Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity

We consider an augmented mixed finite element method applied to the linear elasticity problem and derive a posteriori error estimators that are simpler and easier to implement than the ones available in the literature. In the case of homogeneous Dirichlet boundary conditions, the new a posteriori error estimator is reliable and locally efficient, whereas for non-homogeneous Dirichlet boundary conditions, we derive an a posteriori error estimator that is reliable and satisfies a quasi-efficiency bound. Numerical experiments illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.     

Rough paths and 1d SDE with a time dependent distributional drift: application to polymers

We estimate the support of a uniform density, when it is assumed to be a convex polytope or, more generally, a convex body in \({\mathbb {R}}^d\). In the polytopal case, we construct an estimator achieving a rate which does not depend on the dimension \(d\), unlike the other estimators that have been proposed so far. For \(d\ge 3\), our estimator has a better risk than the previous ones, and it is nearly minimax, up to a logarithmic factor. We also propose an estimator which is adaptive with respect to the structure of the boundary of the unknown support.     

A note on the performance of the gamma kernel estimators at the boundary

The gamma kernel estimator is proposed in Chen [Chen, S.X., 2000. Probability density function estimation using gamma kernels. Annals of the Institute of Statistical Mathematics 52, 471–480] to estimate densities with support $[0,\infty )$. It is shown in his paper that the gamma kernel estimator is non-negative, free of boundary bias, and achieves the optimal rate of convergence for the mean integrated squared error. Numerical results reported in Chen’s paper show that, in the boundary region, the gamma kernel estimator even outperforms some widely used boundary corrected density estimators such as the boundary kernel estimator. However, our study finds that the gamma kernel estimator at $x=0$ is actually the reflection estimator when the double exponential kernel is used and is only boundary problem free when the estimated density has a shoulder at $x=0$ (i.e., the first derivative of the density at $x=0$ is zero). For densities not satisfying the shoulder condition, we show that the gamma kernel estimator has a severe boundary problem and its performance is inferior to that of the boundary kernel estimator.     

Estimation of a Support Curve via Order Statistics

This paper deals with nonparametric estimation of the boundary curve of the support of a bivariate density function. This estimation problem arises in various contexts, such as for example scatterpoint image analysis and frontier estimation in econometrics. The setup in this paper is a general one, allowing the bivariate density function to be infinite, bounded away from zero or zero at the boundary. Two estimators for the boundary curve are introduced, both based on order statistics. The asymptotic distribution of the estimators and their rate of convergence are established. Via a comparison of the rates of convergence we recommend which estimator to use in a particular situation. Both estimators can be used as an initial estimator in a two-stage procedure, designed for getting a better estimation. Simulation studies demonstrate the finite-sample behavior of the estimators and the proposed two-stage procedure. We illustrate the procedure on a data set on American electric utility companies.     

On-line Tracking of a Smooth Regression Function

We construct an on-line estimator with equidistant design for tracking a smooth function from Stone–Ibragimov–Khasminskii’s class. This estimator has the optimal convergence rate of risk to zero in sample size. The procedure for setting coefficients of the estimator is controlled by a single parameter and has a simple numerical solution. The off-line version of this estimator allows to eliminate a boundary layer. Simulation results are given. This work is partially supported by a fellowship from the Yitzhak and Chaya Weinstein Research Institute for Signal Processing at Tel Aviv University.     

Set estimation under convexity type assumptions

The problem of estimating a set S from a random sample of points taken within S is considered. It is assumed that S is r-convex, which means that a ball of radius r can go around from outside the set boundary. Under this assumption, the r-convex hull of the sample is a natural estimator of S. We obtain convergence rates for this estimator under both the distance in measure and the Hausdorff metric between sets. It is also proved that the boundary of the estimator consistently estimates the boundary of S, in Hausdorff's sense.     

A Lattice-Based Smoother for Regions With Irregular Boundaries and Holes

We consider the problem of estimating a smooth function over a spatial region that is delineated by an irregular boundary and potentially contains holes within the boundary. Methods commonly used for spatial function estimation are well-known to suffer from bias along such boundaries. The estimator we propose is a kernel regression estimator, where the kernel is an approximation to a two-dimensional diffusion process contained within the region of interest. The diffusion process is approximated by the distribution of length-k random walks originating from each observation location and constrained to stay within the domain boundaries. We propose using a cross-validation criterion to find the optimal walk length k, which controls the smoothness of the resulting estimate. Simulations show that the method outperforms the soap film smoother of Wood, Bravington, and Hedley in many realistic scenarios, when data are noisy and borders are highly irregular. We illustrate the practical use of the estimator using measurements of soil manganese concentration around Port Moller, Alaska. Supplementary materials for this article are available online.     

Local linear estimator for stochastic differential equations driven by α-stable Lévy motions

We study the local linear estimator for the drift coefficient of stochastic differential equations driven by α-stable Lévy motions observed at discrete instants. Under regular conditions, we derive the weak consistency and central limit theorem of the estimator. Compared with Nadaraya-Watson estimator, the local linear estimator has a bias reduction whether the kernel function is symmetric or not under different schemes. A simulation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator, especially on the boundary.     

Rolling-ball method for estimating the boundary of the support of a point-process intensity

We suggest a generalisation of the convex-hull method, or ‘DEA’ approach, for estimating the boundary or frontier of the support of a point cloud. Figuratively, our method involves rolling a ball around the cloud, and using the equilibrium positions of the ball to define an estimator of the envelope of the point cloud. Constructively, we use these ideas to remove lines from a triangulation of the points, and thereby compute a generalised form of a convex hull. The radius of the ball acts as a smoothing parameter, with the convex-hull estimator being obtained by taking the radius to be infinite. Unlike the convex-hull approach, however, our method applies to quite general frontiers, which may be neither convex nor concave. It brings to these contexts the attractive features of the convex hull: simplicity of concept, rotation-invariance, and ready extension to higher dimensions. It admits bias corrections, which we describe and illustrate through implementation.     

Linearly interpolated FDH efficiency score for nonconvex frontiers

This paper addresses the problem of estimating the monotone boundary of a nonconvex set in a full nonparametric and multivariate setup. This is particularly useful in the context of productivity analysis where the efficient frontier is the locus of optimal production scenarios. Then efficiency scores are defined by the distance of a firm from this efficient boundary. In this setup, the free disposal hull (FDH) estimator has been extensively used due to its flexibility and because it allows nonconvex attainable production sets. However, the nonsmoothness and discontinuities of the FDH is a drawback for conducting inference in finite samples. In particular, it is shown that the bootstrap of the FDH has poor performances and so is not useful in practice. Our estimator, the LFDH, is a linearized version of the FDH, obtained by linear interpolation of appropriate FDH-efficient vertices. It offers a continuous, smooth version of the FDH. We provide an algorithm for computing the estimator, and we establish its asymptotic properties. We also provide an easy way to approximate its asymptotic sampling distribution. The latter could offer bias-corrected estimator and confidence intervals of the efficiency scores. In a Monte Carlo study, we show that these approximations work well even in moderate sample sizes and that our LFDH estimator outperforms, both in bias and in MSE, the original FDH estimator.     

Boundary kernels for adaptive density estimators on regions with irregular boundaries

In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets.     

A Posteriori Estimators for the Finite Volume Discretization of an Elliptic Problem

We analyze a residual error estimator for a finite volume discretization of a linear elliptic boundary value problem. The error estimator consists of the residual of the strong equation and the jumps across the inter-element boundaries of a primal triangulation. Some numerical experiments are presented.     

On estimation of survival function under random censoring model

We study an estimator of the survival function under the random censoring model. Bahadur-type representation of the estimator is obtained and asymptotic expression for its mean squared errors is given, which leads to the consistency and asymptotic normality of the estimator. A data-driven local bandwidth selection rule for the estimator is proposed. It is worth noting that the estimator is consistent at left boundary points, which contrasts with the cases of density and hazard rate estimation. A Monte Carlo comparison of different estimators is made and it appears that the proposed data-driven estimators have certain advantages over the common Kaplan-Meier estmator.     

Ghost Point Diffusion Maps for Solving Elliptic PDEs on Manifolds with Classical Boundary Conditions

In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM) and its local kernel variants to approximate second-order differential operators defined on smooth manifolds with boundaries that naturally arise in elliptic PDE models. To achieve this goal, we introduce the ghost point diffusion maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points, specified along the estimated tangential direction at the sampled points on the boundary. The resulting GPDM estimator restricts the standard DM matrix to a set of extrapolation equations that estimates the function values at the ghost points. This adjustment is analogous to the classical ghost point method in a finite-difference scheme for solving PDEs on flat domains. As opposed to the classical DM, which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary. Applying the consistent GPDM estimator to solve well-posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions. We numerically validate the proposed mesh-free PDE solver on various problems defined on simple submanifolds embedded in Euclidean spaces as well as on an unknown manifold. Numerically, we also found that the GPDM is more accurate compared to DM in solving elliptic eigenvalue problems on bounded smooth manifolds. © 2021 Wiley Periodicals LLC.     

A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.     

Simultaneous confidence bands for nonparametric regression with missing covariate data

We consider a weighted local linear estimator based on the inverse selection probability for nonparametric regression with missing covariates at random. The asymptotic distribution of the maximal deviation between the estimator and the true regression function is derived and an asymptotically accurate simultaneous confidence band is constructed. The estimator for the regression function is shown to be oracally efficient in the sense that it is uniformly indistinguishable from that when the selection probabilities are known. Finite sample performance is examined via simulation studies which support our asymptotic theory. The proposed method is demonstrated via an analysis of a data set from the Canada 2010/2011 Youth Student Survey.

     

Transformation Kernel Density Estimation With Applications

One of the main objectives of this article is to derive efficient nonparametric estimators for an unknown density fX. It is well known that the ordinary kernel density estimator has, despite several good properties, some serious drawbacks. For example, it suffers from boundary bias and it also exhibits spurious bumps in the tails. We propose a semiparametric transformation kernel density estimator to overcome these defects. It is based on a new semiparametric transformation function that transforms data to normality. A generalized bandwidth adaptation procedure is also developed. It is found that the newly proposed semiparametric transformation kernel density estimator performs well for unimodal, low, and high kurtosis densities. Moreover, it detects and estimates densities with excessive curvature (e.g., modes and valleys) more effectively than existing procedures. In conclusion, practical examples based on real-life data are presented.     

Duality Based A Posteriori Error Estimator for the Dirichlet Problem

The present work is devoted to the a posteriori error estimation for 2nd order elliptic problems with Dirichlet boundary conditions. Using the duality technique we derive a reliable and efficient a posteriori error estimator that measures the error in the energy norm. All the derivations are done on continuous level, and the estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H¹, independently of the discretization method chosen. In particular, we make no use of the Galerkin orthogonality, which enables us to implement the estimator for measuring the error of the fictitious domain/penalty finite element method. The estimator is easily computable, and the only constant present in the estimator is the one from Friedrichs' inequality; the constant depends solely on the domain geometry, and the estimator is quite non‐sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.     

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014     

Convergence analysis of a conforming adaptive finite element method for an obstacle problem

The adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps ‘SOLVE’, ‘ESTIMATE’, ‘MARK’, and ‘REFINE’. The techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE—in fact, there is no modification near the discrete free boundary necessary for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle χ and homogeneous Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in the unconstrained case. Numerical results are given to illustrate the performance of the error estimator.