Error estimation with guaranteed accuracy of finite element method in nonconvex polygonal domains

A new method to estimate errors of the finite element method (FEM) for nonconvex polygonal domain is proposed. It gives mathematically rigorous upper bounds for the errors using calculations with guaranteed accuracy. Numerical examples are shown and their orders concerning mesh sizes are compared with theoretical orders.     

Nonuniform covering method as applied to multicriteria optimization problems with guaranteed accuracy

The nonuniform covering method is applied to multicriteria optimization problems. The ?-Pareto set is defined, and its properties are examined. An algorithm for constructing an ?-Pareto set with guaranteed accuracy ? is described. The efficiency of implementing this approach is discussed, and numerical results are presented.     

Construction of quasi-periodic solutions of guaranteed accuracy by the small parameter method

Theorems on the localization of exact solutions are proved for a quasilinear mathematical model describing quasi-periodic processes. Based on these theorems, constructive algorithms are proposed for calculating quasi-periodic solutions with guaranteed accuracy. Quasi-periodic motions play an important role in engineering and physics, where they often represent determining states. Quasi-periodic motions can be found in many ecological, biological, and economic processes.     

Nonparametric estimation with doubly censored and truncated data

Data from longitudinal studies in which an initiating event and a subsequent event occur in sequence are called doubly censored data if the time of both events is interval-censored. In some cases, the second event also suffers left-truncation. This article is concerned with using doubly censored and truncated data to estimate the distribution function F of the duration time, i.e. the elapsed time between the originating event and the subsequent event. An iterative procedure is proposed to obtain the estimate of F. A simulation study is conducted to investigate the performance of the proposed estimator. A modified data set is used to illustrate the proposed approach.     

A truncated Newton method with nonmonotone line search for unconstrained optimization

In this paper, an unconstrained minimization algorithm is defined in which a nonmonotone line search technique is employed in association with a truncated Newton algorithm. Numerical results obtained for a set of standard test problems are reported which indicate that the proposed algorithm is highly effective in the solution of illconditioned as well as of large dimensional problems.     

Asymptotics for estimation of quantile regressions with truncated infinite-dimensional processes

Many processes can be represented in a simple form as infinite-order linear series. In such cases, an approximate model is often derived as a truncation of the infinite-order process, for estimation on the finite sample. The literature contains a number of asymptotic distributional results for least squares estimation of such finite truncations, but for quantile estimation, results are not available at a level of generality that accommodates time series models used as finite approximations to processes of potentially unbounded order. Here we establish consistency and asymptotic normality for conditional quantile estimation of truncations of such infinite-order linear models, with the truncation order increasing in sample size. We focus on estimation of the model at a given quantile. The proofs use the generalized functions approach and allow for a wide range of time series models as well as other forms of regression model. The results are illustrated with both analytical and simulation examples.     

A sparse finite element method with high accuracy Part I

Summary. In this paper, we develop and analyze a new finite element method called the sparse finite element method for second order elliptic problems. This method involves much fewer degrees of freedom than the standard finite element method. We show nevertheless that such a sparse finite element method still possesses the superconvergence and other high accuracy properties same as those of the standard finite element method. The main technique in our analysis is the use of some integral identities. Received October 1, 1995 / Revised version received August 23, 1999 / Published online February 5, 2001     

Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures

We give upper bounds for the deviation of the norm of a perturbed error functional from the norm of the original error of a higher-dimensional spherical cubature formula. The deviation arises as a result of the combined influence on the computation of small variations of the weights of the cubature formula and rounding for the subsequent calculation of the cubature sum in the given standards of approximation to real numbers. We estimate the practical error of the cubature formula for its action on an arbitrary function in the unit ball of the normed space of integrands. The resulting estimates are applied to studying the practical error of spherical cubature formulas in the case of integrands in Sobolev-type spaces on the higher-dimensional unit sphere. We represent the norm of the error functional in the dual space of the Sobolev class as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for spherical cubature formulas, each of which is constructed as the direct product of Gauss’s quadrature formula along the meridian of the sphere and of the rectangle quadrature formula along the equator. The weights of this direct product with 2m ² nodes are positive. The formula itself is exact at all spherical harmonics up to order 2m ? 1.     

A truncated aggregate smoothing Newton method for minimax problems

Aggregate function is a useful smoothing function to the max-function of some smooth functions and has been used to solve minimax problems, linear and nonlinear programming, generalized complementarity problems, etc. The aggregate function is a single smooth but complex function, its gradient and Hessian calculations are time-consuming. In this paper, a truncated aggregate smoothing stabilized Newton method for solving minimax problems is presented. At each iteration, only a small subset of the components in the max-function are aggregated, hence the number of gradient and Hessian calculations is reduced dramatically. The subset is adaptively updated with some truncating criterions, concerning only with computation of function values and not their gradients or Hessians, to guarantee the global convergence and, for the inner iteration, locally quadratic convergence with as few computational cost as possible. Numerical results show the efficiency of the proposed algorithm.     

Wavelet estimation of conditional density with truncated, censored and dependent data

In this paper we define a new nonlinear wavelet-based estimator of conditional density function for a random left truncation and right censoring model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimators, the MISE expression of the wavelet-based estimators is not affected by the presence of discontinuities in the curves. Also, asymptotic normality of the estimator is established.     

A method of truncated codifferential with application to some problems of cluster analysis

A method of truncated codifferential descent for minimizing continuously codifferentiable functions is suggested. The convergence of the method is studied. Results of numerical experiments are presented. Application of the suggested method for the solution of some problems of cluster analysis are discussed. In numerical experiments Wisconsin Diagnostic Breast Cancer database was used.     

A high accuracy method for solving ODEs with discontinuous right-hand side

Summary Ordinary Differential Equations with discontinuities in the state variables require a differential inclusion formulation to guarantee existence [8]. From this formulation a high accuracy method for solving such initial value problems is developed which can give any order of accuracy and tracks the discontinuities. The method uses an active set approach, and determines appropriate active sets from solutions to Linear Complementarity Problems. Convergence results are established under some non-degeneracy assumptions. The method has been implemented, and results compare favourably with previously published methods [7, 21].     

The error and guaranteed accuracy of cubature formulas in multidimensional periodic Sobolev spaces

We give an upper bound for the deviation of the norm of a perturbed error from the norm of the original error of a cubature formula in a multidimensional bounded domain. The deviation arises as a result of the joint influence on the computations of small variations of the weights of a cubature formula and rounding in the subsequent calculations of the cubature sum in the given standards (formats) of approximation to real numbers. We estimate the practical error of a cubature formula acting on an arbitrary function from the unit ball of a normed space of integrands. The resulting estimates are applied to studying the practical error of cubature formulas in the case of integrands in Sobolev spaces on a multidimensional cube. The norm of the error in the dual space of the Sobolev class is represented as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for cubature formulas constructed as the direct product of quadrature formulas of rectangles along the edges of the unit cube. The weights of this direct product are positive.     

An iterative method for solving an inverse problem for a first-order nonlinear partial differential equation with estimates of guaranteed accuracy and the number of steps

For a partial differential equation simulating population dynamics, the inverse problem of determining its nonlinear right-hand side from an additional boundary condition is studied. This inverse problem is reduced to a functional equation, for which the existence and uniqueness of a solution is proven. An iterative method for solving this inverse problem is proposed. The accuracy of the method is estimated, and restrictions on the number of steps are obtained.     

A truncated conjugate gradient method with an inexact Gauss-Newton technique for solving nonlinear systems

In this paper, a truncated conjugate gradient method with an inexact Gauss-Newton technique is proposed for solving nonlinear systems.?The iterative direction is obtained by the conjugate gradient method solving the inexact Gauss-Newton equation.?Global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithm.     

A truncated projected SVD method for linear discrete ill-posed problems

Truncated singular value decomposition is a popular solution method for linear discrete ill-posed problems. However, since the singular value decomposition of the matrix is independent of the right-hand side, there are linear discrete ill-posed problems for which this method fails to yield an accurate approximate solution. This paper describes a new approach to incorporating knowledge about properties of the desired solution into the solution process through an initial projection of the linear discrete ill-posed problem. The projected problem is solved by truncated singular value decomposition. Computed examples illustrate that suitably chosen projections can enhance the accuracy of the computed solution.     

Tail product-limit process for truncated data with application to extreme value index estimation

A weighted Gaussian approximation to tail product-limit process for Pareto-like distributions of randomly right-truncated data is provided and a new consistent and asymptotically normal estimator of the extreme value index is introduced. A simulation study is carried out to evaluate the finite sample behavior of the proposed estimator and compare it to that recently proposed by Gardes and Stupfler (TEST 24, 207–227, 2015). Also, a new approach of estimating extreme quantiles, under random right truncation, is derived and applied to a real dataset of lifetimes of automobile brake pads.     

Numerical experience with the truncated Newton method for unconstrained optimization

The truncated Newton algorithm was devised by Dembo and Steihaug (Ref. 1) for solving large sparse unconstrained optimization problems. When far from a minimum, an accurate solution to the Newton equations may not be justified. Dembo's method solves these equations by the conjugate direction method, but truncates the iteration when a required degree of accuracy has been obtained. We present favorable numerical results obtained with the algorithm and compare them with existing codes for large-scale optimization.     

On the truncated conjugate gradient method

In this paper, we consider the truncated conjugate gradient method for minimizing a convex quadratic function subject to a ball trust region constraint. It is shown that the reduction in the objective function by the solution obtained by the truncated CG method is at least half of the reduction by the global minimizer in the trust region. Received January 19, 1999 / Revised version received October 1, 1999?Published online November 30, 1999