An experimental survey of a posteriori Courant finite element error control for the Poisson equation

This comparison of some a posteriori error estimators aims at empirical evidence for a ranking of their performance for a Poisson model problem with conforming lowest order finite element discretizations. Modified residual-based error estimates compete with averaging techniques and two estimators based on local problem solving. Multiplicative constants are involved to achieve guaranteed upper and lower energy error bounds up to higher order terms. The optimal strategy combines various estimators.     

Sharp a posteriori error estimate for elliptic equation with singular data

We introduce two residual type a posteriori error estimators for second-order elliptic partial differential equations with its right-hand side in L ^p (1 < p ⩽ 2) space. Both estimators are proved to yield global upper and local lower bounds for the W ^1,p seminorm of the error. We adopt the estimators as the indicators in h-mesh adaptive method to solve two typical model problems. It is verified by the numerical results that the estimators lead to optimal orders of convergence.     

A posteriori error estimates of edge residual type of finite element method for nonmonotone quasi‐linear elliptic problems

In this article, we study the edge residual‐based a posteriori error estimates of conforming linear finite element method for nonmonotone quasi‐linear elliptic problems. It is proven that edge residuals dominate a posteriori error estimates. Up to higher order perturbations, edge residuals can act as a posteriori error estimators. The global reliability and local efficiency bounds are established both in H ¹‐norm and L ²‐norm. Numerical experiments are provided to illustrate the performance of the proposed error estimators. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 813–837, 2014     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

A posteriori error estimates for elliptic problems with Dirac delta source terms

The aim of this paper is to introduce residual type a posteriori error estimators for a Poisson problem with a Dirac delta source term, in L ^p norm and W^1,p seminorm. The estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.     

Local a posteriori error estimates and adaptive control of pollution effects

Local a posteriori error estimators are derived for linear elliptic problems over general polygonal domains in 2d. The estimators lead to a sharp upper bound for the energy error in a local region of interest. This upper bound consists of H¹‐type local error indicators in a slightly larger subdomain, plus weighted L²‐type local error indicators outside this subdomain, which account for the pollution effects. This constitutes the basis of a local adaptive refinement procedure. Numerical experiments show a superior performance than the standard global procedure as well as the generation of locally quasi‐optimal meshes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 421–442, 2003     

A posteriori error estimates of constrained optimal control problem governed by convection diffusion equations

In this paper, we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations. The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L ²-norm and for state y and costate p measured in energy norm. Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.      

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H ¹-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.     

A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015     

Global Error Estimators for Order 7, 8 Runge–Kutta Pairs

Dormand, Prince and their colleagues [3–5] showed in a sequence of papers that the approximation of an initial value differential system propagated by a Runge–Kutta pair, together with a continuous approximation obtained using additional derivative values could be utilized to obtain estimates of the global error. They illustrated the results using pairs of orders p–1 and p for several values of p. The current authors [13] have developed a more direct representation of the order conditions, characterized families of global error estimators for Runge–Kutta pairs of arbitrary values of p, and showed that efficient global error estimating Runge–Kutta methods are based on the nodes of a Lobatto quadrature formula. Here, formulas for a good 7, 8 pair, interpolants of each of orders 7 and 8, and global error estimators of orders 10 and 12 illustrate how to obtain global error estimates of orders 9, 10, or 11, for arbitrary initial value systems. One set of graphs indicates that the stated order of the global error estimators is achieved numerically, and a second set illustrates the relative efficiency for several global error estimators when the approximation is propagated with a variable stepsize.     

Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. For hypersingular integral equations in 2D with a positive-order Sobolev space, we analyse the mathematical relation between the (h???h/2)-error estimator from [S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008), pp. 135–162], the two-level error estimator from [M. Maischak, P. Mund, and E. Stephan, Adaptive multilevel BEM for acoustic scattering, 585 Comput. Methods Appl. Mech. Eng. 150 (1997), pp. 351–367], and the averaging error estimator from [C. Carstensen and D. Praetorius, Averaging techniques for the a posteriori bem error control for a hypersingular integral equation in two dimensions, SIAM J. Sci. Comput. 29 (2007), pp. 782–810]. All of these a posteriori error estimators are simple in the following sense: first, the numerical analysis can be done within the same mathematical framework, namely localization techniques for the energy norm. Second, there is almost no implementational overhead for the realization.     

Exact convergence rate of bootstrap quantile variance estimator

Summary It is shown that the relative error of the bootstrap quantile variance estimator is of precise order n ^-1/4, when n denotes sample size. Likewise, the error of the bootstrap sparsity function estimator is of precise order n ^-1/4. Therefore as point estimators these estimators converge more slowly than the Bloch-Gastwirth estimator and kernel estimators, which typically have smaller error of order at most n ^-2/5.     

Statistical estimation in partial linear models with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. A model calibration approach and a weighting approach are developed to define the estimators of the parametric and nonparametric parts in the partial linear model, respectively. It is shown that the estimators for the parametric part are asymptotically normal and the estimators of g(·) converge to g(·) with an optimal convergent rate. Also, a comparison between the proposed estimators and the complete case estimator is made. A simulation study is conducted to compare the finite sample behaviors of these estimators based on bias and standard error.     

Second derivative estimation using harmonic analysis

Simulation sensitivity analysis is an important problem for simulation practitioners analyzing complex systems. The significance of this problem has resulted in the development of various gradient estimators that can be used to address this issue. Although higher derivative estimators have been discussed concurrently, less attention has been given to assess the efficiency and feasibility of computing such estimators. In this paper, two second derivative estimators are presented. The first estimators, called the HFD estimators, combine harmonic gradient estimators with finite differences second derivative estimators. The resulting hybrid estimators requireO(p) fewer simulation runs to implement compared to the straightforward finite differences approach, wherep is the number of input parameters in the simulation model. The second estimators, called the HA estimators, incorporate harmonic analysis directly, requiring one or two simulation runs to implement, depending on whether a control variate simulation run is made. Expressions for the bias and the variance of the HFD and the HA estimators (with and without variance reduction techniques) are derived. Optimal mean squared error convergence rates are also discussed. In particular, the convergence rates for both these estimators are shown to be the same, though the computational performance of the HFD estimators is better than that for the HA estimators on anM/M/1 queue simulation model. Computational results for the HFD estimators on an (s, S) inventory system simulation model are also included.     

Improving estimation in speckled imagery

We propose an analytical bias correction for the maximum likelihood estimators of theG ₁ ⁰ distribution. This distribution is a very powerful tool for speckled imagery analysis, since it is capable of describing a wide range of target roughness. We compare the performance of the corrected estimators with the corresponding original version using Monte Carlo simulation. This second-order bias correction leads to estimators which are better from both the bias and mean square error criteria.     

Minimax estimates of a linear parameter function in a regression model under restrictions on the parameters and variance-covariance matrix

Minimax nonhomogeneous linear estimators of scalar linear parameter functions are studied in the paper under restrictions on the parameters and variance-covariance matrix. The variance-covariance matrix of the linear model under consideration is assumed to be unknown but from a specific set R of nonnegativedefinite matrices. It is shown under this assumption that, without any restriction on the parameters, minimax estimators correspond to the least-squares estimators of the parameter functions for the “worst” variance-covariance matrix. Then the minimax mean-square error of the estimator is derived using the Bayes approach, and finally the exact formulas are derived for the calculation of minimax estimators under elliptical restrictions on the parameter space and for two special classes of possible variance-covariance matrices R. For example, it is shown that a special choice of a constant q ₀ and a matrixW ₀ defining one of the above classes R leads to the well known Kuks—Olman admissible estimator (see [16]) with a known variance-covariance matrixW ₀. Bibliography:32 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 79–92.     

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

State estimation for a dynamical system described by a linear equation with unknown parameters

We investigate the state estimation problem for a dynamical system described by a linear operator equation with unknown parameters in a Hilbert space. In the case of quadratic restrictions on the unknown parameters, we propose formulas for a priori mean-square minimax estimators and a posteriori linear minimax estimators. A criterion for the finiteness of the minimax error is formulated. As an example, the main results are applied to a system of linear algebraic-differential equations with constant coefficients.     

On Mixed Error Estimates for Elliptic Obstacle Problems

We establish in this paper sharp error estimates of residual type for finite element approximation to elliptic obstacle problems. The estimates are of mixed nature, which are neither of a pure a priori form nor of a pure a posteriori form but instead they are combined by an a priori part and an a posteriori part. The key ingredient in our derivation for the mixed error estimates is the use of a new interpolator which enables us to eliminate inactive data from the error estimators. One application of our mixed error estimates is to construct a posteriori error indicators reliable and efficient up to higher order terms, and these indicators are useful in mesh-refinements and adaptive grid generations. In particular, by approximating the a priori part with some a posteriori quantities we can successfully track the free boundary for elliptic obstacle problems.     

Estimating a density and its derivatives via the minimum distance method

Summary This paper deals with minimum distance (MD) estimators and minimum penalized distance (MPD) estimators which are based on the L _p distance. Rates of strong consistency of MPD density estimators are established within the family of density functions which have a bounded m-th derivative. For the case p=2, it is also proved that the MPD density estimator achieves the optimum rate of decrease of the mean integrated square error and the L ₁ error. Estimation of derivatives of the density is considered as well.In a class parametrized by entire functions, it is proved that the rate of convergence of the MD density estimator (and its derivatives) to the unknown density (its derivatives) is of order in expected L ₁ and L ₂ distances. In the same class of distributions, MD estimators of unknown density and its derivatives are proved to achieve an extraordinary rate (log log n/n)^1/2 of strong consistency.