自适应稀疏伪谱逼近新方法

自适应稀疏伪谱逼近法是广义混沌多项式类方法的最新进展,相对于其它方法具有计算精度高、速度快的优点.但它仍存在如下缺点：1）终止判据对逼近误差的估计精度偏低;2）只适用于单输出问题.本文提出了适用于多输出问题且具有更高逼近精度的自适应稀疏伪谱逼近新方法.本文首先提出了新型终止判据及基于此新型终止判据的自适应稀疏伪谱逼近新方法,并以命题的形式证明了新型终止判据相比于现有终止判据具有更高的估计精度,从而使基于此的逼近函数精度更接近于预期精度;进而,本文基于指标集的统一策略和新型终止判据,提出了适用于多输出问题的自适应稀疏伪谱逼近新方法,该方法因能充分利用各输出变量的抽样结果,具有比将单输出方法直接推广到多输出问题更高的计算效率.多个算例验证了本文所提出新方法的有效性和正确性.     

A new a posteriori error estimate for convection–reaction–diffusion problems

A new a posteriori error estimate is derived for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in the usual energy norm, the underlying bilinear form is decomposed into a computable integral and two other terms which can be estimated from above using elementary tools of functional analysis. Two auxiliary parameter-functions are introduced to construct such a splitting and tune the resulting bound. If these functions are chosen in an optimal way, the exact energy norm of the error is recovered, which proves that the estimate is sharp. The presented methodology is completely independent of the numerical technique used to compute the approximate solution. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g. due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors, etc. Moreover, the only constant that appears in the proposed error estimate is global and stems from the Friedrichs–Poincaré inequality. Numerical experiments illustrate the potential of the proposed error estimation technique.     

OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

In this paper, we provide a theoretical analysis of the partition of unity finite elementmethod (PUFEM), which belongs to the family of meshfree methods. The usual erroranalysis only shows the order of error estimate to the same as the local approximations[12].Using standard linear finite element base functions as partition of unity and polynomials aslocal approximation space, in 1-d case, we derive optimal order error estimates for PUFEMinterpolants. Our analysis show that the error estimate is of one order higher than thelocal approximations. The interpolation error estimates yield optimal error estimates forPUFEM solutions of elliptic boundary value problems.     

View factor for inclined plane with Gaussian source

The view factor (angle factor) for a differential inclined plane in the case of a radiating source of radially Gaussian intensity is considered. This information is useful for modelling of solar radiation in certain applications. The view factor is expressed in terms of two integrals, one of which is obtained in closed form in terms of special functions, and the other is approximated. A compact estimate for the view factor is presented which is suitable for machine computation. While the relative error associated with the final estimate is typically less than 0.01%, and in all cases, less than 0.2%, the method is easily extended to yield even greater accuracy.     

Some properties of a simple moving average when applied to forecasting a time series

Simple (equally weighted) moving averages are frequently used to estimate the current level of a time series, with this value being projected as a forecast for future observations. A key measure of the effectiveness of the method is the sampling error of the estimator, which this paper defines in terms of characteristics of the data. This enables the optimal length of the average for any steady state model to be established and the lead time forecast error derived. A comparison of the performance of a simple moving average (SMA) with an exponentially weighted moving average (EWMA) is made. It is shown that, for a steady state model, the variance of the forecast error is typically less than 3% higher than the appropriate EWMA. This relatively small difference may explain the inconclusive results from the empirical studies about the relative predictive performance of the two methods.     

A posteriori error estimations for mixed finite-element approximations to the Navier-Stokes equations

A posteriori estimates for mixed finite element discretizations of the Navier-Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier-Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.     

A posteriori error estimations for mixed finite-element approximations to the Navier–Stokes equations

A posteriori estimates for mixed finite element discretizations of the Navier–Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier–Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided.     

A MODIFIED TIKHONOV REGULARIZATION METHOD FOR THE CAUCHY PROBLEM OF LAPLACE EQUATION

In this paper,we consider the Cauchy problem for the Laplace equation,which is severely ill-posed in the sense that the solution does not depend continuously on the data.A modified Tikhonov regularization method is proposed to solve this problem.An error estimate for the a priori parameter choice between the exact solution and its regularized approximation is obtained.Moreover,an a posteriori parameter choice rule is proposed and a stable error estimate is also obtained.Numerical examples illustrate the validity and effectiveness of this method.     

Multi-stage nonparametric estimation of density function using orthonormal systems

In this note, the authors propose a new nonparametric method of estimation of density using orthonormal systems iteratively. The asymptotic mean integrated square error of the estimate at each stage is less than or equal to that of the preceding stage. The new estimate is better, in some cases, than the traditional estimate based upon orthonormal functions from the point of view of the mean integrated square error in the limit.     

A discontinuous Galerkin method for the Rosenau equation

A priori error estimates for the Rosenau equation, which is a K-dV like Rosenau equation modelled to describe the dynamics of dense discrete systems, have been studied by one of the authors. But since a priori error bounds contain the unknown solution and its derivatives, it is not effective to control error bounds with only a given step size. Thus we need to estimate a posteriori errors in order to control accuracy of approximate solutions using variable step sizes. A posteriori error estimates of the Rosenau equation are obtained by a discontinuous Galerkin method and the stability analysis is discussed for the dual problem. Numerical results on a posteriori error and wave propagation are given, which are obtained by using various spatial and temporal meshes controlled automatically by a posteriori error.     

Parallel Galerkin domain decomposition procedures based on the streamline diffusion method for convection-diffusion problems

Based upon the streamline diffusion method, parallel Galerkin domain decomposition procedures for convection-diffusion problems are given. These procedures use implicit method in the sub-domains and simple explicit flux calculations on the inter-boundaries of sub-domains by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the parallelism can be achieved by these procedures. The explicit nature of the flux calculations induces a time step limitation that is necessary to preserve stability. Artificial diffusion parameters δ are given. By analysis, optimal order error estimate is derived in a norm which is stronger than L²-norm for these procedures. This error estimate not only includes the optimal H¹-norm error estimate, but also includes the error estimate along the streamline direction ‖β⋅∇(u−U)‖, which cannot be achieved by standard finite element method. Experimental results are presented to confirm theoretical results.     

Explicit error estimate for the nonconforming wilson's element

In this article, we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method. We explicitly obtain the approximation error estimate and the consistency error estimate for the Wilson's element without the regular assumption, respectively, which implies the final finite element error estimate. Such explicit a priori error estimates can be used as computable error bounds.     

Bayes估计的进一步研究及其Pitman最优性

对线性模型参数,讨论了Bayes估计的Pitman最优性,将已有结果进行了改进,去掉了附加条件,证明了在Pitman准则下,Bayes估计一致优于最小二乘估计(LSE),在此基础上,提出了一种基于先验信息的方差分量估计,通过和基于LSE的方差分量估计作比较,证明了新估计是无偏估计且有更小的均方误差．最后,证明了在Pitman准则下生长曲线模型参数的Bayes估计优于最佳线性无偏估计．     

AnL 2-error estimate for an approximation of the solution of a parabolic variational inequality

Summary We estimate the order of the difference between the numerical approximation and the solution of a parabolic variational inequality. The numerical approximation is obtained using a finite element discretization in space and a finite difference discretization in time which is more general than is used in the literature. We obtain better error estimates than those given in the literature. The error estimates are compared with numerical experiments.     

A simple method for computing velocities from tracer observations and a model output

A method is suggested for estimating velocities by combining observations of a continuously distributed tracer and a circulation model output (background velocity field). A theoretical error analysis is provided and checked by simulations. The estimate reduces the model error by about 12–32% and the estimation procedure turns out to be of high computational efficiency. In addition, a compatibility measure between data and model is introduced which is also tested by simulations.     

Runge-Kutta Methods with Global Error Estimates

In the numerical solution of ordinary differential initial valueproblems, it is desirable to have a global error estimate. Inthis paper the practical use of a global error estimate introducedby H. J. Stetter is discussed, and specific procedures involvingexplicit Runge-Kutta methods, which effect a certain savingin function evaluations, are presented. Some numerical resultsshow the accuracy of the estimate.     

Choosing a stepsize for Taylor series methods for solving ODE'S

Problem-dependent upper and lower bounds are given for the stepsize taken by long Taylor series methods for solving initial value problems in ordinary differential equations. Taylor series methods recursively generate the terms of the Taylor series and estimate the radius of convergence as well as the order and location of the primary singularities. A stepsize must then be chosen which is as large as possible to minimize the required number of steps, while remaining small enough to maintain the truncation error less than some tolerance.One could use any of four different measures of trunction error in an attempt to control the global error : i) absolute truncation error per step, ii) absolute trunction error per unit step, iii) relative truncation error per step, and iv) relative truncation error per unit step. For each of these measures, we give bounds for error and for the stepsize which yields a prescribed error. The bounds depend on the series length, radius of convergence, order, and location of the primary singularities. The bounds are shown to be optimal for functions with only one singularity of any order on the circle of convergence.     

Efficient Collocation Schemes for Singular Boundary Value Problems

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.     

An a posteriori error estimate for a first-kind integral equation

In this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the first kind. The estimate is local and sharp for quasi-uniform meshes and so improves earlier work of ours. The mesh-dependence of the constants is analyzed and shown to be weaker than expected from our previous work. Besides the Galerkin boundary element method, the collocation method and the qualocation method are considered. A numerical example is given involving an adaptive feedback algorithm.