Quadrature Formulas for Refinable Functions and Wavelets II: Error Analysis

Abstract. A functional type a posteriori error estimator for the finite element discretization of the stationary reaction-convection-diffusion equation is derived. In case of dominant convection, the solution for this class of problems typically exhibits boundary layers and shock-front like areas with steep gradients. This renders the accurate numerical solution very demanding and appropriate techniques for the adaptive resolution of regions with large approximation errors are crucial. Functional error estimators as derived here contain no mesh-dependent constants and provide guaranteed error bounds for any conforming approximation. To evaluate the error estimator, a minimization problem is solved which does not require any Galerkin orthogonality or any specific properties of the employed approximation space. Based on a set of numerical examples, we assess the performance of the new estimator. It is observed that it exhibits a good efficiency also with convection-dominated problem settings.

Orthogonal polynomials and Gaussian quadrature for refinable weight functions

We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving eigenvalue problems associated with second order linear operators. Eigenvalue problems of such types play important roles in scientific and engineering applications, particularly in theoretical chemistry, solid state physics and material science. Based on the framework developed in [{\it L. Lin, B. Stamm, http://dx.doi.org/10.1051/m2an/2015069}] for second order PDEs, we develop residual type upper and lower bound error estimates for measuring the a posteriori error for eigenvalue problems. The main merit of our method is that the method is parameter-free, in the sense that all but one solution-dependent constants appearing in the upper and lower bound estimates are explicitly computable by solving local and independent eigenvalue problems, and the only non-computable constant can be reasonably approximated by a computable one without affecting the overall effectiveness of the estimates in practice. Compared to the PDE case, we find that a posteriori error estimators for eigenvalue problems must neglect certain terms, which involves explicitly the exact eigenvalues or eigenfunctions that are not accessible in numerical simulations. We define such terms carefully, and justify numerically that the neglected terms are indeed numerically high order terms compared to the computable estimators. Numerical results for a variety of problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective for measuring the error of eigenvalues and eigenfunctions.

We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving second order linear PDEs. Our residual type upper and lower bound error estimates measure the error in the energy norm. The main merit of our method is that the method is parameter-free, in the sense that all but one solution-dependent constants appearing in the upper and lower bound estimates are explicitly computable by solving local eigenvalue problems, and the only non-computable constant can be reasonably approximated by a computable one without affecting the overall effectiveness of the estimates in practice. As a side product of our formulation, the penalty parameter in the interior penalty formulation can be automatically determined as well. We develop an efficient numerical procedure to compute the error estimators. Numerical results for a variety of problems in 1D and 2D demonstrate that both the upper bound and lower bound are effective.esaim

ABSTRACTThe estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.

Parametric estimation of a stability factor for certified reduced basis methods via adaptive Gaussian process

The problem of the estimation of a regression function by continuous piecewise linear functions is formulated as a nonconvex, nonsmooth optimization problem. Estimates are defined by minimization of the empirical L 2 risk over a class of functions, which are defined as maxima of minima of linear functions. An algorithm for finding continuous piecewise linear functions is presented. We observe that the objective function in the optimization problem is semismooth, quasidifferentiable and piecewise partially separable. The use of these properties allow us to design an efficient algorithm for approximation of subgradients of the objective function and to apply the discrete gradient method for its minimization. We present computational results with some simulated data and compare the new estimator with a number of existing ones.

I will present three projects that are related to the modeling of covariance structures on the Euclidean space and on sphere. Firstly, we propose a method to model isotropic random field on sphere, where a tapered Matern covariance function is used to capture the local behavior while a nonparametric expansion controls the behavior at large distances. A model selection procedure based on residual sum of squares with penalization is used to reduce over fitting. Secondly, we address the issue of modeling axially symmetric spatial random fields on sphere with a kernel convolution approach. The observed random field is generated by convolving a latent uncorrelated random field with a class of Matern type kernel functions. By allowing the parameters in the kernel functions to vary with locations, we are able to generate a flexible class of covariance functions and capture the nonstationary properties. We use pre-computation tables to speed up the computation. For regular grid data on sphere, the block circulant property of the covariance matrix enables us to use Fast Fourier Transform (FFT) to get the determinant and the inverse efficiently. Thirdly, we proposed a semiparametric variogram estimating method through its spectral representation to model the intrinsically stationary random on R. The low frequency part of the spectral density is estimated by solving a regularized inverse problem through quadratic programming. The behavior at high frequencies, however, is modeled via a parametric tail in the form of a power decaying function. The power parameter in the tail is estimated by a log likelihood method. All proposed methodologies are supplemented with simulation studies and real data analyses.

Estimation in the interval censoring model is considered. A class of smooth functionals is introduced, of which the mean is an example. The asymptotic information lower bound for such functionals can be represented as an inner product of two functions. In case 1, i.e. one observation time per unobservable event time, both functions can be given explicitly. We mainly consider case 2, with two observation times for each unobservable event time, in the situation that the observation times can not become arbitrarily close to each other. For case 2, one of the functions in the inner product can only be given implicitly as solution to a Fredholm integral equation. We study properties of this solution and, in a sequel to this paper, prove that the nonparametric maximum likelihood estimator of the functional asymptotically reaches the information lower bound.

Berkson errors are commonplace in empirical microeconomics and occur whenever we observe an average in a specified group rather than the true individual value. In consumer demand this form of measurement error is present because the price an individual pays is often measured by the average price paid by individuals in a specified group (e.g., a county). We show the importance of such measurement errors for the estimation of demand in a setting with nonseparable unobserved heterogeneity. We develop a consistent estimator using external information on the true distribution of prices. Examining the demand for gasoline in the U.S., accounting for Berkson errors is found to be quantitatively important for estimating price effects and for welfare calculations. Imposing the Slutsky shape constraint greatly reduces the sensitivity to Berkson errors.

<p>We augment Tversky and Khaneman (1992) (TK92) Cumulative Prospect Theory (CPT) function space with a sample space for states of nature, and depict a commutative map of behavior on the augmented space. In particular, we use a homotopy lifting property to mimic behavioral stochastic processes arising from deformation of stochastic choice into outcome. A psychological distance metric (in the class of Dudley-Talagrand inequalities) for stochastic learning, was used to characterize stopping times for behavioral processes. In which case, for a class of nonseparable space-time probability density functions, we find that behavioral processes are uniformly stopped before the goal of fair gamble is attained. Further, we find that when faced with a fair gamble, agents exhibit submartingale [supermartingale] behavior, subjectively, under CPT probability weighting scheme. We show that even when agents have classic von Neuman-Morgenstern preferences over probability distribution, and know that the gamble is a martingale, they exhibit probability weighting to compensate for probability leakage arising from the their stopped behavioral process. </p>

Spectral methods are powerful tools to study and model the dependency structure of spatial temporal processes. However, standard spectral approaches as well as geostatistical methods assume separability and stationarity of the covariance function; these can be very unrealistic assumptions in many settings. In this work, we introduce a general and flexible parametric class of spatial temporal covariance models, that allows for lack of stationarity and separability by using a spectral representation of the process. This new class of covariance models has a unique parameter that indicates the strength of the interaction between the spatial and temporal components; it has the separable covariance model as a particular case. We introduce an application with ambient ozone air pollution data provided by the U.S. Environmental Protection Agency (U.S. EPA).

Research Highlights

Density estimates with methods of uniform distribution mod 1. Some nonparametric multivariate density estimators for continuous functions, based on the Fejer and Jackson kernel, are presented. Uniform strong consistency results are obtained with methods of uniform distribution mod 1.

Dealing with data coming from a space-time inhomogeneous process, there is often the need of semi-parametric estimates of the conditional intensity function; isotropic or anisotropic multivariate kernel estimates can be used, with windows sizes h. The properties of the intensities estimated with this choice of h are not always good for specific fields of application; we could try to choose h in order to have good predictive properties of the estimated intensity function. Since a direct ML approach cannot be followed, we propose an estimation procedure, computationally intensive, based on the subsequent increments of likelihood obtained adding an observation at time. The first results obtained are very encouraging. Some application in statistical seismology is presented.