共查询到20条相似文献,搜索用时 109 毫秒
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基于样本数据来数值模拟函数的高阶导数是数值逼近中遇到的一类重要而且基本的问题, 差商方法是数值微分的传统方法. 但是在实际问题的求解中, 它表现出强烈的不稳定性. 在实际应用中, 由于差商计算的不稳定性, 它仅能用来模拟函数的低阶导数. 为了更好地模拟函数的高阶导数, 本文利用multiquadric 拟插值提出了一种新的方法. 并将multiquadric 拟插值方法模拟函数导数的稳定性与传统差商方法所得结果进行了对比. 数值例子很好地验证了本文的理论. 从理论论证和数值例子比较来看, multiquadric 拟插值方法比差商方法更为稳定. 这个性质也表明, 基于散乱甚至有干扰的数据, 在逼近函数的高阶导数时, multiquadric 拟插值方法是一个有效的工具. 相似文献
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针对非线性Black-Scholes方程,基于quasi-Shannon小波函数给出了一种求解非线性偏微分方程的自适应多尺度小波精细积分法.该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性Black-Scholes方程自适应离散为非线性常微分方程组;然后将用于求解常微分方程组的精细积分法和小波变换的动态过程相结合,并利用非线性处理技术(如同伦分析技术)可有效求解非线性Black-Scholes方程.数值结果表明了该方法在数值精度和计算效率方面的优越性. 相似文献
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《中国科学:数学》2010,(12)
基于样本数据来数值模拟函数的高阶导数是数值逼近中遇到的一类重要而且基本的问题,差商方法是数值微分的传统方法.但是在实际问题的求解中,它表现出强烈的不稳定性.在实际应用中,由于差商计算的不稳定性,它仅能用来模拟函数的低阶导数.为了更好地模拟函数的高阶导数,本文利用multiquadric拟插值提出了一种新的方法.并将multiquadric拟插值方法模拟函数导数的稳定性与传统差商方法所得结果进行了对比.数值例子很好地验证了本文的理论.从理论论证和数值例子比较来看,multiquadric拟插值方法比差商方法更为稳定.这个性质也表明,基于散乱甚至有干扰的数据,在逼近函数的高阶导数时,multiquadric拟插值方法是一个有效的工具. 相似文献
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《应用数学与计算数学学报》2017,(3)
将径向基函数(radial basis function,RBF)插值引入积分方程的求解中,具体将待求函数表示为RBF的线性组合,再通过配点法将积分方程离散为线性或非线性方程组,求得权系数后给出待求函数的近似表示.论文选用的RBF是插值性能优异的多重二次曲面(multiquadric,MQ)函数,能在较少节点下取得较高的近似精度;而且RBF定义为距离的函数,在三维或高维插值时仅需改变距离公式,因而便于推广到高维积分方程求解中.在RBF插值矩阵的构造中,元素的积分计算分别通过高斯积分或基于区域剖分的数值求积完成,实现了一维、二维下Fredholm和Volterra方程的求解.算例结果表明:论文方法具有实施方便和精度较高的优点,是一种适合积分方程求解的新方法. 相似文献
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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. 相似文献
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In this paper, we construct a univariate quasi-interpolation operator to non-uniformly distributed data by cubic multiquadric functions. This operator is practical, as it does not require derivatives of the being approximated function at endpoints. Furthermore, it possesses univariate quadratic polynomial reproduction property, strict convexity-preserving and shape-preserving of order 3 properties, and a higher convergence rate. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operator with that of Wu and Schaback’s quasi-interpolation scheme. 相似文献
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MA LiMin & WU ZongMin Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Sciences Fudan University Shanghai China 《中国科学 数学(英文版)》2010,(4)
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 s... 相似文献
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Based on multiquadric trigonometric quasi-interpolation, the paper proposes a meshless symplectic scheme for Hamiltonian wave equation with periodic boundary conditions. The scheme first discretizes the equation in space using an iterated derivative approximation method based on multiquadric trigonometric quasi-interpolation and then in time with an appropriate symplectic scheme. This in turn yields a finite-dimensional semi-discrete Hamiltonian system whose energy and momentum (approximations of the continuous ones) are invariant with respect to time. The key feature of the scheme is that it conserves both the energy and momentum of the Hamiltonian system for both uniform and scattered centers, while classical energy-momentum conserving schemes are only for uniform centers. Numerical examples provided at the end of the paper show that the scheme is efficient and easy to implement. 相似文献
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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. 相似文献
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This paper discusses the sufficient conditions for the shape preserving quasi-interpolation with multiquadric. Some quasi-interpolation
schema is given such that the interpolation as well as its high derivatives is convergent.
Supported by the National Natural Science Foundation of China. 相似文献
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This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method. 相似文献
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In this paper, by virtue of using the linear combinations of the shifts of f(x) to approximate the derivatives of f(x) and Waldron’s superposition idea (2009), we modify a multiquadric quasi-interpolation with the property of linear reproducing to scattered data on one-dimensional space, such that a kind of quasi-interpolation operator Lr+1f has the property of r+1(r∈Z,r≥0) degree polynomial reproducing and converges up to a rate of r+2. There is no demand for the derivatives of f in the proposed quasi-interpolation Lr+1f, so it does not increase the orders of smoothness of f. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operators with that of Wu-Schaback’s quasi-interpolation scheme and Feng-Li’s quasi-interpolation scheme. 相似文献
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Renhong Wang Jinming Wu 《计算数学(英文版)》2007,25(1):97-103
In this paper, we propose a new approach to solve the approximate implicitization problem based on RBF networks and MQ quasi-interpolation. This approach possesses the advantages of shape preserving, better smoothness, good approximation behavior and relatively less data etc. Several numerical examples are provided to demonstrate the effectiveness and flexibility of the proposed method. 相似文献