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1.
M.J.D. Powell 《Numerische Mathematik》1994,68(1):107-128
Summary.
Let be a function from to that has
square
integrable second derivatives and let be the thin plate spline
interpolant
to at the points in
. We seek
bounds on the error when is in the convex
hull of
the interpolation points or when is close to at least one of
the
interpolation points but need not be in the convex hull. We find, for example,
that, if is inside a triangle whose vertices are any three
of the
interpolation points, then is bounded above by a
multiple of ,
where is the length of the longest side of the triangle and
where the
multiplier is independent of the interpolation points. Further, if
is any
bounded set in that is not a subset of a single straight
line, then we
prove that a sequence of thin plate spline interpolants converges to
uniformly on . Specifically, we require , where
is now the least upper bound on the numbers and where ,
, is
the least Euclidean
distance from to an interpolation point. Our method of
analysis applies
integration by parts and the Cauchy--Schwarz inequality to the scalar product
between second derivatives that occurs in the variational calculation of thin
plate spline interpolation.
Received November 10, 1993 / Revised version received March 1994 相似文献
2.
Michael Griebel Christian Rieger Barbara Zwicknagl 《Foundations of Computational Mathematics》2018,18(2):459-508
We present a theoretical framework for reproducing kernel-based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators. An explicit representation of the reproducing kernel is given in terms of an infinite series. We provide stability estimates for the kernel, including inverse Bernstein-type estimates for kernel-based trial spaces, and we give condition estimates for the interpolation matrix. Then, a deterministic error analysis for regularized reconstruction schemes is presented by means of sampling inequalities. In particular, we provide error bounds for a regularized reconstruction scheme based on a numerically feasible approximation of the kernel. This allows us to derive explicit coupling relations between the series truncation, the regularization parameters and the data set. 相似文献
3.
In this paper, we consider a class of penalized NCP-functions, which includes several existing well-known NCP-functions as
special cases. The merit function induced by this class of NCP-functions is shown to have bounded level sets and provide error
bounds under mild conditions. A derivative free algorithm is also proposed, its global convergence is proved and numerical
performance compared with those based on some existing NCP-functions is reported. 相似文献
4.
N. V. Baidakova 《Proceedings of the Steklov Institute of Mathematics》2015,288(1):29-39
We consider a natural class of composite finite elements that provide the mth-order smoothness of the resulting piecewise polynomial function on a triangulated domain and do not require any information on neighboring elements. It is known that, to provide a required convergence rate in the finite element method, the “smallest angle condition” must be often imposed on the triangulation of the initial domain; i.e., the smallest possible values of the smallest angles of the triangles must be lower bounded. On the other hand, the negative role of the smallest angle can be weakened (but not eliminated completely) by choosing appropriate interpolation conditions. As shown earlier, for a large number of methods of choosing interpolation conditions in the construction of simple (noncomposite) finite elements, including traditional conditions, the influence of the smallest angle of the triangle on the error of approximation of derivatives of a function by derivatives of the interpolation polynomial is essential for a number of derivatives of order 2 and higher for m ≥ 1. In the present paper, a similar result is proved for some class of composite finite elements. 相似文献
5.
Gaurav Thakur 《Journal of Functional Analysis》2011,260(8):2283-2299
We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker-Shannon-Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We prove some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties. 相似文献
6.
本文应用陈怀惠和顾永兴关于Zalcman不正规性条件的改进结果,推广和加强了Lappan的一个正规定则.Lappan证明:若亚纯函数族中的所有函数的球面导数的幂方(大于2)在紧集上的积分一致有界,则该族是正则的.本文证明,把积分限制在函数值的模小于给定常数的子集上,结论仍然成立.同时,用高阶导数的积分替代球面导数的积分,得到十分一般的结果.另外对幂方为2的情形也进行了讨论. 相似文献
7.
Robert Schaback 《Advances in Computational Mathematics》1995,3(3):251-264
For interpolation of scattered multivariate data by radial basis functions, an “uncertainty relation” between the attainable error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the Narcowich-Ward theory of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial basis functions that are currently in use. 相似文献
8.
Robert Schaback 《Advances in Computational Mathematics》1995,3(1):251-264
For interpolation of scattered multivariate data by radial basis functions, an “uncertainty relation” between the attainable
error and the condition of the interpolation matrices is proven. It states that the error and the condition number cannot
both be kept small. Bounds on the Lebesgue constants are obtained as a byproduct. A variation of the Narcowich-Ward theory
of upper bounds on the norm of the inverse of the interpolation matrix is presented in order to handle the whole set of radial
basis functions that are currently in use. 相似文献
9.
Johann Engelbrecht Igor Fedotov Tanya Fedotova Ansie Harding 《International Journal of Mathematical Education in Science & Technology》2013,44(6):831-846
Quadrature methods for approximating the definite integral of a function f(t) over an interval [a,b] are in common use. Examples of such methods are the Newton–Cotes formulas (midpoint, trapezoidal and Simpson methods etc.) and the Gauss–Legendre quadrature rules, to name two types of quadrature. Error bounds for these approximations involve higher order derivatives. For the Simpson method, in particular, the error bound involves a fourth-order derivative. Discounting the fact that calculating a fourth-order derivative requires a lot of differentiation, the main concern is that an error bound for the Simpson method, for example, is only relevant for a function that is four times differentiable, a rather stringent condition. This paper caters for functions for which derivatives exist only of order lower than normally required. A number of quadrature methods are considered and error bounds derived involving only lower order derivatives that can be used depending on the smoothness of the function. 相似文献
10.
The main theme of this paper is the construction of efficient, reliable and affordable error bounds for two families of quadrature methods for highly oscillatory integrals. We demonstrate, using asymptotic expansions, that the error can be bounded very precisely indeed at the cost of few extra derivative evaluations. Moreover, in place of derivatives it is possible to use finite difference approximations, with spacing inversely proportional to frequency. This renders the computation of error bounds even cheaper and, more importantly, leads to a new family of quadrature methods for highly oscillatory integrals that can attain arbitrarily high asymptotic order without computation of derivatives.
AMS subject classification (2000) Primary 65D30, secondary 34E05.Received June 2004. Accepted October 2004. Communicated by Lothar Reichel. 相似文献
11.
Apostolos Gerasoulis 《BIT Numerical Mathematics》1982,22(2):200-210
Nyström's interpolation formula is applied to the numerical solution of singular integral equations. For the Gauss-Chebyshev method, it is shown that this approximation converges uniformly, provided that the kernel and the input functions possess a continuous derivative. Moreover, the error of the Nyström interpolant is bounded from above by the Gaussian quadrature errors and thus convergence is fast, especially for smooth functions. ForC
input functions, a sharp upper bound for the error is obtained. Finally numerical examples are considered. It is found that the actual computational error agrees well with the theoretical derived bounds.This research has been partially supported by a grant from the Rutgers Research Council. 相似文献
12.
BaoHuai Sheng 《中国科学 数学(英文版)》2012,55(6):1243-1256
In the present paper,we provide an error bound for the learning rates of the regularized Shannon sampling learning scheme when the hypothesis space is a reproducing kernel Hilbert space(RKHS) derived by a Mercer kernel and a determined net.We show that if the sample is taken according to the determined set,then,the sample error can be bounded by the Mercer matrix with respect to the samples and the determined net.The regularization error may be bounded by the approximation order of the reproducing kernel Hilbert space interpolation operator.The paper is an investigation on a remark provided by Smale and Zhou. 相似文献
13.
该文考察Banach空间上的远达函数的可导性与远达点的存在性间的关系,指出某些Banach空间上的远达函数(对有界闭集而言)具等于1或-1的单侧方向导数蕴含远达点的存在性,并给出了Banach空间CLUR和LUR的新等价刻划. 相似文献
14.
Renxing Ni 《Journal of Mathematical Analysis and Applications》2006,316(2):642-651
The relationship between directional derivatives of generalized farthest functions and the existence of generalized farthest points in Banach spaces is investigated. It is proved that the generalized farthest function generated by a bounded closed set having a one-sided directional derivative equal to 1 or −1 implies the existence of generalized farthest points. New characterization theorems of (compact) locally uniformly convex sets are given. 相似文献
15.
We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results. 相似文献
16.
WANG Xinghua~ 《中国科学A辑(英文版)》2005,48(1)
The best quadrature formula has been found in the following sense:for afunction whose norm of the second derivative is bounded by a given constant and thebest quadrature formula for the approximate evaluation of integration of that function canminimize the worst possible error if the values of the function and its derivative at certainnodes are known.The best interpolation formula used to get the quadrature formula aboveis also found.Moreover,we compare the best quadrature formula with the open compoundcorrected trapezoidal formula by theoretical analysis and stochastic experiments. 相似文献
17.
Based on Peano kernel technique, explicit error bounds (optimal for the highest order derivative) are proved for the derivatives of cardinal spline interpolation (interpolating at the knots for odd degree splines and at the midpoints between two knots for even degree splines). The results are based on a new representation of the Peano kernels and on a thorough investigation of their zero distributions. The bounds are given in terms of Euler–Frobenius polynomials and their zeros. 相似文献
18.
We study computability and applicability of error bounds for a given semidefinite pro-gramming problem under the assumption that the recession function associated with the constraint system satisfies the Slater condition. Specifically, we give computable error bounds for the distances between feasible sets, optimal objective values, and optimal solution sets in terms of an upper bound for the condition number of a constraint system, a Lipschitz constant of the objective function, and the size of perturbation. Moreover, we are able to obtain an exact penalty function for semidefinite programming along with a lower bound for penalty parameters. We also apply the results to a class of statistical problems. 相似文献
19.
The best quadrature formula has been found in the following sense: for a function whose norm of the second derivative is bounded
by a given constant and the best quadrature formula for the approximate evaluation of integration of that function can minimize
the worst possible error if the values of the function and its derivative at certain nodes are known. The best interpolation
formula used to get the quadrature formula above is also found. Moreover, we compare the best quadrature formula with the
open compound corrected trapezoidal formula by theoretical analysis and stochastic experiments. 相似文献
20.
WANG Xinghua & YANG Shijun Department of Mathematics Zhejiang University Hangzhou China Department of Mathematics Hangzhou Normal College Hangzhou China 《中国科学A辑(英文版)》2006,49(8)
As usual, denote by KWr[a,b] the Sobolev class consisting of every function whose (r-1)th derivative is absolutely continuous on the interval [a,b] and rth derivative is bounded by K a.e. in [a, b]. For a function f∈KWr[a, b], its values and derivatives up to r -1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KWr[a. b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KWr[a, b] is also obtained. 相似文献