponential,β.We per-form a search to find the smallest root mean squared error(RMSE)by varying the exponent,the maximum,the minimum range of the GMQ shape parame-ter,and boundary influence,with the exponential having the most influence.Because the GMQ-RBFs are essentially nonlinear,it follows that the starting point of the simple search influences the end result.The optimal algorithm for high dimensional PDEs is still the subject of much research and may wait for the common place availability of massively parallel quantum computers for even higher dimensional PDEs and integral equations(IEs).
文摘本文针对积分值条件下的拟插值问题,提出了一种基于Multiquadric (MQ)函数的新型高精度数值逼近方法。作为一类条件正定径向基函数,MQ函数凭借其指数级收敛特性在拟插值理论中具有重要的应用价值。现有的MQ拟插值方法主要基于函数值,在实际应用中,函数信息经常以连续区间上的积分值形式呈现,本文重点解决仅知积分值条件下的构造问题。具体地,首先基于积分值的线性组合实现对节点处函数值及二阶导数值的逼近,进而结合利用函数值与二阶导数信息的拟插值方法,构造出新型的高精度积分值型MQ拟插值算子并推导了相应的误差估计表达式。数值实验结果表明,该方法有较好的逼近效果且其数值收敛阶与理论分析是吻合的,验证了所提算法的有效性。This paper proposes a novel high-precision numerical approximation method for quasi-interpolation problems under integral value conditions, utilizing Multiquadric (MQ) functions. As a class of conditionally positive definite radial basis functions, MQ functions hold significant application value in quasi-interpolation theory due to their exponential convergence properties. Existing MQ quasi-interpolation methods primarily rely on function values;however, in practical scenarios, functional information is often presented in the form of integral values over continuous intervals. This work focuses on addressing the construction of quasi-interpolation operators under the condition of known integral values. Specifically, we first approximate the function values and second-order derivative values at nodes through linear combinations of integral values. Subsequently, by integrating a quasi-interpolation framework that incorporates both function values and second-order derivative information, a novel high-precision integral-value-based MQ quasi-interpolation operator is constructed, accompanied by derived error estimation formulas. Numerical experiments demonstrate the favorable approximation performance of the proposed method, with the numerical convergence order aligning well with theoretical analyses, thereby validating the effectiveness of the algorithm.
基金supported by the Major State Basic Research Development Program of China (973 Program) (Grant No.2006CB303102)the Science and Technology Commission of Shanghai Municipality (Grant No.09DZ2272900)
文摘Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation,especially for solving the differential equations numerically.The classical method is the divided difference method.However,it has been shown strongly unstable in practice.Actually,it can only be used to simulate the lower order derivatives in applications.To simulate the high order derivatives,this paper suggests a new method using multiquadric quasi-interpolation.The stability of the multiquadric quasi-interpolation method is compared with the classical divided difference method.Moreover,some numerical examples are presented to confirm the theoretical results.Both theoretical results and numerical examples show that the multiquadric quasi-interpolation method is much stabler than the divided difference method.This property shows that multiquadric quasi-interpolation method is an efficient tool to construct an approximation of high order derivatives based on scattered sampling data even with noise.
文摘In this paper, by using multivariate divided differences to approximate the partial derivative and superposition, we extend the multivariate quasi-interpolation scheme based on dimension-splitting technique which can reproduce linear polynomials to the scheme quadric polynomials. Furthermore, we give the approximation error of the modified scheme. Our multivariate multiquadric quasi-interpolation scheme only requires information of lo- cation points but not that of the derivatives of approximated function. Finally, numerical experiments demonstrate that the approximation rate of our scheme is significantly im- proved which is consistent with the theoretical results.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 11070131 10801024+1 种基金 U0935004)the Fundamental Research Funds for the Central Universities, China
文摘Quasi-interpolation is very useful in the study of approximation theory and its applications,since it can yield solutions directly without the need to solve any linear system of equations.Based on the good performance,Chen and Wu presented a kind of multiquadric （MQ） quasi-interpolation,which is generalized from the L D operator,and used it to solve hyperbolic conservation laws and Burgers’ equation.In this paper,a numerical scheme is presented based on Chen and Wu’s method for solving the Korteweg-de Vries （KdV） equation.The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative,and the forward divided difference to approximate the temporal derivative,where the spatial derivative is approximated by the derivative of the generalized L D quasi-interpolation operator.The algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.
文摘Multiquadric (MQ)函数作为径向基函数的一种,其作为核函数可逼近任何光滑函数,被广泛应用在拟插值的研究中,现有的MQ拟插值大部分都是以离散函数值为已知条件,而在实际应用中,积分值作为已知条件也比较常见,为了让MQ拟插值得到更广泛的应用,本文提出了一种新的基于积分值的MQ拟插值算子。首先利用连续区间上积分值的线性组合来对节点处的导数值进行逼近,然后根据已有的MQ拟插值进一步得到新的积分值型MQ拟插值算子,并给出了误差估计。最后通过数值实验展示了本文构造的积分值型MQ拟插值算子的逼近效果,说明了该方法的可行性和有效性。As a kind of radial basis function, Multiquadric (MQ) function, as a kernel function, can approximate any smooth function, and is widely used in the research of quasi-interpolation. Most of the existing MQ quasi-interpolation is based on the known condition of the discrete function value, and in practical applications, the integral value is also a common-known condition. In order to make MQ quasi-interpolation more widely used, a new MQ quasi-interpolation operator based on integral value is proposed in this paper. First, the derivative value at the node is approximated by the linear combination of integral values on the continuous interval, and then a new integral value MQ quasi-interpolation operator is obtained according to the existing MQ quasi-interpolation, and the error estimate is given. Finally, the approximation effect of the integral-valued MQ quasi-interpolation operator constructed in this paper is demonstrated by numerical experiments, and the feasibility and effectiveness of the proposed method are demonstrated.
