共查询到18条相似文献,搜索用时 67 毫秒
1.
本文针对扩散方程,借助于Halton节点,基于径向基函数和迭代法提出一种新的无网格方法.该方法在时间层上采用全隐离散,在空间层上构造了θ-型迭代格式,然后利用径向基函数去逼近未知函数.由于本文采用紧支集径向基函数,因而导出的离散系统矩阵是稀疏的,且有较好的条件数.最后通过数值实验验证新方法的有效性. 相似文献
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我们知道,n元函数关于某个自变量的偏导数可理解为:固定其余的x-1个自变量xl1…,xi-1,xi+1,…,xn,即令这些自变量为常数,这样几x;,…,xn)就是关于xi的一元函数,天就是f关于xi的导数。这样我们将多元函数的偏导数概念和一元函数的导数之间建立了联系,然后可用求解常微分方程的方法求解一些简单的偏微分方程。以下树中均设未知函数是充分光滑的。例1已知u(0,y)=y,未满足方程的函数y=u(x,y)解:由于正可理解为固定y,即令y为常数时X关于X的导数,故方程两边对X积分可得C(C,…ZC+C式中C为积分常数。由于y为常… 相似文献
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研究了线性微分方程f((n))+An-2f((n-2))+…+A0(z)f=0整函数解的Julia集的径向分布,其中n≥2,Aj(z)(j=0,1,…,n-2)是具有有限下级的整函数,得到了这类方程线性无关解的乘积的Julia集的径向分布的下界. 相似文献
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《数学的实践与认识》2013,(16)
研究了一类非线性偏微分方程的极值原理,发现不含有线性项的方程的极值原理总是成立的,与初值函数的符号无关,而含有线性项的方程的极值原理受初值函数符号的影响,在初值函数非负时可以证明,而初值函数非正的时候则不成立,并利用数值解进行了验证. 相似文献
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从高等数学教材课后习题的偏导数恒等式变换求解,引导学生讨论一类偏微分方程的求解.在拓展课程内容、应用和常微分方程变量分离方法的基础上,巩固多元复合函数求导法则,常系数线性微分方程求解方法和傅里叶级数的相关理论与方法. 相似文献
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为提高偏微分方程的计算求解精度,设计了以多元二次径向基神经网络为求解单元的偏微分计算方法,给出了多元二次径向基神经网络的具体求解结构,并以此神经网络为求解基础,给出了具体的偏微分计算步骤.通过具体的偏微分求解实例验证方法的有效性,并以3种不同设计样本数构建的多元二次径向基神经网络为计算单元,从实例求解所需的计算时间以及解的精度作对比,结果表明,采用基于多元二次径向基神经网络的偏微分方程求解方法具有求解精度高以及计算效率低等特点. 相似文献
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基于径向基逼近理论,本文为KdV方程构造了一个无网格辛算法.首先借助径向基空间离散Hamilton函数以及Poisson括号,把KdV方程转化成一个有限维的Hamilton系统.然后用辛积分子离散有限维系统,得到辛算法.文章进一步讨论了所构造辛算法的收敛性和误差界.数值例子验证了理论分析. 相似文献
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Constrained Global Optimization of Expensive Black Box Functions Using Radial Basis Functions 总被引:1,自引:0,他引:1
We present a new strategy for the constrained global optimization of expensive black box functions using response surface models. A response surface model is simply a multivariate approximation of a continuous black box function which is used as a surrogate model for optimization in situations where function evaluations are computationally expensive. Prior global optimization methods that utilize response surface models were limited to box-constrained problems, but the new method can easily incorporate general nonlinear constraints. In the proposed method, which we refer to as the Constrained Optimization using Response Surfaces (CORS) Method, the next point for costly function evaluation is chosen to be the one that minimizes the current response surface model subject to the given constraints and to additional constraints that the point be of some distance from previously evaluated points. The distance requirement is allowed to cycle, starting from a high value (global search) and ending with a low value (local search). The purpose of the constraint is to drive the method towards unexplored regions of the domain and to prevent the premature convergence of the method to some point which may not even be a local minimizer of the black box function. The new method can be shown to converge to the global minimizer of any continuous function on a compact set regardless of the response surface model that is used. Finally, we considered two particular implementations of the CORS method which utilize a radial basis function model (CORS-RBF) and applied it on the box-constrained Dixon–Szegö test functions and on a simple nonlinearly constrained test function. The results indicate that the CORS-RBF algorithms are competitive with existing global optimization algorithms for costly functions on the box-constrained test problems. The results also show that the CORS-RBF algorithms are better than other algorithms for constrained global optimization on the nonlinearly constrained test problem. 相似文献
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B. J. C. Baxter 《Foundations of Computational Mathematics》2008,8(3):395-407
A radial basis function (RBF) has the general form
where the coefficients a
1,…,a
n
are real numbers, the points, or centres, b
1,…,b
n
lie in ℝ
d
, and φ:ℝ
d
→ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance
(Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline
spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported
when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy
certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities,
with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions.
Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore,
we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite
functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we
are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore,
the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric
integration.
Dedicated to Arieh Iserles on the occasion of his 60th birthday. 相似文献
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Li Zha Renzhong Feng~* 《高等学校计算数学学报(英文版)》2007,16(4):348-357
In this paper,a new quasi-interpolation with radial basis functions which satis- fies quadratic polynomial reproduction is constructed on the infinite set of equally spaced data.A new basis function is constructed by making convolution integral with a constructed spline and a given radial basis function.In particular,for twicely differ- entiable function the proposed method provides better approximation and also takes care of derivatives approximation. 相似文献
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In paper [1],it was shown that an explicit expression of the cardinal basis functions for two-point Hermite interpolation. This paper will show the explicit expression of Hermite interpolation under the Ball basis. 相似文献
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Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated
reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply
to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes
of functions still were not "large" enough to be applicable to a number of practical situations. In this paper we obtain Sobolev-type
error estimates on compact regions of Rn when the RBFs have Fourier transforms that decay algebraically. In addition, we derive a Bernstein inequality for spaces
of finite shifts of an RBF in terms of the minimal separation parameter. 相似文献
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Refined Error Estimates for Radial Basis Function Interpolation 总被引:1,自引:0,他引:1
We discuss new and refined error estimates for radial-function scattered-data interpolants and their derivatives. These estimates hold on R
d
, the d-torus, and the 2-sphere. We employ a new technique, involving norming sets, that enables us to obtain error estimates, which in many cases give bounds orders of magnitude smaller than those previously known. 相似文献