Accurate a posteriori error evaluation in the reduced basis method

In the reduced basis method, the evaluation of the a posteriori estimator can become very sensitive to round-off errors. In this Note, the origin of the loss of accuracy is revealed, and a solution to this problem is proposed and illustrated on a simple example.     

A posteriori boundary element error estimation

An a posteriori error estimator is presented for the boundary element method in a general framework. It is obtained by solving local residual problems for which a local concept is introduced to accommodate the fact that integral operators are nonlocal operators. The estimator is shown to have an upper and a lower bound by the constant multiples of the exact error in the energy norm for Symm's and hypersingular integral equations. Numerical results are also given to demonstrate the effectiveness of the estimator for these equations. It can be used for adaptive h,p, and hp methods.     

An integration procedure including error estimation

An ALGOL procedure for integration with error estimation of initial value problems for an ordinary differential equation is presented and analyzed. A three-step method of order four is utilized. No special starting procedure is required.     

An operator-theoretical approach to error estimation

Summary Error estimates (as opposed to bounds) may involve unspecified constants, in which case they have no direct value for numerical purposes. A practical method for obtaining realistic upper bounds of such generic constants is devised here. Directly applicable as it stands to a wide class of pointwise approximation problems, this method can further be adapted technically to that alternative class of important problems which is concerned with mean-square approximation. It is based on an explicitly motivated analysis of the general problem of bounding errors, which essentially evolves in the setting of operator theory while referring to such classical tools as the Peano kernel theorem and its generalization known as the Bramble-Hilbert lemma.     

On residual-based a posteriori error estimation in hp-FEM

A family , [0,1], of residual-based error indicators for the hp-version of the finite element method is presented and analyzed. Upper and lower bounds for the error indicators are established. To do so, the well-known Clément/Scott–Zhang interpolation operator is generalized to the hp-context and new polynomial inverse estimates are presented. An hp-adaptive strategy is proposed. Numerical examples illustrate the performance of the error indicators and the adaptive strategy.     

Restricted estimation in multivariate measurement error regression model

We study a multivariate ultrastructural measurement error (MUME) model with more than one response variable. This model is a synthesis of multivariate functional and structural models. Three consistent estimators of regression coefficients, satisfying the exact linear restrictions have been proposed. Their asymptotic distributions are derived under the assumption of a non-normal measurement error and random error components. A simulation study is carried out to investigate the small sample properties of the estimators. The effect of departure from normality of the measurement errors on the estimators is assessed.     

Asymptotic error estimation in linear elastic beam models

In this work, we establish that the error in norm H¹ between the solution of the three-dimensional linear elasticity system and that of the classical Bernoulli-Navier model, for a clamped rod with transversal section having a diameter of order s. is ()(ɛ^1/2).     

Accurate estimation of the number of binary partitions

Many authors have worked with the problem of binary partitions, but all estimates for the total number obtained so far are restricted to the exponential part only and hence very crude. The present paper is intended to give a final solution of the whole problem.     

Approximating the extreme Ritz values and upper bounds for the A-norm of the error in CG

Numerical Algorithms - In this paper, we investigate the generalized eigenvalue problem Ax = λBx arising from economic models. Under certain conditions, there is a simple generalized...     

A posteriori error estimation for elliptic eigenproblems

An a posteriori error estimator is presented for a subspace implementation of preconditioned inverse iteration, which derives from the well‐known inverse iteration in such a way that the associated system of linear equations is solved approximately by using a preconditioner. The error estimator is integrated in an adaptive multigrid algorithm to compute approximations of a modest number of the smallest eigenvalues together with the eigenfunctions of an elliptic differential operator. Error estimation is applied both within the actual finite element space (in order to estimate the iteration error) as well as in its hierarchical refinement of higher‐order elements (to estimate the discretization error) which gives rise to a balanced reduction of the iteration error and of the discretization error in the adaptive multigrid algorithm. Copyright © 2002 John Wiley & Sons, Ltd.     

Long-time error estimation for semi-linear parabolic equations

The long-time error estimation approach of Sun and Ewing (Dyn. Contin. Discrete Impuls. Systems Ser. B Appl. Algorithms, 9 (2002) 115–129) is applied here for the error analysis and estimation of linear and semi-linear parabolic partial differential equations. The analysis is carried out using the stability–smoothing indicator, the smoothing assumption, the moving attractor, the exact error propagation and the two-level error propagation analysis introduced by Sun and Ewing (Dyn. Contin. Discrete Impuls. Systems Ser. B Appl. Algorithms, 9 (2002) 115–129). Moreover, an inverse elliptic projection is employed here as a key technique in dealing with the spatial discretization error. The error estimates obtained are uniform in time. The results are substantiated by a complete mathematical analysis and numerical experiments.     

Local error estimation for multistep collocation methods

Multistep collocation methods for initial value problems in ordinary differential equations are known to be a subclass of multistep Runge-Kutta methods and a generalisation of the well-known class of one-step collocation methods as well as of the one-leg methods of Dahlquist. In this paper we derive an error estimation method of embedded type for multistep collocation methods based on perturbed multistep collocation methods. This parallels and generalizes the results for one-step collocation methods by Nørsett and Wanner. Simple numerical experiments show that this error estimator agrees well with a theoretical error estimate which is a generalisation of an error estimate first derived by Dahlquist for one-leg methods.     

Defect-based a-posteriori error estimation for differential-algebraic equations

We show how the QDeC estimator, an efficient and asymptotically correct a-posteriori error estimator for collocation solutions to ODE systems, can be extended to differential-algebraic equations of index 1. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic theory of the global error and some techniques of error estimation

Summary The error of the approximate solution obtained by discretising a functional equation can be shown under certain conditions to possess an asymptotic expansion in terms of some parameter which is usually a representative step-length. We consider the case of two-parameter expansions, which is particularly relevant to parabolic equations. We derive results for the existence of the expansion and for the application of the classical difference correction and of defect correction. The theory is illustrated by the discussion of a simple parabolic problem     

Stability in the presence of degeneracy and error estimation

Received February 10, 1997 / Revised version received June 6, 1998 Published online October 9, 1998     

Optimal error estimation for Petrov-Galerkin methods in two dimensions

Summary We examine the optimality of conforming Petrov-Galerkin approximations for the linear convection-diffusion equation in two dimensions. Our analysis is based on the Riesz representation theorem and it provides an optimal error estimate involving the smallest possible constantC. It also identifies an optimal test space, for any choice of consistent norm, as that whose image under the Riesz representation operator is the trial space. By using the Helmholtz decomposition of the Hilbert space [L ²()]², we produce a construction for the constantC in which the Riesz representation operator is not required explicitly. We apply the technique to the analysis of the Galerkin approximation and of an upwind finite element method.     

Consistent nonparametric estimation of error distributions in linear model

For the linear modely _i =x _i +e _i ,i=1, 2 ···, let the error sequence {e _i } _i=1 be iid r.v.'s, with unknown densityf(x). In this paper, a nonparametric estimation method based on the residuals is proposed for estimatingf(x) and the consistency of the estimators is obtained.The project supported by National Natural Science Foundation of China Crant 18971061.     

A probabilistic theory for error estimation in automatic integration

Summary A probabilistic theory for derivation and analysis of error criteria for automatic quadrature is presented. In particular, conditional average error criteria are derived for quadratures which have derivative-bound error estimates. These probabilistic error criteria are compared to variations of heuristic error criteria derived by discretizing the derivative in the original error bound. It is shown that the theory provides a mathematical foundation and a quantitative model for these discrete error criteria. It is also shown that estimating the conditional average error is equivalent to testing error with the spline interpolation as a sample integrand, and that this process can be made implicit by using appropriate error criteria with local error-checks.This paper is based on the author's Ph.D. thesis in computational complexity and numerical analysis, completed at the University of California, Berkeley     

Improved estimation of regression parameters in measurement error models

The problem of simultaneous estimation of the regression parameters in a multiple regression model with measurement errors is considered when it is suspected that the regression parameter vector may be the null-vector with some degree of uncertainty. In this regard, we propose two sets of four estimators, namely, (i) the unrestricted estimator, (ii) the preliminary test estimator, (iii) the Stein-type estimator and (iv) the postive-rule Stein-type estimator. In an asymptotic setup, properties of these estimators are studied based on asymptotic distributional bias, MSE matrices, and risks under a quadratic loss function. In addition to the asymptotic dominance of the Stein-type estimators, the paper contains discussion of dominating confidence sets based on the Stein-type estimation. Asymptotic analysis is considered based on a sequence of local alternatives to obtain the desired results.     

Time integration error estimation for continuous Galerkin schemes

The purpose of this contribution is the time integration error estimation for continuous Galerkin schemes applied to the linear semi-discrete equation of motion. A special focus is on the effort for the error estimation for large finite element models. Error estimators for the global time integration error as well as for the local error in the last time interval are presented. The Galerkin formulation in time allows the application of the well-known duality based error estimation techniques for the estimation of the time integration error. The main effort of these error estimators is the computation of the dual solution. In order to diminish the computational effort for solving the dual problem the error estimation is carried out in a reduced modal basis. The relevant modes which have to remain in the basis can be determined via the initial conditions of the dual problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)