An adaptive boundary collocation method for linear PDEs

The article proposes an adaptive algorithm based on a boundary collocation method for linear PDEs satisfying the maximal principle with possibly nonlinear boundary conditions. Given the error tolerance and an initial number of terms in the solution expansion, the algorithm computes expansion coefficients by collocation of boundary conditions and evaluates the maximum absolute error on the boundary. If error exceeds the error tolerance, additional expansion terms and boundary collocation points are added and the process repeated until the tolerance is satisfied. The performance of the algorithm is illustrated by an example of the potential flow past a cylinder placed between parallel walls. © 1995 John Wiley & Sons, Inc.     

Local theory of extrapolation methods

In this paper we discuss the theory of one-step extrapolation methods applied both to ordinary differential equations and to index 1 semi-explicit differential-algebraic systems. The theoretical background of this numerical technique is the asymptotic global error expansion of numerical solutions obtained from general one-step methods. It was discovered independently by Henrici, Gragg and Stetter in 1962, 1964 and 1965, respectively. This expansion is also used in most global error estimation strategies as well. However, the asymptotic expansion of the global error of one-step methods is difficult to observe in practice. Therefore we give another substantiation of extrapolation technique that is based on the usual local error expansion in a Taylor series. We show that the Richardson extrapolation can be utilized successfully to explain how extrapolation methods perform. Additionally, we prove that the Aitken-Neville algorithm works for any one-step method of an arbitrary order s, under suitable smoothness.     

Asymptotic error expansion of wavelet approximations of smooth functions II

Summary. We generalize earlier results concerning an asymptotic error expansion of wavelet approximations. The properties of the monowavelets, which are the building blocks for the error expansion, are studied in more detail, and connections between spline wavelets and Euler and Bernoulli polynomials are pointed out. The expansion is used to compare the error for different wavelet families. We prove that the leading terms of the expansion only depend on the multiresolution subspaces and not on how the complementary subspaces are chosen. Consequently, for a fixed set of subspaces , the leading terms do not depend on the fact whether the wavelets are orthogonal or not. We also show that Daubechies' orthogonal wavelets need, in general, one level more than spline wavelets to obtain an approximation with a prescribed accuracy. These results are illustrated with numerical examples. Received May 3, 1993 / Revised version received January 31, 1994     

A Small Noise Asymptotic Expansion for Young SDE Driven by Fractional Brownian Motion: A Sharp Error Estimate With Malliavin Calculus

This article shows an analytically tractable small noise asymptotic expansion with a sharp error estimate for the expectation of the solution to Young’s pathwise stochastic differential equations (SDEs) driven by fractional Brownian motions with the Hurst index H > 1/2. In particular, our asymptotic expansion can be regarded as small noise and small time asymptotics by the error estimate with Malliavin culculus. As an application, we give an expansion formula in one-dimensional general Young SDE driven by fractional Brownian motion. We show the validity of the expansion through numerical experiments.     

Approximation of an Eigenvalue Problem Associated with the Stokes Problem by the Stream Function-Vorticity-Pressure Method

By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported. The first author was supported by China Postdoctoral Sciences Foundation.     

An asymptotic expansion for product integration applied to Cauchy principal value integrals

Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. It is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed.     

High-dimensional Edgeworth expansion of a test statistic on independence and its error bound

In this paper, we calculate Edgeworth expansion of a test statistic on independence when some of the parameters are large, and simulate the goodness of fit of its approximation. We also calculate an error bound for Edgeworth expansion. Some tables of the error bound are given, which show that the derived bound is sufficiently small for practical use.     

Drawdown in prolate spheroidal–spherical coordinates obtained via Green’s function and perturbation methods

When investigating aquifer behaviour it is important to note that there exists a close relationship between the geometrical properties of the aquifer and the behaviour of the solution. In this paper our concern is to solve the flow equation described by prolate spheroidal coordinates by means of perturbation and the Green’s function method, where the spheroid is considered to be a perturbation of a sphere. We transformed the spheroidal coordinates to spherical polar coordinates in the limit, as the shape factor tends to zero. The new groundwater flow equation is solved via an asymptotic parameter expansion and the Green’s function method. The approximate solution of the new equation is compared with experimental data from real world. To take into account the error committed while approximating, we estimate the error in the asymptotic expansion. The error functions obtained suggest that the error would be very small for the shape factor tending to zero if the first two terms of the expansion are taken as an approximation.     

一类广义插值函数与广义有限元方法的后验估计

１．问题的提出具有快速振荡系数的微分方程大量出现在复合材料、多孔介质渗流等实际问题中．因为这类方程系数的激烈振荡性（受小尺度ε控制）,通常的有限元方法需花费巨大的计算工作量才能获得有意义的数值解（文［７］）,这对多维问题是无法承受的．８０年代发展起来的广义有限元法（文［１］［３］［５］）,为这类问题的解决提供了一条有效的新途径,它可在剖分步长ｈ＞＞　ε的情况下得到令人满意的数值结果．在广义有限元的理论分析中,因方程解的正则性估计通常与小尺度ε　有关,所以通常的有限元分析方法有一定的困难,目前尚未…     

Investigation of the effect of the measurement with the abrupt noise and the number of approximated terms in conductional boundary estimation problem

A direct error vector analysis of inverse heat conduction problem is presented to detect the measured noise. Based on the reverse matrix approach that the inverse problem is solved directly in a linear domain, and the error vector is formulated from the difference between the measured temperature and the estimated temperature. There is no prior knowledge on the exact solution while the error vector is constructed. The error vector is used to investigate the consistence of the measured data in the domain and lead to detect the noise data. Furthermore, the proper number of the undetermined variable is able to suggest based on the mean value of the error vector and the value of the condition number of the reverse matrix. In the first example, a test problem with the measurement noise is presented. The estimated result is influent by the noise globally. The result shows that the value of error vector is changed significantly at the coordinate of the measurement noise appeared. In other words, the error vector analysis is able to identify the noise data. In the second example, the proper number of series expansion terms is investigated. From the result, it shows that the number of expansion terms with the small mean value and condition number can better approximate to the unknown condition. It means that the proposed method is able to suggest a proper number of expansion terms when the function of the recovered boundary is unknown.     

强相关数据回归函数的小波估计

讨论了在强相关数据情形下对回归函数的小波估计,并且给出了估计量的均方误差的一个渐近展开表示式． 对研究估计量的优劣,所推导的近似表示式显得非常重要．对一般的回归函数核估计,如果回归函数不是充分光滑,这个均方误差表示式并不成立A·D2但对小波估计,即使回归函数间断连续,这个均方误差表示式仍然成立．因此,小波估计的收敛速度要比核估计来得快,从而小波估计在某种程度上改进了现有的核估计．     

Asymptotic error expansion for the Nyström method for nonlinear Fredholm integral equations of the second kind

In this paper the asymptotic error expansion for the Nyström method for one-dimensional nonlinear Fredholm integral equations of the second kind is considered. We show that the Nyström solution admits an error expansion in powers of the step-sizeh. Thus Richardson's extrapolation can be performed on the solution, and this will greatly increase the accuracy of the numerical solution.The project has been supported by the National Natural Science Foundation of China.     

图形的同调类中封闭图形数量的渐近估计

In this paper we give an asymptotic expansion including error terms for the number of cycles in homology classes for connected graphs.Mainly,we obtain formulae about the coefficients of error terms which depend on the homology classes and give two examples of how to calculate the coefficient of first error term.     

Extrapolation methods in Lie groups

Summary. The numerical solution of differential equations on Lie groups by extrapolation methods is investigated. The main principles of extrapolation for ordinary differential equations are extended on the general case of differential equations in noncommutative Lie groups. An asymptotic expansion of the global error is given. A symmetric method is given and quadratic asymptotic expansion of the global error is proved. The theoretical results are verified by numerical experiments. Received September 27, 1999 / Revised version received February 14, 2000 / Published online April 5, 2001     

Distributional expansion of maximum from logarithmic general error distribution

Logarithmic general error distribution is an extension of lognormal distribution. In this paper, with optimal norming constants the higher-order expansion of distribution of partial maximum of logarithmic general error distribution is derived.     

Extrapolation of the Hood–Taylor elements for the Stokes problem

An asymptotic error expansion concerning of Hood–Taylor elements for the Stokes problem is established on uniform rectangular meshes. The extrapolation based on such an expansion increases the rate of convergence by one order.     

Spatial approximation of stochastic convolutions

We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary frame. We show how to obtain error estimates when the truncated expansion is used in the equation. For the stochastic heat and wave equations, we combine the truncated expansion with a standard finite element method and derive a priori bounds for the mean square error. Specializing the frame to biorthogonal wavelets in one variable, we show how the hierarchical structure, support and cancelation properties of the primal and dual bases lead to near sparsity and can be used to simplify the simulation of the noise and its update when new terms are added to the expansion.     

A MULTI-PARAMETER SPLITTING EXTRAPOLATION AND A PARALLEL ALGORITHM FOR ELLIPTIC EIGENVALUE PROBLEM

1.IntroductionTheextrapolationmethodhasbecomeanimportanttechniquetoobtainmoreaccuratenumericalsolutionssinceitwasfirstestablishedbyruchardsonin1926.Theapplicationsofextrapolationmethodinthefinitdifferencecanbefoundin[14].In1983,Q.Lin,T.LhandS.Shen[8]intro…     

Error bounds for asymptotic approximations of the partition function

We consider Markovian queueing models with a finite number of states and a product form solution for its steady state probability distribution. Starting from the integral representation for the partition function in complex space we construct error bounds for its asymptotic expansion obtained by the saddle point method. The derivation of error bounds is based on an idea by Olver applicable to integral transforms with an exponentially decaying kernel. The bounds are expressed in terms of the supremum of a certain function and are asymptotic to the absolute value of the first neglected term in the expansion as the large parameter approaches infinity. The application of these error bounds is illustrated for two classes of queueing models: loss systems and single chain closed queueing networks.     

Asymptotic properties of integrated square error and cross-validation for kernel estimation of a regression function

Summary We derive an asymptotic expansion of integrated square error in kernel-type nonparametric regression. A similar result is obtained for a cross-validatory estimate of integrated square error. Together these expansions show that cross-validation is asymptotically optimal in a certain sense.